Takahiro Kato May 24, 2017 - viXravixra.org/pdf/1503.0085v6.pdf · The term profunctor is quite appropriate in this regard. However, the notion has another, probably more important,

Modules and Universal Constructions

Takahiro Kato

May 24, 2017

ABSTRACT. Modules (also known as profunctors or distributors) and morphisms amongthem subsume categories and functors and provide more general and abstract framework toexplore the theory of structures, in particular, the notion of universal property. In this bookwe generalize and redevelop the basic notions and results of various universal constructions incategory theory using this framework of modules.

About modules and profunctors

The terms module and profunctor are synonymous in the sense that both are dened as afunctor of the form F ∶ Xop × A → Set. The notion given by this denition is seen by manyas a generalization of the notion of functor. The term profunctor is quite appropriate inthis regard. However, the notion has another, probably more important, aspect; it is alsoseen as an extension of categories. We use the term module when this aspect of the notionis emphasized. Profunctors constitute the morphisms of the category Prof, while modulesconstitute the objects of the category MOD. This book studies modules, not profunctors; inother words, this book studies an extension of categories.

Topics

Chapter 1 introduces modules and cells among them. A moduleM ∶ X A from a categoryX to A is a functor of the formM ∶Xop ×A→ Set, assigning a set of arrows to each pair ofobjects x ∈ X and a ∈ A. A cell from a module M to N sends each arrow of M to an arrowof N . The hom-functor of a category C forms an endomodule ⟨C⟩ ∶ C C called the hom ofC, and the arrow function of a functor F ∶ C → D forms a cell ⟨F⟩ ∶ ⟨C⟩ ⟨D⟩ from the homof C to the hom of D. Modules and cells thus subsume categories and functors in this wayand set up a more general and conceptual framework to explore the structure of mathematics.Presheaves and copresheaves are called right and left modules respectively in this book andstudied as special instances of modules.Chapter 2 discusses the action of a module on its domain and codomain, the operation

that yields the Yoneda embedding functor in the case of a hom endomodule. The chapterintroduces an important class of modules called representable, which each functor produces bycomposition with the hom of its codomain.Chapter 3 presents two variants of modules, namely collages and commas1, which are special

sorts of cospans and spans between two categories. We will establish an isomorphism betweenthe category of modules and the category of collages, and, later in Chapter 9, construct anadjoint equivalence between the category of commas and the category of collages. Two for-getful functors from MOD (the category of modules and cells) to CAT are dened throughconstruction of collages and commas, and it is shown that they form left and right adjoints ofthe embedding CATMOD given by the hom assignment C↦ ⟨C⟩.Chapter 4 introduces the notion of frames of a module. A cylindrical frame of an endomodule

abstracts a natural transformation between two functors, and a conical frame of a right (resp.left) module abstracts a cone between an object and a functor. Ordinary and extraordinarycylinders, which are, so to speak, ordinary and extraordinary natural transformations spanning

1The term two-sided discrete bration is used in the literature to refer to what this book calls a comma.The term comma is adopted because every two-sided discrete bration is given by a comma category andits projection functors.

2

a module2, are dened using cylindrical frames. Likewise, cones are dened along a moduleusing conical frames.Chapter 5 succeeds Chapter 2 and discusses the actions of the domain and codomain of a

module on its arrows and frames. It is shown, as a generalization of the Yoneda embedding,that these actions embed a module XA in (the hom of) the category of right modules (i.e.presheaves) over X and the category of left module (i.e. copresheaves) over A. The Yonedalemma is presented in a general form to state that the morphisms from the representablemodule of a functor F ∶ X → A to an arbitrary module M ∶ X A correspond one-to-onewith the cylinders dened between F andM. Using the lemma, we show a variety of bijectivecorrespondences between frames and cells.The remaining chapters explore universal constructions (such as limits, ends, lifts, extensions,

adjoint functors, etc) in the framework of modules using the machinery developed so far.Universality is formulated in terms of universal arrows in a module; the embedding of a modulein the category of presheaves makes the denition of a universal simple enough: an arrow in amodule is universal if and only if its image under the embedding is an isomorphism. Universalconstructions are dened along a module and we study if they are preserved by a cell. Hereare some highlights.

Limits and ends are dened along a module as universal cones and universal extraordinarycylinders, respectively.

Lifts (of which Kan lifts are special instances) are dened as universal ordinary cylinders,and extensions (of which Kan extensions are special instances) are dened as universalcells. We introduce the notion of pointwise lifts and show that the notion subsumes that ofpointwise extensions and vice versa.

The concept of an adjoint is extended to a cell. We show that a cell preserves universalsif it has adjoints, and deduce RAPL (right adjoints preserve limits) as a corollary of thisgeneral mechanism.

The notion of an equivalence of categories is extended to that of an equivalence of modules.It is show that an equivalence cell preserves and reects universals.

Epicity and monicity are dened not only for arrows in a category but for arrows in a module.A proof of the special adjoint functor theorem is given using epi-mono-factorizations formodules.

The concept of density is also generalized for modules (in fact, density is most naturallydened with modules). The fact every presheaf is a colimit of representables is provedusing the concept of density to show how the concept works in the framework of modules.

Advice on reading

The exposition is carried out within the framework of set-enriched 1-dimensional categories.Chapter 0 presents the notations and some elementary facts of category theory used in thesequel.Size issues are not treated rigorously. The specication of a universe is almost always implicit;

a universe U is chosen so that given categories and modules become locally U-small, i.e. allhom-sets become U-small. We just say small instead of U-small for U chosen implicitly.

2The notion is called het natural transformation in [Ell07].

3

This book is made up of sequences of Denition - Proposition - Theorem - Corollary,with each optionally preceded by Note and followed by Remark. Each Proposition istightly associated with the preceding Denition and states some straightforward consequencesof it. Two statements dual to each other are indicated by the symbols and . Proofs aregiven for the assertions indicated by .Parentheses and brackets serve mainly as punctuation to enhance readability; parentheses

are used to delimit objects and arrows, whereas square (resp. angle) brackets are used todelimit categories and functors (resp. modules and cells).

4

Contents

0. Preliminaries 7

1. Modules and Cells 131.1. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2. Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3. Cell morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2. Slicing and Action 372.1. Slicing of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2. Action of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3. Yoneda functors and representations . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3. Collages and Commas 503.1. Collages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2. Commas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4. Frames 674.1. Cylindrical frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2. Conical frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3. Ordinary cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4. Extraordinary cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.5. Weighted cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.6. Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7. Bicylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.8. Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.9. Cones and wedges in a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.10. Cones and wedges in Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5. Yoneda Lemma 1445.1. Yoneda modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.2. Yoneda morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3. Yoneda morphisms for cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.4. Yoneda morphisms for cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.5. Correspondences between frames and cells . . . . . . . . . . . . . . . . . . . . . . . 183

6. Universals 1966.1. Units of one-sided modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1966.2. Universal arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.3. Units of two-sided modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.4. Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.5. Kan lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

5

Contents

7. Limits 2207.1. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207.2. Limits with parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.3. Limits in a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2307.4. Limits of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8. Adjunctions 2448.1. Adjunctions for modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2448.2. Adjunctions for categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.3. Transformations of adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2548.4. Adjoints of a cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2588.5. Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2628.6. Equivalence of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

9. Adjoint Functor Theorem 2719.1. Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.2. General adjoint functor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2779.3. Epimorphisms and Monomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 2809.4. Subobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2829.5. Epi-mono factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2859.6. Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.7. Special adjoint functor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

10.Collages and Commas (continued) 29110.1. Cylinder modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29110.2. Equivalence COM ≃CLG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.3. Equivalence [X ↓A] ≃ [X ↑A] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30110.4. Equivalences [X ↓] ≃ [X ∶] and [↓A] ≃ [∶A] . . . . . . . . . . . . . . . . . . . . . . 306

11.Extensions 31411.1. coYoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31411.2. Weighted limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32411.3. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.4. Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34411.5. Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

12.Ends 35612.1. Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35612.2. Ends with parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35912.3. Ends of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

A. List of Symbols 373

6

0. Preliminaries

1. Given a universe U, SetU, CatU, and CATU denote the categories of U-small sets, U-smallcategories, and locally U-small categories respectively. By Set we mean SetU for U chosenimplicitly, and similarly for Cat and CAT.

2. A category is dened in terms of hom-sets. Pairwise disjointness of hom-sets is not required.Given a category C, an element f of a hom-set homC (a,b) or a triple (a, f,b) such that

f ∈ homC(a,b) is called an arrow of C (or a C-arrow) and written f ∶ a → b, afÐ→ b, or

just f if its domain and codomain are understood or unimportant. The composite of arrowsf ∶ a→ b and g ∶ b→ c is written as

f g = g f

. An invertible arrow is called iso (or an isomorphism).

3. The set of objects of a category C is denoted by ∥C∥ and the object function of a functorF is denoted by ∥F∥.

4. Italic small letters a, b, c, ... vary over objects and arrows. When we write c ∈ C for acategory C, c stands for an object or arrow of C.

5. The value of a functor F ∶C→ B at c ∈C is written as

c ⋅ F = F (c) = Fc = F ⋅ c

, and the component of a natural transformation τ ∶ S → T ∶ C → B at an object c ∈ ∥C∥ iswritten as

c ⋅ τ = τc = τ ⋅ c.

6. The opposite of a category C is denoted by C∼, and the opposite of a functor F ∶ C → Bis denoted by F∼ ∶ C∼ → B∼, or often by F ∶ C∼ → B∼ using the same name as its originalcounterpart.

7. Given categories A and B, the product, coproduct, and functor categories are denoted byA×B, A+B and [A,B] respectively. The terminal category and its sole object are denotedby ∗; the identity arrow ∗ → ∗ is also denoted by ∗. The interval category is denoted by 2and its objects are denoted by 0 and 1.

8. Italic capital letters F , G, ... vary over both functors and natural transformations. When

we write F ∈ [A,B], F ∶A→ B, or AFÐ→ B for categories A and B, F stands for an object

or an arrow of the functor category [A,B], i.e. a functor or a natural transformation.

9. Let A, B, and C be categories. The composite of F ∶A→ B and G ∶ B→C is written as

F G = G F

7

0. Preliminaries

, and its value a ⋅ [F G] = (a ⋅ F ) ⋅G at a ∈A is often written as

a ⋅ F ⋅G = G ⋅ F ⋅ a

. The composition (F,G) ↦ F G denes a bifunctor [A,B] × [B,C] → [A,C], and theright and left exponential transpositions yield functors

[A,−] ∶ [B,C]→ [[A,B] , [A,C]] ; G↦ (F ↦ F G)

and[−,C] ∶ [A,B]→ [[B,C] , [A,C]] ; F ↦ (G↦ F G)

. The functor [A,−] sends each functor G ∶ B→C to the postcomposition functor

[A,G] ∶ [A,B]→ [A,C] ; F ↦ F G

and sends each natural transformation τ ∶ S → T ∶ B → C to the postcomposition naturaltransformation

[A, τ] ∶ [A,S]→ [A,T] ∶ [A,B]→ [A,C]dened by

[A, τ]F = F τ ∶ F S→ F T ∶A→C

for F a functor A → B. Dually, the functor [−,C] sends each functor F ∶ A → B to theprecomposition functor

[F,C] ∶ [B,C]→ [A,C] ; G↦ F G

and sends each natural transformation σ ∶ S → T ∶ A → B to the precomposition naturaltransformation

[σ,C] ∶ [S,C]→ [T,C] ∶ [B,C]→ [A,C]dened by

[σ,C]G = σ G ∶ S G→ T G ∶A→C

for G a functor B→C.

10. The bifunctorial operation [−,−] on CAT∼ ×CAT is dened in the following way:

a) for a pair of categories X and A, [X,A] is the category of functors X→A;

b) for a functor P ∶X→Y and a categoryA, [P,A] ∶ [Y,A]→ [X,A] is the precompositionfunctor;

c) for a functor Q ∶A→ B and a category X, [X,Q] ∶ [X,A]→ [X,B] is the postcomposi-tion functor.

The commutativity of

[Y,A] [P,A] //

[Y,Q]

[X,A][X,Q]

[Y,B][P,B]

// [X,B]

expresses the associativity of functor composition.

8

0. Preliminaries

11. The right and left exponential transposes of a bifunctor K ∶ E ×D→C are denoted by

K⊣ ∶D→ [E,C] K⊢ ∶ E→ [D,C]

, and the (simple) transpose of a functor K ∶ E → [D,C] is denoted by K⊺ ∶ D → [E,C].These transpositions form iso functors as indicated in the commutative diagram

[E ×D,C]⊣

vv⊢

((

[D, [E,C]]⊺ //

⊢

66

[E, [D,C]]⊺

oo⊣

hh

, natural in E, D, and C.

12. The identity functor 1E ∶ E→ E is also denoted by E using the name of the category. Moregenerally, if D is a subcategory E, the inclusion functor D E is also denoted by D usingthe name of its domain; the restriction of F ∶ E→C to D is written as

D F = F D

. Precomposition with the inclusion

DD // E

yields a functor

[E,C] [D,C] // [D,C], restriction to D, for any category C.

13. By the obvious isomorphism [∗,E] ≅ E, a functor (resp. a natural transformation) ∗ → Eis identied with an object (resp. an arrow) of E. Given categories E and D and given anobject e ∈ ∥E∥, the product functor e ×D as in

∗E

∗ ×D

e×D

oo //D

d

E E ×Doo //D

gives the sectione ×D ∶D→ E ×D

of E ×D at e, under the identication ∗ ×D ≅D. The section

E × d ∶ E→ E ×D

of E ×D at d ∈ ∥D∥ is dened symmetrically.

14. Given categories E and C, the evaluation

(e,F )↦ e ⋅ F ∶ E × [E,C]→C

is identied with the composition

(e,F )↦ e F ∶ [∗,E] × [E,C]→ [∗,C]

9

0. Preliminaries

. Given an object e ∈ ∥E∥, the precomposition functor

[e,C] ∶ [E,C]→C

, evaluation at e, takes each F ∈ [E,C] and yields the composite

∗ e // EF // C

, i.e. the value of F at e.

15. The left slice of a bifunctor K ∶ E ×D → C at e ∈ ∥E∥ is the functor D → C given by thevalue of the left exponential transpose K⊢ ∶ E→ [D,C] at e. The naturality square

[E ×D,C][e×D,C]

⊢ // [E, [D,C]][e,[D,C]]

[∗ ×D,C] ⊢ // [∗, [D,C]]

shrinks to the commutative triangle

[E ×D,C] ⊢ //

[e×D,C] &&

[E, [D,C]]

[e,[D,C]]ww[D,C]

by the identications [∗ ×D,C] ≅ [D,C] ≅ [∗, [D,C]]; the precomposition

De×D // E ×D

K // C

of K with the section e×D thus yields the same functor as the one given by the evaluation

∗ e // EK⊢ // [D,C]

, i.e. the left slice of K at e.

16. Similarly, the slice of an (exponentially transposed) bifunctor K ∶ D → [E,C] at e ∈ ∥E∥is given by the value of the transpose K⊺ ∶ E → [D,C] at e (i.e. by the left slice of thebifunctor K⊢ ∶ E ×D→C at e). The naturality square

[D, [E,C]][D,[e,C]]

⊺ // [E, [D,C]][e,[D,C]]

[D, [∗,C]] ⊺// [∗, [D,C]]

shrinks to the commutative triangle

[D, [E,C]]

[D,[e,C]] ''

⊺ // [E, [D,C]]

[e,[D,C]]ww[D,C]

by the identications [D, [∗,C]] ≅ [D,C] ≅ [∗, [D,C]]; the postcomposition

DK // [E,C] [e,C] // C

10

0. Preliminaries

of K with the evaluation [e,C] thus yields the same functor as the one given by the evalu-ation

∗ e // EK⊺

// [D,C], i.e. the slice of K at e.

17. Given a category E, the unique functor E → ∗ is denoted by ∆E or just by ∆. Givencategories E and D, the product functor ∆E ×D as in

E

∆E

E ×D

∆E×D

oo //D

D

∗ ∗ ×Doo //D

gives the projection∆E ×D ∶ E ×D→D

of E ×D on D, under the identication ∗ ×D ≅D. The projection

E ×∆D ∶ E ×D→ E

of E ×D on E is dened symmetrically.

18. Given categories D, E, and C, the D-ary diagonal functor of the functor category [E,C]is the precomposition functor

[E ×∆D,C] ∶ [E,C]→ [E ×D,C]

sending each functor F ∶ E→C to the composite bifunctor

E ×DE×∆D // E

F // C

. The diagonal functor [E ×∆D,C] thus duplicates each F ∶ E→C across D by introducingto it a dummy variable varying over D. As a special case, the D-ary diagonal functor

[∆D,C] ∶C→ [D,C]

of a category C sends each object c ∈ ∥C∥ to the constant functor ∆c ∶D→C given by thecomposition

D∆D // ∗ c // C

. The commutativity of

[E,C][E×∆D,C]

xx

[∆D,[E,C]]''

[E,C][E×∆D,C]

xx

[E,[∆D,C]]''

[E ×D,C] ⊣ // [D, [E,C]]⊢

oo [E ×D,C] ⊢ // [E, [D,C]]⊣

oo

follows from the commutativity of the naturality squares

[E × ∗,C][E×∆D,C]

⊣ // [∗, [E,C]]⊢

oo

[∆D,[E,C]]

[E × ∗,C][E×∆D,C]

⊢ // [E, [∗,C]]⊣

oo

[E,[∆D,C]]

[E ×D,C] ⊣ // [D, [E,C]]⊢

oo [E ×D,C] ⊢ // [E, [D,C]]⊣

oo

by the identications [E × ∗,C] ≅ [E,C] ≅ [∗, [E,C]] and [E × ∗,C] ≅ [E,C] ≅ [E, [∗,C]].

11

0. Preliminaries

19. A subcategory D ⊆ E is called essentially wide if for every object e ∈ ∥E∥ there exists anobject d ∈ ∥D∥ isomorphic to e. A functor is called essentially surjective if its image is anessentially wide subcategory of the codomain. Note that a subcategory is essentially wideif and only if the inclusion functor is essentially surjective. We use the following lemma inthe sequel.Lemma. Let τ ∶ S → T ∶ E → C be a natural transformation and D be an essentially widesubcategory of E. Then τ is a natural isomorphism if and only if so is its restriction to D;that is, if and only if the component of τ at each d ∈ ∥D∥ is an isomorphism.

Proof. Since a natural transformation is iso i each component is an isomorphism, theforward implication is obvious. Assume now that τ is iso on D. We need to show that thecomponent τe at each e ∈ ∥E∥ is an isomorphism. Since D is essentially wide in E, there isan object d ∈ ∥D∥ and an iso E-arrow h ∶ d→ e, giving a naturality square

d ⋅ S τd //

h ⋅ S

T ⋅ dT ⋅ h

e ⋅ S τe// T ⋅ e

with h ⋅ S and T ⋅ h iso (because any functor preserves isomorphisms). Now, since τd anisomorphism by assumption, so is τe = (h ⋅ S)−1 τd (T ⋅ h) as required.

12

1. Modules and Cells

1.1. Modules

Denition 1.1.1.

A right module M over a category X, written M ∶ X ∗, is a functor M ∶ X∼ → Set.Given a pair of right modules M,N ∶ X ∗, a morphism Φ from M to N , writtenΦ ∶M→ N ∶X ∗, is a natural transformation Φ ∶M→ N ∶X∼ → Set.

A left moduleM over a category A, writtenM ∶ ∗A, is a functorM ∶A→ Set. Given apair of left modulesM,N ∶ ∗A, a morphism Φ fromM toN , written Φ ∶M→ N ∶ ∗A,is a natural transformation Φ ∶M→ N ∶A→ Set.

Remark 1.1.2.(1)

IfM ∶X ∗ is a right module, then the image of an object (resp. an arrow) x ∈X underthe functorM ∶X∼ → Set is written (x) ⟨M⟩ or just x ⟨M⟩. For an object x ∈ ∥X∥, theset x ⟨M⟩ is called the hom-set at x. An element m of a hom-set x ⟨M⟩, or a pair (x,m)such that m ∈ x ⟨M⟩, is called an arrow of M (or an M-arrow) and written m ∶ x ∗or x

m // ∗ (or just m if its domain is understood or unimportant). IfM ∶ ∗A is a left module, then the image of an object (resp. an arrow) a ∈A under

the functorM ∶ A → Set is written ⟨M⟩ (a) or just ⟨M⟩a. For an object a ∈ ∥A∥, theset ⟨M⟩a is called the hom-set at a. An element m of a hom-set ⟨M⟩a, or a pair (m,a)such that m ∈ ⟨M⟩a, is called an arrow of M (or an M-arrow) and written m ∶ ∗ a

or ∗ m // a (or just m if its codomain is understood or unimportant).(2)

If Φ ∶M → N ∶ X ∗ is a right module morphism, then the component of the naturaltransformation Φ ∶M → N ∶ X∼ → Set at x ∈ ∥X∥ is written x ⟨Φ⟩. The image of anM-arrow m ∶ x ∗ under the function x ⟨Φ⟩ ∶ x ⟨M⟩→ x ⟨N ⟩ is written variously as

m ⋅ x ⟨Φ⟩ = m ⋅Φ = Φ (m) = Φ ⋅m = x ⟨Φ⟩ ⋅m

. If Φ ∶M → N ∶ ∗ A is a left module morphism, then the component of the natural

transformation Φ ∶ M → N ∶ A → Set at a ∈ ∥A∥ is written ⟨Φ⟩a. The image of anM-arrow m ∶ ∗ a under the function ⟨Φ⟩a ∶ ⟨M⟩a→ ⟨N ⟩a is written variously as

m ⋅ ⟨Φ⟩a = m ⋅Φ = Φ (m) = Φ ⋅m = ⟨Φ⟩a ⋅m

.

Notation 1.1.3. The notations [X∼,Set] and [A,Set] (see Preliminary 10) are abbreviated to[X ∶] and [∶ A] respectively;(1)

for a category X, [X ∶] (an abbreviation of [X∼,Set]) denotes the category of rightmodules over X;

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for a categoryA, [∶A] (an abbreviation of [A,Set]) denotes the category of left modulesover A;

(2) for a functor K ∶ E →X, [K ∶] ∶ [X ∶] → [E ∶] (an abbreviation of [K∼,Set] ∶ [X∼,Set] →

[E∼,Set]) denotes the precomposition functor given by the assignment M ↦ K M . for a functor K ∶ E → A, [∶ K] ∶ [∶A] → [∶ E] (an abbreviation of [K,Set] ∶ [A,Set] →

[E,Set]) denotes the precomposition functor given by the assignment M ↦ K M .

Remark 1.1.4. .(1) If X (resp. A) is a small category, then the category [X ∶] (resp. [∶A]) is locally small.(2) The assignment K ↦ [K ∶] (resp. K ↦ [∶K]) is contravariant functorial.(3) By denition,

[∶A] = [A∼ ∶]; a left module over A is the same thing as a right module over the opposite category A∼.

Denition 1.1.5.

A right module morphism Φ ∶M → N ∶ X ∗ is called iso if it is an isomorphism in thecategory [X ∶].

A left module morphism Φ ∶M → N ∶ ∗ A is called iso if it is an isomorphism in thecategory [∶A].

Proposition 1.1.6.

A right module morphism Φ ∶ M → N ∶ X ∗ is iso if and only if each componentx ⟨Φ⟩ ∶ x ⟨M⟩→ x ⟨N ⟩ is a bijection.

A left module morphism Φ ∶M → N ∶ ∗ A is iso if and only if each component ⟨Φ⟩a ∶⟨M⟩a→ ⟨N ⟩a is a bijection.

Proof. Since a right module morphism Φ ∶ M → N ∶ X ∗ is the same thing as a naturaltransformation Φ ∶M → N ∶ X∼ → Set, this is an instance of the general fact that a naturaltransformation is iso i each component is an isomorphism.

Denition 1.1.7. A [two-sided] module M from a category X to a category A, writtenM ∶ X A, is a bifunctor M ∶ X∼ ×A → Set. Given a pair of modules M,N ∶ X A,a morphism Φ from M to N , written Φ ∶ M → N ∶ X A, is a natural transformationΦ ∶M→ N ∶X∼ ×A→ Set.

Remark 1.1.8.(1) If M ∶ X A is a module, then the image of an object (resp. an arrow) (x, a) ∈ X∼ ×A

under the functor M ∶ X∼ ×A → Set is written (x) ⟨M⟩ (a) or just x ⟨M⟩a. For a pairof objects x ∈ ∥X∥ and a ∈ ∥A∥, the set x ⟨M⟩a is called the hom-set from x to a. Anelement m of a hom-set x ⟨M⟩a, or a triple (x,m,a) such that m ∈ x ⟨M⟩a, is called an

arrow of M (or an M-arrow) and written m ∶ x a or xm // a (or just m if its domain

and codomain are understood or unimportant).(2) If Φ ∶ M → N ∶ X A is a module morphism, then the component of the natural

transformation Φ ∶M → N ∶ X∼ ×A → Set at (x,a) ∈ ∥X∼ ×A∥ is written x ⟨Φ⟩a. Theimage of an M-arrow m ∶ x a under the function x ⟨Φ⟩a ∶ x ⟨M⟩a → x ⟨N ⟩a is writtenvariously as

m ⋅ x ⟨Φ⟩a = m ⋅Φ = Φ (m) = Φ ⋅m = x ⟨Φ⟩a ⋅m.

(3) Given a universe U, a moduleM ∶XA is called

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a) locally U-small if X and A are locally U-small and all hom-sets ofM are U-small; thatis, if all hom-sets of X, A, andM are U-small;

b) U-small if it is locally U-small and if X and A are U-small.We just say small instead of U-small for U chosen implicitly.

Notation 1.1.9. The notation [X∼ ×A,Set] (see Preliminary 10) is abbreviated to [X ∶ A];(1) for a pair of categories X and A, [X ∶A] (an abbreviation of [X∼ ×A,Set]) denotes the

category of modules XA;(2) for a pair of functors S ∶ E→X and T ∶D→A, [S ∶ T] ∶ [X ∶A]→ [E ∶D] (an abbreviation

of [S∼ ×T,Set] ∶ [X∼ ×A,Set]→ [E∼ ×D,Set]) denotes the precomposition functor givenby the assignment M ↦ [S∼ ×T] M .

Remark 1.1.10.(1) If X and A are small categories, then the category [X ∶A] is locally small.(2) The assignment (S,T)↦ [S ∶ T] is contravariant bifunctorial.(3) By denition,

[X ∶A] = [∶X∼ ×A]; a two-sided moduleM ∶XA is the same thing as a left moduleM ∶ ∗X∼ ×A. Thecanonical isomorphism

[A∼ ×X]∼ ≅X∼ ×A

yields[X ∶A] ≅ [A∼ ×X ∶]

; by this isomorphism, a two-sided moduleM ∶X A is regarded as the same thing as aright moduleM ∶A∼ ×X ∗.

(4) The canonical isomorphism X∼ ≅X∼ × ∗ yields a canonical isomorphism

[X ∶] ≅ [X ∶ ∗]

natural in X. By this isomorphism, a right module over X is identied with a two-sidedmodule from X to the terminal category.

The canonical isomorphism A ≅ ∗ ×A yields a canonical isomorphism

[∶A] ≅ [∗ ∶A]

natural in A. By this isomorphism, a left module over A is identied with a two-sidedmodule from the terminal category to A.

(5) There are obvious isomorphisms

Set ≅ [∗ ∶] ≅ [∶ ∗] ≅ [∗ ∶ ∗]

, by which a set is identied with a module ∗ ∗ over the terminal category.

Denition 1.1.11. A module morphism Φ ∶M → N ∶ X A is called iso if it is an isomor-phism in the category [X ∶A].

Proposition 1.1.12. A module morphism Φ ∶ M → N ∶ X A is iso if and only if eachcomponent x ⟨Φ⟩a ∶ x ⟨M⟩a→ x ⟨N ⟩a is a bijection.

Proof. See the proof of Proposition 1.1.6.

Denition 1.1.13.

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(1) LetM ∶X ∗ be a right module. The composite g m = m g of an X-arrow g ∶ y → x

and anM-arrow m ∶ x ∗, as indicated in

xm // ∗

y

g

OO

g m

99

, is theM-arrow y ∗ dened by

g m = g ⟨M⟩ ⋅m

, the image of m under the function g ⟨M⟩ ∶ x ⟨M⟩→ y ⟨M⟩. LetM ∶ ∗ A be a left module. The composite m f = f m of anM-arrow m ∶ ∗ a

and an A-arrow f ∶ a→ b, as indicated in

∗ m //

m f %%

a

fb

, is theM-arrow ∗ b dened by

m f = m ⋅ ⟨M⟩ f

, the image of m under the function ⟨M⟩ f ∶ ⟨M⟩a→ ⟨M⟩b.(2) LetM ∶XA be a module.

The composite g m = m g of an X-arrow g ∶ y → x and an M-arrow m ∶ x a, asindicated in

xm // a

y

g

OO

g m

99

, is theM-arrow y a dened by

g m = g ⟨M⟩a ⋅m

, the image of m under the function g ⟨M⟩a ∶ x ⟨M⟩a→ y ⟨M⟩a. The composite m f = f m of an M-arrow m ∶ x a and an A-arrow f ∶ a → b, as

indicated inx

m //

m f %%

a

fb

, is theM-arrow x b dened by

m f = m ⋅ x ⟨M⟩ f

, the image of m under the function x ⟨M⟩ f ∶ x ⟨M⟩a→ x ⟨M⟩b.

Remark 1.1.14.(1) A module M ∶ X A thus induces a composition law among the arrows of X, A, andM. Conversely, a moduleM ∶ X A may be dened by giving a set ofM-arrows and acomposition law of them with X-arrows and A-arrows satisfying the associativity axiom,i.e. by giving a collage (see Section 3.1) from X to A.

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(2) The composite(g m) f = g m f = g (m f)

of an X-arrow g ∶ y → x, anM-arrow m ∶ x a, and an A-arrow f ∶ a→ b, as indicated in

xm // a

f

y

g

OO

g m f// b

, is theM-arrow y b dened by

g m f = m ⋅ g ⟨M⟩ f

, the image of m under the function g ⟨M⟩ f ∶ x ⟨M⟩a→ y ⟨M⟩b.(3) The naturality of a module morphism Φ ∶M→ N ∶XA is expressed by the identity

Φ (g m f) = g Φ (m) f

for every composable triple of an X-arrow g, anM-arrow m, and an A-arrow f.(4) When a two-sided moduleM ∶XA is regarded as a right moduleM ∶A∼×X ∗ (resp.

a left module M ∶ ∗ X∼ ×A), an M-arrow m ∶ x a is written as m ∶ (a,x) ∗ (resp.m ∶ ∗ (x,a)), and the compositions

xm // a x

m //

m f %%

a

f

xm // a

f

y

g

OO

g m

99

b y

g

OO

g m f// b

are written as

(a,x) m // ∗ (a,x) m // ∗ (a,x) m // ∗

(a,y)(a,g)

OO

(a,g) m

99

(b,x)(f,x)

OO

(f,x) m

99

(b,y)(f,g)

OO

(f,g) m

88

(resp.

∗ m //

m (g,a) %%

(x,a)(g,a)

∗ m //

m (x,f) &&

(x,a)(x,f)

∗ m //

m (g,f) &&

(x,a)(g,f)

(y,a) (x,b) (y,b)).

Denition 1.1.15. The hom of a category C is the endomodule ⟨C⟩ ∶ C C given by theassignment (x, a)↦ homC (x, a) for x, a ∈C.

Remark 1.1.16.(1) For a pair of objects a,b ∈ ∥C∥, the hom-set homC (a,b) of a category C is the same thing

as the hom-set a ⟨C⟩b of the endomodule ⟨C⟩.(2) Hereafter a hom-set is written as a ⟨C⟩b rather than homC (a,b); likewise, for an object

c ∈ ∥C∥ and a C-arrow f ∶ a→ b, the functions

homC (c, f) ∶ homC (c,a)→ homC (c,b) homC (f,c) ∶ homC (b,c)→ homC (a,c)are written as

c ⟨C⟩ f ∶ c ⟨C⟩a→ c ⟨C⟩b f ⟨C⟩c ∶ b ⟨C⟩c→ a ⟨C⟩c.

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(3) When C is understood, the hom-set between objects a and b is also written as [a,b].For example, the set of functions from a small set S to a small set T is written as [S,T ]rather than S ⟨Set⟩T ; this notation is also consistent with that in Preliminary 10, withsets regarded as discrete categories.

Notation 1.1.17.(1) In a diagram, a module M ∶ X A and a module morphism Φ ∶M → N ∶ X A are

depicted as

XM //A X

M++

N33Φ A

.(2) Italic capital letters M , N , ... vary over both modules and module morphisms. When we

write M ∈ [X ∶A], M ∶ X A, or XM //A , M stands for an object or an arrow of the

module category [X ∶A], i.e. a module or a module morphism (cf. Preliminary 8).

Denition 1.1.18.

Given a right module (or module morphism) M and a functor (or natural transformation)S as in

ES //X

M // ∗, their composite, written [S] ⟨M⟩ or just S ⟨M⟩, is the right module (or module morphism)E ∗ dened by the composition

E∼ S //X∼ M // Set

. Given a left module (or module morphism) M and a functor (or natural transformation) T

as in

∗ M //A DToo

, their composite, written ⟨M⟩ [T ] or just ⟨M⟩T , is the left module (or module morphism)∗D dened by the composition

DT //A

M // Set

.

Remark 1.1.19. The composition

(S,M)↦ S ⟨M⟩ ∶ [E,X] × [X ∶]→ [E ∶]is functorial in each variable, contravariant in S and covariant in M . If S ∶ E → X is afunctor, then

S ⟨M⟩ = S M =M ⋅ [S ∶]for any M ∈ [X ∶].

The composition(M,T )↦ ⟨M⟩T ∶ [∶A] × [D,A]→ [∶D]

is functorial in each variable, covariant in both M and T . If T ∶D→A is a functor, then

⟨M⟩T = T M =M ⋅ [∶ T]

for any M ∈ [∶A].

18


Denition 1.1.20. Given a module (or module morphism) M , a functor (or natural transfor-mation) S, and a second functor (or natural transformation) T , all as in

ES //X

M //A DToo

, their composite, written [S] ⟨M⟩ [T ] or just S ⟨M⟩T , is the module (or module morphism)ED dened by the composition

E∼ ×DS×T //X∼ ×A

M // Set

.

Remark 1.1.21.(1) The composition

(S,M,T )↦ S ⟨M⟩T ∶ [E,X] × [X ∶A] × [D,A]→ [E ∶D]

is functorial in each variable, contravariant in S and covariant in M and T . If S ∶ E → Xand T ∶D→A are functors, then

S ⟨M⟩T = [S ×T] M =M ⋅ [S ∶ T]

for any M ∈ [X ∶A].(2) As a special case,

given

ES //X

M //A

, their composite S ⟨M⟩ ∶ EA is dened by the composition

E∼ ×AS×A //X∼ ×A

M // Set

. If A is the terminal category, then the composite S ⟨M⟩ ∶ E ∗ coincides with thatdened in Denition 1.1.18 under the identication [X ∶] ≅ [X ∶ ∗] and [E ∶] ≅ [E ∶ ∗].

given

XM //A D

Too

, their composite ⟨M⟩T ∶XD is dened by the composition

X∼ ×DX∼×T //X∼ ×A

M // Set

. If X is the terminal category, then the composite ⟨M⟩T ∶ ∗ D coincides with thatdened in Denition 1.1.18 under the identication [∶A] ≅ [∗ ∶A] and [∶D] ≅ [∗ ∶D].

Proposition 1.1.22.

(1) Given

E′ S′ // ES //X

M //A DToo D′T ′oo

, the associative lawS′ ⟨S ⟨M⟩T ⟩T ′ = [S′ S] ⟨M⟩ [T T ′]

holds.

19


(2) Given

ES //X

M //A DToo

, the associative lawS ⟨⟨M⟩T ⟩ = S ⟨M⟩T = ⟨S ⟨M⟩⟩T

holds.

Proof.(1) Indeed,

S′ ⟨S ⟨M⟩T ⟩T ′ = [S′ × T ′] [[S × T ] M]= [[S′ × T ′] [S × T ]] M= [[S′ S] × [T ′ T ]] M= [S′ S] ⟨M⟩ [T T ′]

.(2) By what we have just seen,

S ⟨⟨M⟩T ⟩ = S ⟨[1X] ⟨M⟩T ⟩ = [S 1X] ⟨M⟩T = S ⟨M⟩T

and⟨S ⟨M⟩⟩T = ⟨S ⟨M⟩ [1A]⟩T = S ⟨M⟩ [1A T ] = S ⟨M⟩T

.

Example 1.1.23.

(1) Given a module and functors as in

ES //X

M //A DToo

, their composite is the module S ⟨M⟩T ∶ ED dened by

e ⟨S ⟨M⟩T⟩d = (e ⋅ S) ⟨M⟩ (T ⋅ d)

for e ∈ E and d ∈D. An S ⟨M⟩T-arrow m ∶ e d is given by anM-arrow m ∶ e ⋅ S T ⋅ d.For an E-arrow h ∶ e′ → e, an S ⟨M⟩T-arrow m ∶ e d, and a D-arrow k ∶ d → d′, theircomposite h m k ∶ e′ d′ is given by theM-arrow (h ⋅ S) m (T ⋅ h) ∶ e′ ⋅ S T ⋅ d′ asindicated in

e e ⋅ S m // T ⋅ dT ⋅ k

d

k

e′h

OO

e′ ⋅ Sh ⋅ SOO

h m k// T ⋅ d′ d′

.(2) Given a module morphism and functors as in

ES //X

M))

N55Φ A D

Too

, their composite is the module morphism S ⟨Φ⟩T ∶ S ⟨M⟩T→ S ⟨N ⟩T ∶ ED dened by

e ⟨S ⟨Φ⟩T⟩d = (e ⋅ S) ⟨Φ⟩ (T ⋅ d)

for e ∈ ∥E∥ and d ∈ ∥D∥.

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(3) Given a natural transformation σ ∶ S′ → S, a moduleM, and a second natural transforma-tion τ ∶ T→ T′ as in

ES

))

S′55↑σ X

M //A DT

uu

T′ii ↓τ

, their composite is the module morphism σ ⟨M⟩ τ ∶ S ⟨M⟩T → S′ ⟨M⟩T′ ∶ E D whichmaps each S ⟨M⟩T-arrow m ∶ e d to the S′ ⟨M⟩T′-arrow σe m τd ∶ e d as indicatedin

e ⋅ S m // T ⋅ dτd

e ⋅ S′σe

OO

m ⋅ σ⟨M⟩τ// T′ ⋅ d

.(4) The restriction of a moduleM ∶XA to subcategories Y ⊆X and B ⊆A is given by the

composition

YY //X

M //A BBoo

of M with the two inclusion functors. Similarly, the restriction of a module morphismΦ ∶M→ N ∶XA to Y and B is given by the composition

YY //X

M))

N55Φ A B

Boo

.(5) The evaluation of a moduleM ∶XA at (x, a) ∈X∼×A is identied with the composition

∗ x //XM //A ∗aoo

.(6) Given a moduleM ∶X ×Y A ×B and an object (x,a) ∈ ∥X ×A∥, the composition

Yx×Y //X ×Y

M //A ×B Ba×Boo

of M with the sections x ×Y and a ×B yields the slice of M at (x,a), i.e. the module[x ×Y] ⟨M⟩ [a ×B] ∶Y B such that

y ⟨[x ×Y] ⟨M⟩ [a ×B]⟩ b = (x, y) ⟨M⟩ (a, b)

for y ∈Y and b ∈ B.(7)

Given a right moduleM ∶X ∗ and a category E, the composition

XM // ∗ E

∆Eoo

yields a two-sided module ⟨M⟩∆E ∶ X E by duplicating M across E. The E-arydiagonal functor of [X ∶] is the precomposition functor

[X ∶ ∆E] ∶ [X ∶]→ [X ∶ E]

sending each right moduleM ∶X ∗ to the composite module ⟨M⟩∆E ∶X E. As aspecial case, the E-ary diagonal functor

[∶ ∆E] ∶ Set→ [∶ E]

21


of Set sends each small set S to the constant left module ⟨S⟩∆E ∶ ∗ E given by thecomposition

∗ S // ∗ E∆Eoo

. Given a left moduleM ∶ ∗A and a category E, the composition

E∆E // ∗ M //A

yields a two-sided module ∆E ⟨M⟩ ∶ E A by duplicating M across E. The E-arydiagonal functor of [∶A] is the precomposition functor

[∆E ∶A] ∶ [∶A]→ [E ∶A]

sending each left module M ∶ ∗ A to the composite module ∆E ⟨M⟩ ∶ E A. As aspecial case, the E-ary diagonal functor

[∆E ∶] ∶ Set→ [E ∶]

of Set sends each small set S to the constant right module ∆E ⟨S⟩ ∶ E ∗ given by thecomposition

E∆E // ∗ S // ∗

.(8)

Given a right moduleM ∶X ∗ and a category E, the composition

E∼ ×X∆E∼×X //X

M // ∗

yields a right module [∆E∼ ×X] ⟨M⟩ ∶ E∼ × X ∗ by introducing to M a dummyvariable varying over E∼. Note that this right module is the same thing as the two-sidedmodule ⟨M⟩∆E ∶X E in the preceding example.

Given a left moduleM ∶ ∗A and a category E, the composition

∗ M //A E∼ ×A∆E∼×Aoo

yields a left module ⟨M⟩ [∆E∼ ×A] ∶ ∗ E∼×A by introducing toM a dummy variablevarying over E∼. Note that this left module is the same thing as the two-sided module∆E ⟨M⟩ ∶ EA in the preceding example.

(9) Given a category E, the compositions

E∆E // ∗ ⟨∗⟩ // ∗ ∗ ⟨∗⟩ // ∗ E

∆Eoo

, where ⟨∗⟩ is the hom of the terminal category, yield right and left modules

∆E∗ ∶= ∆E ⟨∗⟩ ∆∗E ∶= ⟨∗⟩∆E

over E such thate ⟨∆E∗⟩ = ∗ = ⟨∆∗E⟩e

for every object e ∈ ∥E∥. In fact, ∆E∗ and ∆∗E are given by the same functor ∆E∗ = ∆∗E ∶E → Set which sends every object of E to the singleton set ∗ and sends every arrow ofE to the identity function ∗→ ∗.

22


Theorem 1.1.24. Let Φ ∶ M → N ∶ X A be a module morphism and let Y and B beessentially wide (see Preliminary 19) subcategories of X and A. Then Φ is iso if and only ifits restriction to Y and B (see Example 1.1.23(4)) is iso; that is, if and only if the componenty ⟨Φ⟩b ∶ y ⟨M⟩b→ y ⟨N ⟩b at each y ∈ ∥Y∥ and b ∈ ∥B∥ is a bijection (cf. Proposition 1.1.12).

Proof. Since a module morphism Φ ∶M→ N ∶XA is the same thing as a natural transfor-mation Φ ∶M→ N ∶X∼ ×A→ Set, this is an instance of the lemma in Preliminary 19.

Theorem 1.1.25. If Φ in Example 1.1.23(2) is a module isomorphism, so is the compositeS ⟨Φ⟩T. The converse holds if S and T are essentially surjective.

Proof. Note that S ⟨Φ⟩T is given by the image of Φ under the precomposition functor [S ∶ T].Since any functor preserves isomorphisms, if Φ is an isomorphism, so is S ⟨Φ⟩T. Since theimage of an essentially surjective functor is an essentially wide subcategory of its codomain,the second assertion follows from Theorem1.1.24.

Remark 1.1.26. Theorem1.1.24 is a special case of Theorem1.1.25 where S and T are inclusionfunctors.

Denition 1.1.27. Given a module M ∶ X A, the opposite module M∼ ∶ A∼ X∼ isdened by the composition

A ×X∼ ∼ //X∼ ×AM // Set

, where ∼ denotes the simple transposition (a, x)↦ (x, a); similarly, given a module morphismsΦ ∶M → N ∶ X A, the opposite module morphism Φ∼ ∶M∼ → N ∼ ∶ A∼ X∼ is dened bythe composition

A ×X∼ ∼ //X∼ ×AM //

N//Φ Set

.

Remark 1.1.28.(1) The assignmentsM↦M∼ and Φ↦ Φ∼ yield an isomorphism

[X ∶A] ≅ [A∼ ∶X∼].

(2) For any moduleM and any module morphism Φ,

⟨M∼⟩∼ =M ⟨Φ∼⟩∼ = Φ

.(3) For any category C,

⟨C∼⟩ = ⟨C⟩∼

; that is, the hom of the opposite category is the opposite module of the hom.(4) For any composite module S ⟨M⟩T,

⟨S ⟨M⟩T⟩∼ = T ⟨M∼⟩S

; that is, the opposite of the composite

ES //X

M //A DToo

is given by the composite

D∼ T //A∼ M∼//X∼ E∼Soo

.

23


(5) The opposite of the module morphisms Φ ∶M→ N ∶XA is often denoted by Φ ∶M∼ →N ∼ ∶A∼ X∼ using the same name as its original counterpart (cf. Preliminary 6).

(6) The opposite of a right moduleM ∶X ∗ is the left moduleM∼ ∶ ∗X∼; in fact,M andM∼ are dened by the same functor

M∼ =M ∶X∼ → Set

.

1.2. Cells

Denition 1.2.1. Given a pair of modules M ∶ X A and N ∶ Y B, and given a pair offunctors P ∶ X → Y and Q ∶ A → B, a module cell (or just a cell) Φ ∶ P Q ∶M N , writtendiagrammatically as

X

P

M //A

Q

Y N//

Φ

B

, is dened by a module morphism Φ ∶M→ P ⟨N ⟩Q ∶XA.

Remark 1.2.2.(1) The functors P and Q are called the left and right components of a cell Φ ∶ P Q ∶M N .

A cell is sometimes denoted just by Φ ∶ M N and its left and right components arereferred to by Φ0 ∶M0 → N0 and Φ1 ∶M1 → N1 respectively.

(2) A cell Φ ∶ P Q ∶M N sends eachM-arrow m ∶ x a to the N -arrow m ⋅Φ ∶ x ⋅P Q ⋅a,the image of m under the function

x ⟨M⟩a x⟨Φ⟩a // x ⟨P ⟨N ⟩Q⟩a = (x ⋅ P) ⟨N ⟩ (Q ⋅ a)

.(3) Cells and module morphisms are regarded as special instances of each other. A cell Φ ∶

P Q ∶M N is thought of as a module morphism from M to the composite moduleP ⟨N ⟩Q. Conversely, a module morphism Φ ∶M→ N ∶XA is expressed by a cell

X

1

M //A

1

X N//

Φ

A

.(4) The identity module morphismM→M yields the identity cell

X

1

M //A

1

X M//

1

A

.(5) Any composition

XP //Y

N // B AQoo

24


trivially yields a cell

X

P

P⟨N ⟩Q //A

Q

Y N//

1

B

; a cell

X

P

M //A

Q

Y N//

1

B

expresses an identityM = P ⟨N ⟩Q.

Proposition 1.2.3. A cell is compatible with composition of arrows. Specically, a cell Φ ∶P Q ∶M N sends a commutative diagram

xm // a

f

y

g

OO

g m f// b

in M to a commutative diagram

x ⋅ P m ⋅Φ // Q ⋅ aQ ⋅ f

y ⋅ Pg ⋅ POO

(g m f) ⋅Φ// Q ⋅ b

in N ; that is, the identity

(g m f) ⋅Φ = (g ⋅ P) (m ⋅Φ) (Q ⋅ f)

holds.

Proof. The commutativity of the second diagram follows from the naturality of Φ (see Re-mark 1.1.14(3)) and noting Example 1.1.23(1).

Denition 1.2.4.

LetM ∶X ∗ and N ∶Y ∗ be right modules. Given a functor P ∶X→Y, a right modulecell Φ ∶ P ∗ ∶M N , written diagrammatically as

XM //

P

∗1

Y N//

Φ

∗

, is dened by a right module morphism Φ ∶M→ P ⟨N ⟩ ∶X ∗. Let M ∶ ∗ A and N ∶ ∗ B be left modules. Given a functor Q ∶ A → B, a left module

cell Φ ∶ ∗ Q ∶M N , written diagrammatically as

∗ M //

1

A

Q

∗ N//

Φ

B

, is dened by a left module morphism Φ ∶M→ ⟨N ⟩Q ∶ ∗A.

25


Remark 1.2.5. A right (resp. left) module cell in Denition 1.2.4 is regarded as a special instanceof a two-sided module cell in Denition 1.2.1 where A and B (resp. X and Y) are the terminalcategory under the identication in Remark 1.1.10(4). Conversely, by Remark 1.1.10(3), a two-sided module cell in Denition 1.2.1 is the same thing as a right (resp. left) module cell depictedbelow:

A∼ ×XM //

Q×P

∗1

∗ M //

1

X∼ ×A

P×Q

B∼ ×Y N//

Φ

∗ ∗ N//

Φ

Y∼ ×B

Denition 1.2.6.

LetM ∶ X ∗ be a right module and N ∶ Y B be a two-sided module. Given a functorP ∶ X → Y and an object b ∈ ∥B∥, a right conical cell Φ ∶ P b ∶ M N , writtendiagrammatically as

X

P

M // ∗b

Y N//

Φ

B

, is dened by a right module morphism Φ ∶M→ P ⟨N ⟩b ∶X ∗. Let M ∶ ∗ A be a left module and N ∶ Y B be a two-sided module. Given an

object y ∈ ∥Y∥ and a functor Q ∶ A → B, a left conical cell Φ ∶ y Q ∶M N , writtendiagrammatically as

∗y

M //A

Q

Y N//

Φ

B

, is dened by a left module morphism Φ ∶M→ y ⟨N ⟩Q ∶ ∗A.

Remark 1.2.7.(1) A conical cell in Denition 1.2.6 is regarded as a special instance of a two-sided module cell

in Denition 1.2.1 where A (resp. X) is the terminal category under the identication inRemark 1.1.10(4).

(2) A right (resp. left) module cell in Denition 1.2.4 is regarded as a special instance of aright (resp. left) conical cell in Denition 1.2.6 where B (resp. Y) is the terminal category.Conversely, a right (resp. left) conical cell in Denition 1.2.6 is the same thing as a right(resp. left) module cell depicted below:

X

P

M // ∗1

∗1

M //A

Q

Y⟨N ⟩b

//

Φ

∗ ∗y⟨N ⟩

//

Φ

B

Denition 1.2.8. Given a pair of modules J ∶ E D and M ∶ X A, the module of cellsJ M,

⟨J ,M⟩ ∶ [E,X] [D,A], is dened by

(S) ⟨J ,M⟩ (T ) = (J ) ⟨E ∶D⟩ (S ⟨M⟩T )for S ∈ [E,X] and T ∈ [D,A], where ⟨E ∶D⟩ is the hom of the module category [E ∶D].

26


Remark 1.2.9.(1) For a pair of functors S ∶ E→X and T ∶D→A, the set (S) ⟨J ,M⟩ (T) consists of all cells

S T ∶ J M.(2) Given a natural transformation σ ∶ S′ → S, a cell Θ ∶ S T, and a second natural transfor-

mation τ ∶ T→ T′ as in

E

S′ σ

S

J //D

Tτ T′

X M//

Θ

A

, their composite is the cell

E

S′

J //D

T′

X M//

σ Θ τ

A

dened by the module morphism σ Θ τ ∶ J → S′ ⟨M⟩T′ given by the composition

J Θ // S ⟨M⟩Tσ⟨M⟩τ // S′ ⟨M⟩T′

. By Example 1.1.23(3), the cell σ Θ τ sends each J -arrow j ∶ e d to theM-arrow

j ⋅ ⟨σ Θ τ⟩ = σe (j ⋅Θ) τd ∶ e ⋅ S′ T′ ⋅ d

as indicated in

ej // d

e ⋅ S j ⋅Θ // T ⋅ dτd

e ⋅ S′σe

OO

j ⋅ ⟨σ Θ τ⟩// T′ ⋅ d

.(3) If J is small andM is locally small, then the module ⟨J ,M⟩ is locally small.

Proposition 1.2.10. Given a module J and a composite module P ⟨N ⟩Q as in

EJ //D

X

P

P⟨N ⟩Q //A

Q

Y N//

1

B

, the identity

[E,X][E,P]

⟨J ,P⟨N ⟩Q⟩ // [D,A][D,Q]

[E,Y]⟨J ,N ⟩

//

1

[D,B]

, i.e.⟨J ,P ⟨N ⟩Q⟩ = [E,P] ⟨J ,N ⟩ [D,Q]

, holds.

27


Proof. For any S ∈ [E,X] and T ∈ [D,A],

S ⟨J ,P ⟨N ⟩Q⟩T = (J ) ⟨E ∶D⟩ (S ⟨P ⟨N ⟩Q⟩T )= (J ) ⟨E ∶D⟩ ([S P] ⟨N ⟩ [Q T ])= (S P) ⟨J ,N ⟩ (Q T )= (S ⋅ [E,P]) ⟨J ,N ⟩ ([D,Q] ⋅ T )= (S) ⟨[E,P] ⟨J ,N ⟩ [D,Q]⟩ (T )

Remark 1.2.11. The cell ⟨J ,P ⟨N ⟩Q⟩ 1 ⟨J ,N ⟩ above sends each cell

E

S

J //D

T

XP⟨N ⟩Q

//

Θ

A

to the cell

E

S P

J //D

Q T

Y N//

Θ

B

dened by the same module morphism.

Denition 1.2.12. If Θ ∶ J M is a cell and Φ ∶M→ N is a module morphism as in

E

S

J //D

T

XM //

Θ

NΦ 77A

, then their composite Θ Φ = Φ Θ is the cell

E

S

J //D

T

X N//

Θ Φ

A

dened by the module morphism Θ Φ ∶ J → S ⟨N ⟩T given by the composition

J Θ // S ⟨M⟩TS⟨Φ⟩T // S ⟨N ⟩T

.

Remark 1.2.13. See Example 1.1.23(2) for the module morphism S ⟨Φ⟩T. The cell Θ Φ sendseach J -arrow j ∶ e d to the N -arrow

j ⋅ ⟨Θ Φ⟩ = j ⋅Θ ⋅Φ ∶ e ⋅ S T ⋅ d

, the image of j under the composite function

e ⟨J ⟩d e⟨Θ⟩d // e ⟨S ⟨M⟩T⟩d = (e ⋅ S) ⟨M⟩ (T ⋅ d) (e ⋅ S)⟨Φ⟩(T ⋅ d) // (e ⋅ S) ⟨N ⟩ (T ⋅ d)

.

28


Denition 1.2.14. Given a module J ∶ ED and a module morphism Φ ∶M→ N ∶XA,the module morphism

⟨J ,Φ⟩ ∶ ⟨J ,M⟩→ ⟨J ,N ⟩ ∶ [E,X] [D,A]

, postcomposition with Φ, is dened by

(S) ⟨J ,Φ⟩ (T) = (J ) ⟨E ∶D⟩ (S ⟨Φ⟩T)

for each pair of functors S ∶ E→X and T ∶D→A.

Remark 1.2.15.(1) The module morphism ⟨J ,Φ⟩ maps each cell Θ ∶ S T ∶ J M to the cell Θ Φ ∶ S T ∶J N dened in Denition 1.2.12.

(2) The assignment Φ→ ⟨J ,Φ⟩ is functorial; indeed the functor

⟨J ,−⟩ ∶ [X ∶A]→ [[E,X] ∶ [D,A]]

is dened byS ⟨J ,M⟩T = (J ) ⟨E ∶D⟩ (S ⟨M⟩T )

for S ∈ [E,X], T ∈ [D,A], and M ∈ [X ∶A].Note. By Remark 1.2.2(3), the following denition is regarded as a special case of Deni-tion 1.2.12, and vice versa.

Denition 1.2.16. Given a pair of cells as in

E

S

J //D

T

X

P

M //

Θ

A

Q

Y N//

Φ

B

, their composite Θ Φ = Φ Θ is the cell

E

S P

J //D

Q T

Y N//

Θ Φ

B

dened by the module morphism Θ Φ ∶ J → [S P] ⟨N ⟩ [Q T] given by the composition

J Θ // S ⟨M⟩TS⟨Φ⟩T // S ⟨P ⟨N ⟩Q⟩T = [S P] ⟨N ⟩ [Q T]

.

Remark 1.2.17. The cell Θ Φ sends each J -arrow j ∶ e d to the N -arrow

j ⋅ ⟨Θ Φ⟩ = j ⋅Θ ⋅Φ ∶ e ⋅ S ⋅ P Q ⋅T ⋅ d

29


, the image of j under the composite function

e ⟨J ⟩de⟨Θ⟩d

e ⟨S ⟨M⟩T⟩d = (e ⋅ S) ⟨M⟩ (T ⋅ d)(e ⋅ S)⟨Φ⟩(T ⋅ d)

(e ⋅ S) ⟨P ⟨N ⟩Q⟩ (T ⋅ d) = (e ⋅ S ⋅ P) ⟨N ⟩ (Q ⋅T ⋅ d).

Proposition 1.2.18. Modules and cells among them form a category with the compositiongiven in Denition 1.2.16 and the identities given in Remark 1.2.2(4).

Proof. The only non-trivial part is the verication of the associativity of the composition.Consider cells as in

X

P

M //A

Q

X′

P′

M′ //

Φ

A′

Q′

X′′

P′′

M′′ //

Φ′

A′′

Q′′

X′′′M′′′

//

Φ′′

A′′′

. The composites (Φ Φ′) Φ′′ and Φ (Φ′ Φ′′) are dened by the module morphisms⟨Φ P ⟨Φ′⟩Q⟩ [P P′] ⟨Φ′′⟩ [Q′ Q] and Φ P ⟨Φ′ P′ ⟨Φ′′⟩Q′⟩Q respectively. But by the func-toriality (see Remark 1.1.21(1)) and associativity (see Proposition 1.1.22) of the composition,we have

⟨Φ P ⟨Φ′⟩Q⟩ [P P′] ⟨Φ′′⟩ [Q′ Q] = Φ ⟨P ⟨Φ′⟩Q P ⟨P′ ⟨Φ′′⟩Q′⟩Q⟩= Φ P ⟨Φ′ P′ ⟨Φ′′⟩Q′⟩Q

.

Remark 1.2.19.(1) Given a universe U, MODU denotes the category consisting of all locally U-small modules

and all cells among them. By MOD we mean MODU for U chosen implicitly.(2) Given a pair of categories X and A, there is a canonical embedding [X ∶A]MOD, iden-

tical on objects, given by the obvious arrow function (see Remark 1.2.2(3)). The embeddingis not, in general, full.

Proposition 1.2.20. A cell Φ ∶ P Q ∶M N is an isomorphism in the category MOD ifand only if the functors P and Q are iso and the module morphism Φ ∶M→ P ⟨N ⟩Q is iso.

Proof. If Φ ∶ P Q ∶M N has an inverse Ψ ∶ S T ∶ N M, then P (resp. Q) and S (resp.T) are inverse to each other and Φ ∶M → P ⟨N ⟩Q and P ⟨Ψ⟩Q ∶ P ⟨N ⟩Q →M are inverse toeach other. Conversely, if P, Q, and Φ ∶M→ P ⟨N ⟩Q are iso, the inverse of Φ ∶ P Q ∶M Nis given by the cell Ψ ∶ S T ∶ N M with S = P−1, T = Q−1, and the module morphismΨ ∶ N → S ⟨M⟩T dened by Ψ = [P−1] ⟨Φ−1⟩ [Q−1].

Denition 1.2.21. A cell Φ ∶ P Q ∶M N is called

30


(1) iso if it satises the equivalent conditions in Proposition 1.2.20;(2) fully faithful if the module morphism Φ ∶M→ P ⟨N ⟩Q is iso.

Remark 1.2.22. In Proposition 1.2.29 we will see the relation between the notion of fully faith-fulness for cells and that for functors.

Note. Proposition 1.2.10 allows the following denition.

Denition 1.2.23. Given a module J ∶ ED and a cell

X

P

M //A

Q

Y N//

Φ

B

, the cell

[E,X] ⟨J ,M⟩ //

[E,P]

[D,A][D,Q]

[E,Y]⟨J ,N ⟩

//

⟨J ,Φ⟩

[D,B]

, postcomposition with Φ, is dened by the module morphism

⟨J ,M⟩ ⟨J ,Φ⟩ // ⟨J ,P ⟨N ⟩Q⟩ = [E,P] ⟨J ,N ⟩ [D,Q]

, postcomposition (see Denition 1.2.14) with Φ ∶M→ P ⟨N ⟩Q.

Remark 1.2.24. The cell ⟨J ,Φ⟩ sends each cell Θ ∶ S T ∶ J M to the cell Θ Φ ∶ S P Q T ∶ J N dened in Denition 1.2.16.

Proposition 1.2.25. The assignment Φ↦ ⟨J ,Φ⟩ is functorial.

Proof. Clearly, the assignment Φ↦ ⟨J ,Φ⟩ preserves the identities. To verify that it preservesthe composition, let Φ and Ψ be a composable pair of cells and consider the cells ⟨J ,Φ⟩, ⟨J ,Ψ⟩,and ⟨J ,Φ Ψ⟩ depicted in

XM //

P

A

Q

[E,X] ⟨J ,M⟩ //

[E,P]

[D,A][D,Q]

Y N //

P′

Φ

B

Q′

[E,Y] ⟨J ,N ⟩ //

[E,P′]

⟨J ,Φ⟩

[D,B][D,Q′]

Z L//

Ψ

C [E,Z]⟨J ,L⟩

//

⟨J ,Ψ⟩

[D,C]

XM //

P P′

A

Q′ Q

[E,X] ⟨J ,M⟩ //

[E,P P′]

[D,A][D,Q′ Q]

Z L//

Φ Ψ

C [E,Z]⟨J ,L⟩

//

⟨J ,Φ Ψ⟩

[D,C]

. We need to verify that the composition of the cells ⟨J ,Φ⟩ and ⟨J ,Ψ⟩ yields the cell ⟨J ,Φ Ψ⟩.First note that [E,P P′] = [E,P] [E,P′] and [D,Q′ Q] = [D,Q′] [D,Q] by the functo-riality of the operations [E,−] and [D,−]. The cell ⟨J ,Φ⟩ ⟨J ,Ψ⟩ is dened by the module

31


morphism ⟨J ,Φ⟩ [E,P] ⟨J ,Ψ⟩ [D,Q] and the cell ⟨J ,Φ Ψ⟩ is dened by the module mor-phism ⟨J ,Φ P ⟨Ψ⟩Q⟩. But by the functoriality of the operation ⟨J ,−⟩ (see Remark 1.2.15(2))and Proposition 1.2.10,

⟨J ,Φ P ⟨Ψ⟩Q⟩ = ⟨J ,Φ⟩ ⟨J ,P ⟨Ψ⟩Q⟩ = ⟨J ,Φ⟩ [E,P] ⟨J ,Ψ⟩ [D,Q]

.

Remark 1.2.26. Given a small module J ∶ ED, the functor

⟨J ,−⟩ ∶MOD→MOD

is dened by the object functionM ↦ ⟨J ,M⟩ and the arrow function Φ ↦ ⟨J ,Φ⟩, extendingthe functor ⟨J ,−⟩ in Remark 1.2.15(2) as shown in

[X ∶A]

⟨J ,−⟩ // [[E,X] ∶ [D,A]]

MOD⟨J ,−⟩

//MOD

, where denotes the canonical embedding in Remark 1.2.19(2).

Denition 1.2.27. The hom of a functor H ∶C→ B is the cell

C⟨C⟩ //

H

C

H

B⟨B⟩

//

⟨H⟩

B

given by the arrow function of H; that is, for each pair of objects x,a ∈ ∥C∥, the function

a ⟨H⟩b ∶ a ⟨C⟩b→ (a ⋅H) ⟨B⟩ (H ⋅ b)

is given byHa,b ∶ homC (a,b)→ homB (H ⋅ a,H ⋅ b) ; f ↦ H ⋅ f

.

Remark 1.2.28.(1) The naturality of the module morphism ⟨H⟩ ∶ ⟨C⟩ → H ⟨B⟩H ∶ C C follows from the

functoriality of H.(2) Hereafter, given a functor H, each component of the arrow function of H is written as

a ⟨H⟩b.

Proposition 1.2.29. A functor H ∶ C → B is iso (resp. fully faithful) if and only if the homcell ⟨H⟩ ∶ H H ∶ ⟨C⟩ ⟨B⟩ is iso (resp. fully faithful) (see Denition 1.2.21).

Proof. Immediate from the denitions.

Theorem 1.2.30. The assignment of the hom dened in Denition 1.1.15 and Denition 1.2.27embeds CAT in MOD. Specically, the assignment C ↦ ⟨C⟩ forms a faithful functor ⟨−⟩ ∶CAT→MOD, injective on objects.

32


Proof. The verication of the functoriality is straightforward. The faithfulness and the injec-tivity on objects are evident.

Denition 1.2.31. Given a cell and commutative squares of functors as in

X′ R //

P′

XM //

P

A

Q

A′Foo

Q′

Y′S

//Y N//

Φ

B B′G

oo

, their pasting composite

X′

P′

R⟨M⟩F //A′

Q′

Y′S⟨N ⟩G

//

R⟨Φ⟩F

B′

is dened by module morphism

R ⟨Φ⟩F ∶ R ⟨M⟩F→ R ⟨P ⟨N ⟩Q⟩F = P′ ⟨S ⟨N ⟩G⟩Q′

given by the composition

X′ R //XM

**

P⟨N ⟩Q44Φ A A′Foo

.

Remark 1.2.32. A pasting composition

X′

1

R //XM //

P

A

Q

A′Foo

1

X′S

//Y N//

Φ

B A′G

oo

with both ends being identities, yields a cell

X′

1

R⟨M⟩F //A′

1

X′S⟨N ⟩G

//

R⟨Φ⟩F

A′

, i.e. a module morphism R ⟨Φ⟩F ∶ R ⟨M⟩F→ S ⟨N ⟩G ∶X′ A′.

Proposition 1.2.33. In Denition 1.2.31, if Φ is fully faithful, so is the cell R ⟨Φ⟩F.

Proof. Immediate from Theorem1.1.25.

Proposition 1.2.34. Cells and commutative squares of functors as in

X′ R //

P′

XM //

P

A

Q

A′Foo

Q′

Y′ S //

K′

Y N //

K

Φ

B

L

B′Goo

L′

Z′T

// Z L//

Ψ

C C′H

oo

33


yield the same cell

R ⟨Φ Ψ⟩F = R ⟨Φ⟩F S ⟨Ψ⟩G ∶ P′ K′ L′ Q′ ∶ R ⟨M⟩F T ⟨L⟩H

irrespective of the order of the horizontal and vertical compositions.

Proof. The horizontal composition followed by the vertical composition yields the cell R ⟨Φ⟩FS ⟨Ψ⟩G,and the vertical composition followed by the horizontal composition yields the cell R ⟨Φ Ψ⟩F,as shown in

X′ R⟨M⟩F //

P′

A′

Q′

Y′ S⟨N ⟩G //

K′

R⟨Φ⟩F

B′

L′

X′ R //

P′ K′

XM //

P K

A

L Q

BFoo

L′ Q′

Z′T⟨L⟩H

//

S⟨Ψ⟩G

C′ Z′T

// Z L//

Φ Ψ

C C′H

oo

X′ R⟨M⟩F //

P′ K′

A′

L′ Q′

X′ R⟨M⟩F //

P′ K′

A′

L′ Q′

Z′T⟨L⟩H

//

R⟨Φ⟩F S⟨Ψ⟩G

C′ Z′T⟨L⟩H

//

R⟨Φ Ψ⟩F

C′

. The cell R ⟨Φ⟩F S ⟨Ψ⟩G is dened by the module morphism R ⟨Φ⟩F P′ ⟨S ⟨Ψ⟩G⟩Q′ and thecell R ⟨Φ Ψ⟩F is dened by the module morphism R ⟨Φ P ⟨Ψ⟩Q⟩F. But

R ⟨Φ P ⟨Ψ⟩Q⟩F = R ⟨Φ⟩F R ⟨P ⟨Ψ⟩Q⟩F

= R ⟨Φ⟩F [R P] ⟨Ψ⟩ [Q F]= R ⟨Φ⟩F [P′ S] ⟨Ψ⟩ [G Q′]= R ⟨Φ⟩F P′ ⟨S ⟨Ψ⟩G⟩Q′

.

1.3. Cell morphisms

Note. In the following, the left and right components of a cell Φ ∶ M N are denoted byΦ0 ∶ M0 → N0 and Φ1 ∶ M1 → N1 respectively (see Remark 1.2.2(1)). This is a commonpractice when discussing cell morphisms.

Denition 1.3.1. Given a parallel pair of cells Φ; Ψ ∶ M N , a morphism from Φ to Ψ,written τ ∶ Φ→ Ψ ∶M N , is dened by a pair of natural transformations

τ0 ∶ Φ0 → Ψ0 τ1 ∶ Φ1 → Ψ1

such that the square

x ⋅Φ0

x ⋅ τ0

m ⋅Φ // Φ1 ⋅ aτ1 ⋅ a

x ⋅Ψ0 m ⋅Ψ// Ψ1 ⋅ a

commutes for everyM-arrow m ∶ x a.

34


Remark 1.3.2.(1) The natural transformations τ0 and τ1 are called the left and right components of a cell

morphism τ .(2) The identity morphism of a cell Φ ∶ M N is given by the pair of identity natural

transformations; that is,1Φ ∶= (1Φ0 ,1Φ1)

.(3) The composition of two cell morphisms τ ∶ Φ → Ψ ∶M N and σ ∶ Ψ → Ω ∶M N is

given componentwise; that is

τ σ ∶= (τ0 σ0, τ1 σ1)

.(4) Given a pair of modules M and N , all cells M N and morphisms among them dene

the cell category [M ∶ N ] with the identities and the composition dened above.(5) IfM is small and N is locally small, then the category [M ∶ N ] is locally small.

Denition 1.3.3. A cell morphism τ ∶ Φ→ Υ ∶M N is called iso if it is an isomorphism inthe category [M ∶ N ].

Proposition 1.3.4. A cell morphism τ ∶ Φ → Υ ∶M N is iso if and only if its componentsτ0 and τ1 are natural isomorphisms.


Denition 1.3.5. The hom of a natural transformation τ ∶ F→ G ∶C→ B is the cell morphism⟨τ⟩ ∶ ⟨F⟩→ ⟨G⟩ ∶ ⟨C⟩ ⟨B⟩, given by the pair (τ, τ).

Theorem 1.3.6. Given a pair of categories C and B, the assignment of the hom dened inDenition 1.2.27 and Denition 1.3.5 embeds the functor category [C,B] in the cell category[⟨C⟩ ∶ ⟨B⟩]. Specically, the assignment H ↦ ⟨H⟩ forms a faithful functor [C,B]→ [⟨C⟩ ∶ ⟨B⟩],injective on objects.

Proof. The verication of the functoriality is straightforward. The faithfulness and the injec-tivity on objects are evident.

Denition 1.3.7. Given a module J and a cell morphisms τ ∶ Φ → Ψ ∶ M N , the cellmorphism

⟨J , τ⟩ ∶ ⟨J ,Φ⟩→ ⟨J ,Ψ⟩ ∶ ⟨J ,M⟩ ⟨J ,N ⟩, postcomposition with τ , is dened by the pair of postcomposition natural transformations

[J0, τ0] ∶ [J0,Φ0]→ [J0,Ψ0] [J1, τ1] ∶ [J1,Φ1]→ [J1,Ψ1]

(see Preliminary 9).

Remark 1.3.8.(1) Given a cell Θ ∶ J M, the commutativity of

Θ0 ⋅ [J0,Φ0]Θ0 ⋅ [J0,τ0]

Θ ⋅ ⟨J ,Φ⟩ // [J1,Φ1] ⋅Θ1

[J1,τ1] ⋅Θ1

Θ0 ⋅ [J0,Ψ0]Θ ⋅ ⟨J ,Ψ⟩

// [J1,Ψ1] ⋅Θ1

35


, i.e.

Θ0 Φ0

Θ0 τ0

Θ Φ // Φ1 Θ1

τ1 Θ1

Θ0 Ψ0 Θ Ψ// Ψ1 Θ1

follows from the commutativity of

e ⋅Θ0 ⋅Φ0

e ⋅Θ0 ⋅ τ0

m ⋅Θ ⋅Φ // Φ1 ⋅Θ1 ⋅ dτ1 ⋅Θ1 ⋅ d

e ⋅Θ0 ⋅Ψ0 m ⋅Θ ⋅Ψ// Ψ1 ⋅Θ1 ⋅ d

for each J -arrow m ∶ e d.(2) The assignment τ ↦ ⟨J , τ⟩ denes the functor

⟨J ,−⟩ ∶ [M ∶ N ]→ [⟨J ,M⟩ ∶ ⟨J ,N ⟩]

; indeed, by the denition of the cell morphism ⟨J , τ⟩, the functoriality of ⟨J ,−⟩ is reducedto that of [Ji,−] ∶ [Mi,Ni]→ [[Ji,Mi] , [Ji,Ni]] for i = 0,1.

36

2. Slicing and Action

2.1. Slicing of modules

Notation 2.1.1. Let X and A be categories. The right exponential transposition

[X∼ ×A,Set]⊣ // [A, [X∼,Set]]⊢

oo

(see Preliminary 11) is denoted by

[X ∶A] // [A, [X ∶]]

oo

; the right exponential transpose of a moduleM ∶XA is a covariant functor

[M] ∶A→ [X ∶]

and the right exponential transpose of a module morphism Φ ∶M→ N ∶XA is a naturaltransformation

[Φ] ∶ [M]→ [N] ∶A→ [X ∶]. The value of M at a ∈ A is written Ma, ⟨M⟩ (a), or just ⟨M⟩a. Similarly, thecomponent of Φ at a ∈ ∥A∥ is written Φa or ⟨Φ⟩a.

The left exponential transposition

[X∼ ×A,Set]⊢ // [X∼, [A,Set]]⊣

oo

(see Preliminary 11) is denoted by

[X ∶A] // [X∼, [∶A]]

oo

or

[X ∶A]∼ // [X, [∶A]∼]

oo

; the left exponential transpose of a moduleM ∶XA is a contravariant functor

[M] ∶X→ [∶A]∼

and the left exponential transpose of a module morphism Φ ∶M → N ∶X A is a naturaltransformation

[Φ] ∶ [N ]→ [M] ∶X→ [∶A]∼

. The value of M at x ∈ X is written xM, (x) ⟨M⟩, or just x ⟨M⟩. Similarly, thecomponent of Φ at x ∈ ∥X∥ is written xΦ or x ⟨Φ⟩.

37


Remark 2.1.2.(1) Given a moduleM ∶XA,

the right exponential transpose [M] ∶ A → [X ∶] sends each object a ∈ ∥A∥ to theright slice ofM at a, i.e. the right module

⟨M⟩a ∶X ∗

given byx ⟨⟨M⟩a⟩ = x ⟨M⟩a

for each x ∈X, and sends each A-arrow f ∶ s→ t to the right module morphism

⟨M⟩ f ∶ ⟨M⟩ s→ ⟨M⟩ t ∶X ∗

which maps eachM-arrow m ∶ x s to theM-arrow m f ∶ x t as indicated in

xm //

m ⋅ ⟨M⟩f %%

s

ft

. the left exponential transpose [M] ∶X → [∶A]∼ sends each object x ∈ ∥X∥ to the left

slice ofM at x, i.e. the left module

x ⟨M⟩ ∶ ∗A

given by⟨x ⟨M⟩⟩a = x ⟨M⟩a

for each a ∈A, and sends each X-arrow f ∶ s→ t to the left module morphism

f ⟨M⟩ ∶ t ⟨M⟩→ s ⟨M⟩ ∶ ∗A

which maps eachM-arrow m ∶ t a to theM-arrow f m ∶ s a as indicated in

s

f

f⟨M⟩ ⋅m

%%t m

// a

.(2) Given a module morphism Φ ∶M→ N ∶XA,

the component of the right exponential transpose [Φ] ∶ [M]→ [N] ∶A→ [X ∶] ata ∈ ∥A∥ gives the right slice of Φ at a, i.e. the right module morphism

⟨Φ⟩a ∶ ⟨M⟩a→ ⟨N ⟩a ∶X ∗

given byx ⟨⟨Φ⟩a⟩ = x ⟨Φ⟩a

for each x ∈ ∥X∥. the component of the left exponential transpose [Φ] ∶ [N ] → [M] ∶X → [∶A]∼ at

x ∈ ∥X∥ gives the left slice of Φ at x, i.e. the left module morphism

x ⟨Φ⟩ ∶ x ⟨M⟩→ x ⟨N ⟩ ∶ ∗A

given by⟨x ⟨Φ⟩⟩a = x ⟨Φ⟩a

for each a ∈ ∥A∥.

38


(3) The evaluation at a ∈ A of the right exponential transpose M of a module M ∶ X Ais identied with the composition

XM //A ∗aoo

(cf. Example 1.1.23(5)). Dually, the evaluation at x ∈ X of the left exponential transposeM is identied with the composition

∗ x //XM //A

.(4) For any moduleM,

[M]∼ = [M∼]; that is, the opposite of the right exponential transpose of M is the left exponentialtranspose of the opposite moduleM∼.

(5) The diagonal functors in Example 1.1.23(7) and the exponential transpositions form thefollowing commutative diagrams (cf. Preliminary 18):

[X ∶][X∶∆E]ww

[∆E,[X∶]]((

[∶A]∼[∆E∶A]ww

[∆E,[∶A]∼]((

[X ∶ E] // [E, [X ∶]]

oo [E ∶A]∼ // [E, [∶A]∼]

oo

[X ∶] = [X∼,Set][X∶∆E]ww

[X∼,[∶∆E]]((

[∶A] = [A,Set][∆E∶A]ww

[A,[∆E∶]]((

[X ∶ E] // [X∼, [∶ E]]

oo [E ∶A] // [A, [E ∶]]

oo

Proposition 2.1.3. For a module morphism Φ ∶M → N ∶ X A, the following conditionsare equivalent:(1) Φ is a module isomorphism;(2) the right exponential transpose [Φ] ∶ [M] → [N] ∶ A → [X ∶] is a natural isomor-

phism;(3) each right slice ⟨Φ⟩a ∶ ⟨M⟩a→ ⟨N ⟩a ∶X ∗ is a right module isomorphism;(4) the left exponential transpose [Φ] ∶ [N ]→ [M] ∶X→ [∶A]∼ is a natural isomorphism;(5) each left slice x ⟨Φ⟩ ∶ x ⟨M⟩→ x ⟨N ⟩ ∶ ∗A is a left module isomorphism.

Proof. Since the right and left exponential transpositions give isomorphisms [A, [X ∶]] ≅ [X ∶A] ≅[X, [∶A]∼], the equivalence (1) ⇔ (2) ⇔ (4) holds. Since a natural transformation is iso ieach component is an isomorphism, the equivalences (2)⇔ (3) and (4)⇔ (5) hold.

Remark 2.1.4. Proposition 2.1.3 also follows from Proposition 1.1.6 and Proposition 1.1.12, not-ing that x ⟨⟨Φ⟩a⟩ = x ⟨Φ⟩a = ⟨x ⟨Φ⟩⟩a.

Proposition 2.1.5. Let Φ ∶M→ N ∶XA be a module morphism. If B is an essentially wide subcategory of A, then Φ is an isomorphism if and only if its

right slice ⟨Φ⟩b ∶ ⟨M⟩b→ ⟨N ⟩b ∶X ∗ at each b ∈ ∥B∥ is a right module isomorphism. If Y is an essentially wide subcategory of X, then Φ is an isomorphism if and only if its

left slice y ⟨Φ⟩ ∶ y ⟨M⟩→ y ⟨N ⟩ ∶ ∗A at each y ∈ ∥Y∥ is a left module isomorphism.

39


Proof. By Proposition 2.1.3, Φ is an isomorphism i its right exponential transpose Φ is anatural isomorphism, and by the lemma in Preliminary 19, this is the case i the restriction ofΦ to B is a natural isomorphism; that is, i the component of Φ at each b ∈ ∥B∥, i.e. theright slice of Φ at each b ∈ ∥B∥, is an isomorphism.

Proposition 2.1.6.

Given a module (or module morphism) M and a functor (or natural transformation) K asin

XM //A E

Koo

, the right exponential transpose of the composite ⟨M⟩K is given by the composition

[X ∶] AMoo E

Koo

; that is,[⟨⟨M⟩K⟩] = [M] K

. Given a module (or module morphism) M and a functor (or natural transformation) K as

in

EK //X

M //A

, the left exponential transpose of the composite K ⟨M⟩ is given by the composition

EK //X

M // [∶A]∼

; that is,[⟨K ⟨M⟩⟩] =K [M]

.

Proof. For any e ∈ E,

[⟨⟨M⟩K⟩] ⋅ e = ⟨⟨M⟩K⟩ e = ⟨M⟩ (K ⋅ e) = [M] ⋅ (K ⋅ e) = [[M] K] ⋅ e

.

Proposition 2.1.7.

Given a functor K and a module M as in

EK //X

M //A

, the right exponential transpose of the composite module K ⟨M⟩ is given by the composition

[E ∶] [X ∶][K∶]oo AMoo

; that is,[⟨K ⟨M⟩⟩] = [K ∶] [M]

.

40


Given a functor K and a module M as in

XM //A E

Koo

, the left exponential transpose of the composite module ⟨M⟩K is given by the composition

XM // [∶A]∼ [∶K] // [∶ E]∼

; that is,[⟨⟨M⟩K⟩] = [M] [∶ K]

.

Proof. For any a ∈A,

[⟨K ⟨M⟩⟩] ⋅a = ⟨K ⟨M⟩⟩a = K ⟨⟨M⟩a⟩ = [K ∶] ⋅ ⟨⟨M⟩a⟩ = [K ∶] ⋅ ([M] ⋅ a) = [[K ∶] [M]] ⋅a

.

Denition 2.1.8. Given a cell

X

P

M //A

Q

Y N//

Φ

B

the right slice of Φ at a ∈ ∥A∥ is the right conical cell

X

P

⟨M⟩a // ∗Q ⋅ a

Y N//

⟨Φ⟩a

B

dened by the right slice of the module morphism Φ ∶M → P ⟨N ⟩Q ∶ X A at a, i.e. bythe right module

⟨Φ⟩a ∶ ⟨M⟩a→ ⟨P ⟨N ⟩Q⟩a ∶X ∗.

the left slice of Φ at x ∈ ∥X∥ is the left conical cell

∗x ⋅ P

x⟨M⟩ //A

Q

Y N//

x⟨Φ⟩

B

dened by the left slice of the module morphism Φ ∶M→ P ⟨N ⟩Q ∶XA at x, i.e. by theleft module

x ⟨Φ⟩ ∶ x ⟨M⟩→ x ⟨P ⟨N ⟩Q⟩ ∶ ∗A

.

Proposition 2.1.9. A cell is fully faithful if and only if each right (resp. left) slice is fullyfaithful.

Proof. Immediate from Proposition 2.1.3.

41


2.2. Action of modules

Denition 2.2.1. Let E be a category andM ∶XA be a module. The right action ofM on the functor category [E,A] is the covariant functor

[ME] ∶ [E,A]→ [X ∶ E]

dened by[ME] ⋅K = ⟨M⟩K

for K ∈ [E,A]. The left action ofM on the functor category [E,X] is the contravariant functor

[EM] ∶ [E,X]→ [E ∶A]∼

dened byK ⋅ [EM] =K ⟨M⟩

for K ∈ [E,X].

Remark 2.2.2.(1) A moduleM ∶XA acts on a functor K ∶ E→A by the composition

XM //A E

Koo

and yields a module⟨M⟩K ∶X E

such thatx ⟨⟨M⟩K⟩ e = x ⟨M⟩ (K ⋅ e)

for x ∈X and e ∈ E. Dually,M acts on a functor K ∶ E→X by the composition

EK //X

M //A

and yields a moduleK ⟨M⟩ ∶ EA

such thate ⟨K ⟨M⟩⟩a = (e ⋅K) ⟨M⟩a

for e ∈ E and a ∈A.(2) A module M ∶ X A acts on a natural transformation τ ∶ S → T ∶ E → A by the

composition

XM //A E

Suu

T

ii τ

and yields a module morphism

⟨M⟩ τ ∶ ⟨M⟩S→ ⟨M⟩T ∶X E

which maps each ⟨M⟩S-arrow m ∶ x e to the ⟨M⟩T-arrow m τe ∶ x e as indicated in

xm //

m ⋅ ⟨M⟩τ &&

S ⋅ eτe

T ⋅ e

42


. Dually,M acts on a natural transformation τ ∶ S→ T ∶ E→X by the composition

ES

))

T

55τ XM //A

and yields a module morphism

τ ⟨M⟩ ∶ T ⟨M⟩→ S ⟨M⟩ ∶ EA

which maps each T ⟨M⟩-arrow m ∶ e a to the S ⟨M⟩-arrow τe m ∶ e a as indicated in

e ⋅ Sτe

τ⟨M⟩ ⋅m

&&e ⋅T m

// a

.(3) By Remark 2.1.2(3), the right exponential transpose

[M] ∶A→ [X ∶]

of a moduleM ∶XA is identied with the right action

[M∗] ∶ [∗,A]→ [X ∶ ∗]

ofM on the functor category [∗,A]. Dually, the left exponential transpose

[M] ∶X→ [∶A]∼

is identied with the left action

[∗M] ∶ [∗,X]→ [∗ ∶A]∼

ofM on the functor category [∗,X].(4) For any category E and any moduleM ∶XA,

[ME]∼ ≅ [E∼M∼]

; that is, the opposite of the right action ofM ∶ X A on [E,A] is (identied with) theleft action of the opposite moduleM∼ ∶A∼ X∼ on [E∼,A∼].

Proposition 2.2.3. Given a category E and a module M ∶XA, the diagrams

[E,A]ME

yy

[E,M]&&

[E,X]EMyy

[E,M]&&

[X ∶ E] // [E, [X ∶]]

oo [E ∶A]∼ // [E, [∶A]∼]

oo

commute.


43


Proposition 2.2.4. Given a category E and a module M ∶XA, the diagrams

A[∆E,A] //

M

[E,A]ME

[E,X]EM

X

M

[∆E,X]oo

[X ∶][X∶∆E]

// [X ∶ E] [E ∶A]∼ [∶A]∼[∆E∶A]oo

commute.

Proof. The diagrams

A[∆E,A] //

M

[E,A] 1 //

[E,M]

[E,A]ME

[X ∶][∆E,[X∶]]

// [E, [X ∶]] // [X ∶ E]

[E,X]EM

[E,X]1oo

[E,M]

X

M

[∆E,X]oo

[E ∶A]∼ [E, [∶A]∼]oo [∶A]∼

[∆E,[∶A]∼]oo

commute by Proposition 2.2.3, and together with the commutative diagrams in Remark 2.1.2(5),yield the desired commutative diagrams.

2.3. Yoneda functors and representations

Denition 2.3.1.

The right Yoneda functor for a category X is the functor

[X] ∶X→ [X ∶]

given by the right exponential transpose of the hom ⟨X⟩ ∶XX; in short,

[X] ∶= [⟨X⟩]

. The left Yoneda functor for a category A is the functor

[A] ∶A→ [∶A]∼

given by the left exponential transpose of the hom ⟨A⟩ ∶AA; in short,

[A] ∶= [⟨A⟩]

.

Remark 2.3.2.

44


The right Yoneda functor X sends each object x ∈ ∥X∥ to the right module

⟨X⟩x ∶X ∗

, called the representable right module of x, and sends each X-arrow f ∶ s → t to the rightmodule morphism

⟨X⟩ f ∶ ⟨X⟩ s→ ⟨X⟩ t ∶X ∗which maps each X-arrow h ∶ x→ s to the X-arrow h f ∶ x→ t as indicated in

xh //

h ⋅ ⟨X⟩f %%

s

ft

(cf. Remark 2.1.2(1)). The left Yoneda functor A sends each object a ∈ ∥A∥ to the left module

a ⟨A⟩ ∶ ∗A

, called the representable left module of a, and sends each A-arrow f ∶ s → t to the leftmodule morphism

f ⟨A⟩ ∶ t ⟨A⟩→ s ⟨A⟩ ∶ ∗A

which maps each A-arrow h ∶ t→ a to the A-arrow f h ∶ s→ a as indicated in

s

f

f⟨A⟩ ⋅ h

%%t

h// a

(cf. Remark 2.1.2(1)).

Denition 2.3.3.

A representation of a right moduleM ∶X ∗ is a pair (r,Υ) consisting of an object r ∈ ∥X∥and a right module isomorphism Υ ∶M ≅ ⟨X⟩ r.

A representation of a left moduleM ∶ ∗A is a pair (r,Υ) consisting of an object r ∈ ∥A∥and a left module isomorphism Υ ∶M ≅ r ⟨A⟩.

Remark 2.3.4.(1)

A right moduleM ∶X ∗ is called representable if it has a representation; that is, if itis isomorphic to the representable right module ⟨X⟩ r for some object r ∈ ∥X∥, whichis said to representM.

A left moduleM ∶ ∗ A is called representable if it has a representation; that is, if itis isomorphic to the representable left module r ⟨A⟩ for some object r ∈ ∥A∥, which issaid to representM.

(2) A representation (r,Υ) of a right module M ∶ X ∗ is expressed by a fully faithful

conical cell:

X

1

M // ∗r

X⟨X⟩

//

Υ

X

.

45


A representation (r,Υ) of a left module M ∶ ∗ A is expressed by a fully faithfulconical cell

∗r

M //A

1

A⟨A⟩

//

Υ

A

.

Denition 2.3.5. Let X and A be categories. The right general Yoneda functor for the functor category [A,X] is the functor

[XA] ∶ [A,X]→ [X ∶A]

given by the right action of the hom ⟨X⟩ ∶XX on the functor category [A,X]; in short,

[XA] ∶= [⟨X⟩A]

. The left general Yoneda functor for the functor category [X,A] is the functor

[XA] ∶ [X,A]→ [X ∶A]∼

given by the left action of the hom ⟨A⟩ ∶AA on the functor category [X,A]; in short,

[XA] ∶= [X⟨A⟩]

.

Remark 2.3.6.(1)

The right general Yoneda functor XA sends each functor G ∶A→X to the module

⟨X⟩G ∶XA

, called the corepresentable module of G, and sends each natural transformation τ ∶ S→T ∶A→X to the module morphism

⟨X⟩ τ ∶ ⟨X⟩S→ ⟨X⟩T ∶XA

which maps each ⟨X⟩S-arrow h ∶ x a to the ⟨X⟩T-arrow h τa ∶ x a as indicated in

xh //

h ⋅ ⟨X⟩τ &&

S ⋅ aτa

T ⋅ a

(cf. Remark 2.2.2(2)). The left general Yoneda functor XA sends each functor F ∶X→A to the module

F ⟨A⟩ ∶XA

, called the representable module of F, and sends each natural transformation τ ∶ S →T ∶X→A to the module morphism

τ ⟨A⟩ ∶ T ⟨A⟩→ S ⟨A⟩ ∶XA

46


which maps each T ⟨A⟩-arrow h ∶ x a to the S ⟨A⟩-arrow τx h ∶ x a as indicated in

x ⋅ Sτx

τ⟨A⟩ ⋅ h

&&x ⋅T

h// a

(cf. Remark 2.2.2(2)).(2)

The right Yoneda functor[X] ∶X→ [X ∶]

for a category X is identied with the right general Yoneda functor

[X∗] ∶ [∗,X]→ [X ∶ ∗]

for the functor category [∗,X]. The left Yoneda functor

[A] ∶A→ [∶A]∼

for a category A is identied with the left general Yoneda functor

[∗A] ∶ [∗,A]→ [∗ ∶A]∼

for the functor category [∗,A].

Proposition 2.3.7. Given categories E, X, and A, the diagrams

X[∆E,X] //

X

[E,X]XE

[E,A]EA

A

A

[∆E,A]oo

[X ∶][X∶∆E]

// [X ∶ E] [E ∶A]∼ [∶A]∼[∆E∶A]oo

commute.

Proof. This is a special case of Proposition 2.2.4 where M is given by the hom of X (resp.A).

Denition 2.3.8. LetM ∶XA be a module. A corepresentation of M is a pair (R,Υ) consisting of a functor R ∶ A → X and a module

isomorphism Υ ∶M ≅ ⟨X⟩R. A representation of M is a pair (R,Υ) consisting of a functor R ∶ X → A and a module

isomorphism Υ ∶M ≅ R ⟨A⟩.

Remark 2.3.9.(1) A moduleM ∶XA is called

corepresentable if it has a corepresentation; that is, if it is isomorphic to the corep-resentable module ⟨X⟩R for some functor R ∶ A → X, which is said to corepresentM.

representable if it has a representation; that is, if it is isomorphic to the representablemodule R ⟨A⟩ for some functor R ∶X→A, which is said to representM.

(2)

47


A representation of a right moduleM ∶X ∗ is identied with a corepresentation of atwo-sided moduleM ∶XA where A is the terminal category.

A representation of a left module M ∶ ∗ A is identied with a representation of atwo-sided moduleM ∶XA where X is the terminal category.

(3) A corepresentation (R,Υ) of a moduleM ∶XA is expressed by a fully faithful cell

X

1

M //A

R

X⟨X⟩

//

Υ

X

. A representation (R,Υ) of a moduleM ∶XA is expressed by a fully faithful cell

X

R

M //A

1

A⟨A⟩

//

Υ

A

.(4) Example 2.3.11 shows that not all modules are representable.

Proposition 2.3.10. Let M ∶XA be a module. A functor R ∶ A → X and a module morphism Υ ∶M → ⟨X⟩R ∶ X A form a corepresen-

tation ofM if and only if for every a ∈ ∥A∥ the object R ⋅ a and the right module morphism⟨Υ⟩a ∶ ⟨M⟩a→ ⟨X⟩ (R ⋅ a) ∶X ∗ form a representation of the right module ⟨M⟩a ∶X ∗.

A functor R ∶X→A and a module morphism Υ ∶M→ R ⟨A⟩ ∶XA form a representationof M if and only if for every x ∈ ∥X∥ the object x ⋅ R and the left module morphism x ⟨Υ⟩ ∶x ⟨M⟩→ (x ⋅ R) ⟨A⟩ ∶ ∗A form a representation of the left module x ⟨M⟩ ∶ ∗A.

Proof. By Proposition 2.1.3, Υ ∶M→ ⟨X⟩R is iso i

⟨Υ⟩a ∶ ⟨M⟩a→ ⟨⟨X⟩R⟩a = ⟨X⟩ (R ⋅ a)

is iso for every a ∈ ∥A∥.

Example 2.3.11.

(1) Let X and A be discrete categories. A correspondence R from ∥X∥ to ∥A∥, i.e. a subsetof ∥X∥ × ∥A∥, denes a module R ∶XA by

x ⟨R⟩a =⎧⎪⎪⎨⎪⎪⎩

∗ if (x,a) ∈R∅ otherwise

. R is representable (resp. corepresentable) if and only if R is a function ∥X∥→ ∥A∥ (resp.∥A∥ → ∥X∥); R is both corepresentable and representable if and only if R is a one-to-onecorrespondence.

(2) Let X and A be thin categories, i.e. preordered sets. A correspondence R from ∥X∥ to∥A∥ denes a module R ∶XA if and only if it satises the following conditions:a) (x,a) ∈R and a ≤ b implies that (x,b) ∈R,b) (x,a) ∈R and y ≤ x implies that (y,a) ∈R.When this is the case,

48


R is corepresentable if and only if for every a ∈ ∥A∥ the set x ∈ ∥X∥ ∣ (x,a) ∈R has amaximum element.

R is representable if and only if for every x ∈ ∥X∥ the set a ∈ ∥A∥ ∣ (x,a) ∈R has aminimum element.

If R is both corepresentable and representable, then a corepresenting functor G ∶ A → Xand a representing functor F ∶X→A form a Galois connection between X and A, with

x ≤ G ⋅ a⇔ (x,a) ∈R⇔ x ⋅ F ≤ a

.

49

3. Collages and Commas

3.1. Collages

Denition 3.1.1. A collage M from a category X to a category A, written M ∶ X A, isdened by a category ⌊M⌋, collage category, satisfying the following conditions:(1) the coproduct category X +A is a wide full subcategory of ⌊M⌋;(2) a ⟨⌊M⌋⟩x = ∅ if x ∈ ∥X∥ and a ∈ ∥A∥.

Remark 3.1.2.(1) The inclusion of the coproduct category X +A into the collage category ⌊M⌋ is denoted

byM ∶X +A→ ⌊M⌋

or by

XM0ÐÐ→ ⌊M⌋ M1←ÐÐA

.(2)

A collageM ∶X ∗ from a category X to the terminal category is called a right collageover X; the inclusion of X into the collage category ⌊M⌋ is denoted by

M ∶X→ ⌊M⌋

. A collageM ∶ ∗A from the terminal category to a category A is called a left collage

over A; the inclusion of A into the collage category ⌊M⌋ is denoted by

M ∶A→ ⌊M⌋

.(3) A collageM is called locally small if the collage category ⌊M⌋ is locally small.(4) Any collageM ∶XA denes the unique functor ⌊∆M⌋ ∶ ⌊M⌋→ 2 making the diagram

XM0 //

∆X

⌊M⌋⌊∆M⌋

AM1oo

∆A

∗0

// 2 ∗1

oo

commute. Conversely, any functor H ∶ C → 2 denes a unique collage M ∶ X A with⌊M⌋ =C and ⌊∆M⌋ = H.

Denition 3.1.3. Given a pair of collages M,N ∶ X A, a morphism Φ from M to N ,written Φ ∶ M → N ∶ X A, is dened by a functor ⌊Φ⌋ ∶ ⌊M⌋ → ⌊N ⌋, collage functor,satisfying the following equivalent conditions:(1) ⌊Φ⌋ is identity on X +A;

50


(2) the triangleX +A

Mxx

N&&

⌊M⌋⌊Φ⌋

// ⌊N ⌋

commutes;(3) the two triangles

⌊M⌋

⌊Φ⌋

X

M066

N0 ((

A

M1hh

N1vv⌊N ⌋commute;

(4) the triangle

⌊M⌋

⌊∆M⌋ $$

⌊Φ⌋ // ⌊N ⌋

⌊∆N ⌋2

commutes.

Remark 3.1.4. Given a pair of categories X and A, all locally small collages X A andmorphisms among them dene the category [X ↑A] with the obvious identities and the com-position. Indeed, [X ↑A] is fully embedded into the coslice category under X +A.

Denition 3.1.5. Given a pair of collages M ∶ X A and N ∶ Y B, and given a pair offunctors P ∶ X → Y and Q ∶ A → B, a collage cell Φ ∶ P Q ∶M N is dened by a functor⌊Φ⌋ ∶ ⌊M⌋→ ⌊N ⌋, collage functor, satisfying the following equivalent conditions:(1) ⌊Φ⌋ is identity on X +A;(2) the diagram

X

P

M0 // ⌊M⌋⌊Φ⌋

A

Q

M1oo

Y N0

// ⌊N ⌋ BN1

oo

commutes;(3) the triangle

⌊M⌋

⌊∆M⌋ $$

⌊Φ⌋ // ⌊N ⌋

⌊∆N ⌋2

commutes.

Remark 3.1.6.(1) All locally small collages and cells among them dene the category CLG with the obvious

identities and the composition. Indeed, there is an obvious isomorphism between CLG

and the slice category CAT/2.(2) There is an obvious forgetful functor ⌊−⌋ ∶ CLG → CAT, sending each collage M to the

collage category ⌊M⌋.

51


(3) Given a pair of categories X and A, there is a canonical embedding [X ↑A] CLG,identical on objects, dened by the arrow function Φ ↦ (Φ ∶ 1X 1A). The embedding isnot, in general, full.

Denition 3.1.7. Given a parallel pair of collage cells Φ ∶ M N and Ψ ∶ M N , amorphism from Φ to Ψ, written τ ∶ Φ → Ψ ∶M N , is dened by a natural transformation⌊τ⌋ ∶ ⌊Φ⌋→ ⌊Ψ⌋ ∶ ⌊M⌋→ ⌊N ⌋, collage natural transformation.

Remark 3.1.8. Given a pair of collages M and N , all collage cells M N and morphismsamong them dene the category [M ↑ N ] with the obvious identities and the composition.Indeed, [M ↑ N ] is fully embedded into the functor category [⌊M⌋ , ⌊N ⌋].Note. In what follows we show a one-to-one correspondence between modules and collages. InDenition 3.1.9 and Denition 3.1.13, a module and a collage corresponding to each other aregiven the same name.

Denition 3.1.9.

(1) Given a module M ∶ X A, the corresponding collage is dened by the collage category⌊M⌋ given in the following way:a) the objects of ⌊M⌋ consist of all objects of the coproduct category X +A;b) the arrows of ⌊M⌋ consist of all X-arrows, A-arrows andM-arrows:

x ⟨⌊M⌋⟩y = x ⟨X⟩y for x,y ∈ ∥X∥; a ⟨⌊M⌋⟩b = a ⟨A⟩b for a,b ∈ ∥A∥; x ⟨⌊M⌋⟩a = x ⟨M⟩a for x ∈ ∥X∥ and a ∈ ∥A∥; a ⟨⌊M⌋⟩x = ∅ for a ∈ ∥A∥ and x ∈ ∥X∥.

c) the composition law of ⌊M⌋ is those of X, A, andM (see Denition 1.1.13(2)).(2) Given a module morphism Φ ∶M → N ∶ X A, the corresponding collage morphism is

dened by the collage functor ⌊Φ⌋ ∶ ⌊M⌋→ ⌊N ⌋ given in the following way:a) ⌊Φ⌋ is the identity on X +A;b) x ⟨⌊Φ⌋⟩a = x ⟨Φ⟩a for x ∈ ∥X∥ and a ∈ ∥A∥.

(3) Given a module cell

XM //

P

A

Q

Y N//

Φ

B

, the corresponding collage cell

X

P

M0 // ⌊M⌋⌊Φ⌋

A

Q

M1oo

Y N0

// ⌊N ⌋ BN1

oo

is dened by the collage functor ⌊Φ⌋ ∶ ⌊M⌋→ ⌊N ⌋ given in the following way:a) ⌊Φ⌋ is identical to P +Q on X +Y;b) x ⟨⌊Φ⌋⟩a = x ⟨Φ⟩a for x ∈ ∥X∥ and a ∈ ∥A∥.

(4) Given a module cell morphism τ ∶ Φ→ Ψ ∶M N , the corresponding collage cell morphismis dened by the collage natural transformation ⌊τ⌋ ∶ ⌊Φ⌋→ ⌊Ψ⌋ ∶ ⌊M⌋→ ⌊N ⌋ given by: ⌊τ⌋x = [τ0]x for x ∈ ∥X∥; ⌊τ⌋a = [τ1]a for a ∈ ∥A∥.

Remark 3.1.10.

52


(1) For a module morphism Φ ∶M→ N , the functoriality of ⌊Φ⌋ ∶ ⌊M⌋→ ⌊N ⌋ follows from thenaturality of Φ (see Remark 1.1.14(3)), and for a module cell Φ ∶M N , the functorialityof ⌊Φ⌋ ∶ ⌊M⌋→ ⌊N ⌋ follows from Proposition 1.2.3.

(2) For a right module M ∶ X ∗, the corresponding right collage (see Remark 3.1.2(2)) isconstructed as above by identifyingM with the two-sided module from X to the terminalcategory. Dually for a left moduleM ∶ ∗A.

Proposition 3.1.11. The module-to-collage correspondence given in Denition 3.1.9 is func-torial and denes the following functors:

(1) [X ∶A] ♯Ð→ [X ↑A] for categories X and A;

(2) MOD♯Ð→CLG;

(3) [M ∶ N ] ♯Ð→ [M ↑ N ] for modules M and N .

Proof. The functoriality is easily veried.

Remark 3.1.12. The composition of the functor MOD♯Ð→ CLG and the forgetful functor

⌊−⌋ ∶CLG→CAT in Remark 3.1.6(2) yields the functor ⌊−⌋ ∶MOD→CAT.

Denition 3.1.13.

(1) Given a collageM ∶XA, the corresponding module is dened by the composition

XM0 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ A

M1oo

, where ⟨⌊M⌋⟩ is the hom of the collage category ofM.(2) Given a collage morphism Φ ∶M → N ∶ X A, the corresponding module morphism is

dened by the pasting composition

⌊M⌋⌊Φ⌋

⟨⌊M⌋⟩ // ⌊M⌋⌊Φ⌋

X

M0 44

N0**

A

M1jj

N1tt⌊N ⌋

⟨⌊N ⌋⟩//

⟨⌊Φ⌋⟩

⌊N ⌋

(see Remark 1.2.32), where ⟨⌊Φ⌋⟩ is the hom of the collage functor of Φ.(3) Given a collage cell

X

P

M0 // ⌊M⌋⌊Φ⌋

A

Q

M1oo

Y N0

// ⌊N ⌋ BN1

oo

, the corresponding module cell

XM //

P

A

Q

Y N//

Φ

B

is dened by the pasting composition

X

P

M0 // ⌊M⌋⌊Φ⌋

⟨⌊M⌋⟩ // ⌊M⌋⌊Φ⌋

A

Q

M1oo

Y N0

// ⌊N ⌋⟨⌊N ⌋⟩

//

⟨⌊Φ⌋⟩

⌊N ⌋ BN1

oo

, where ⟨⌊Φ⌋⟩ is the hom of the collage functor of Φ.

53


(4) Given a collage cell morphism τ ∶ Φ→ Ψ ∶M N , the corresponding module cell morphismis dened by the pair of natural transformations

τ0 ∶ Φ0 → Ψ0 τ1 ∶ Φ1 → Ψ1

, the restrictions of the collage natural transformation ⌊τ⌋ ∶ ⌊Φ⌋ → ⌊Ψ⌋ ∶ ⌊M⌋ → ⌊N ⌋ to thedomain and the codomain of the collageM.

Proposition 3.1.14. The collage-to-module correspondence given in Denition 3.1.13 is func-torial and denes the following functors:

(1) [X ↑A] Ð→ [X ∶A] for categories X and A;

(2) CLGÐ→MOD;

(3) [M ↑ N ] Ð→ [M ∶ N ] for modules M and N .


Theorem 3.1.15. The corresponding functors

[X ∶A]♯

// [X ↑A]oo

MOD♯

// CLGoo [M ∶ N ]

♯// [M ↑ N ]

oo

in Proposition 3.1.11 and Proposition 3.1.14 are isomorphisms inverse to each other.

Proof. Easily veried.

Remark 3.1.16.(1) As noted earlier, a module and a collage corresponding to each other are given the same

name and freely identied with each other.(2) A moduleM ∶XA is recovered from its collage by the composition

XM0 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ A

M1oo

; that is,M =M0 ⟨⌊M⌋⟩M1

. This identity yields a cell

X

M0

M //A

M1⌊M⌋

⟨⌊M⌋⟩//

IM

⌊M⌋

called the unit cell ofM.

Theorem 3.1.17. There is a canonical adjunction between the functor ⌊−⌋ ∶ MOD → CAT

(see Remark 3.1.12) and the embedding ⟨−⟩ ∶CAT→MOD (see Theorem1.2.30), with the unitgiven by the family of unit cells IM ∶M ⟨⌊M⌋⟩. Specically, for each module M ∶ X Aand each category E,(1) the adjunct of a cell

XM //

S

A

T

E⟨E⟩

//

Φ

E

is given by the functor F ∶ ⌊M⌋→ E dened in the following way:

54


a) F is identical to S +T on X +A;b) x ⟨F⟩a = x ⟨Φ⟩a for x ∈ ∥X∥ and a ∈ ∥A∥.

(2) the adjunct of a functor F ∶ ⌊M⌋ → E is given by the cell Φ ∶ M ⟨E⟩ dened by thecomposition

X

M0

M //A

M1⌊M⌋F

IM

⟨⌊M⌋⟩ // ⌊M⌋F

E⟨E⟩

//

⟨F⟩

E

of unit cell of M and the hom of F.

Proof. It is easily veried that the correspondences are inverse to each other. It remains toprove that the family of the unit cells satises the naturality condition. For this we need tosee that, given a cell

XM //

P

A

Q

Y N//

Φ

B

, the square

M IM //

Φ

⟨⌊M⌋⟩⟨⌊Φ⌋⟩

NIN

// ⟨⌊N ⌋⟩

commutes, i.e. the compositions

XM //

P

A

Q

X

M0

M //A

M1Y

N0

N //

Φ

B

N1

⌊M⌋ ⟨⌊M⌋⟩ //

⌊Φ⌋

IM

⌊M⌋⌊Φ⌋

⌊N ⌋⟨⌊N ⌋⟩

//

IN

⌊N ⌋ ⌊N ⌋⟨⌊N ⌋⟩

//

⟨⌊Φ⌋⟩

⌊N ⌋

yield the same cell. First note that P N0 =M0 ⌊Φ⌋ and N1 Q = ⌊Φ⌋ M1 by the constructionof the collage functor ⌊Φ⌋. Since IN and IM are dened by the identity module morphismsN N and M M, the cells Φ IN and IM ⟨⌊Φ⌋⟩ are given by the module morphisms ΦandM0 ⟨⌊Φ⌋⟩M1. But Φ =M0 ⟨⌊Φ⌋⟩M1 by Theorem3.1.15.

Remark 3.1.18. The functor ⌊−⌋ ∶ MOD → CAT is thus a left adjoint of the embedding⟨−⟩ ∶CAT→MOD.

3.2. Commas

Denition 3.2.1.

A right comma K over a category X, written K ∶ X ∗, consists of a category denotedby ⌈K⌉ and a right comma bration1 K ∶ ⌈K⌉ → X, i.e. a functor satisfying the following

1A comma bration is called a discrete bration in the literature.

55


condition: for every object k in ⌈K⌉ and every X-arrow g ∶ y → K (k), there is a unique⌈K⌉-arrow gk ∶ s→ k, the lift of g at k, such that K (gk) = g.

A left comma K over a category A, written K ∶ ∗A, consists of a category denoted by ⌈K⌉and a left comma bration K ∶ ⌈K⌉ → A, i.e. a functor satisfying the following condition:for every object k in ⌈K⌉ and every A-arrow f ∶ K (k) → b, there is a unique ⌈K⌉-arrowfk ∶ k→ t, the lift of f at k, such that K (fk) = f.

Remark 3.2.2.(1) A left comma over a category A is the same thing as a right comma over the opposite

category A∼.(2) The bre of a right comma K ∶ X ∗ at x ∈ ∥X∥, written as x ⟨K⟩, is the subcategory of

⌈K⌉ consisting of all objects k with K (k) = x and all arrows h with K (h) = 1x. The breof a left comma K ∶ ∗A at a ∈ ∥A∥ is dened dually and written as ⟨K⟩a.

(3) A right comma K ∶ X ∗ is called locally small if X is locally small and all its bres aresmall. Dually, a left comma K ∶ ∗ A is called locally small if A is locally small and allits bres are small.

Denition 3.2.3.

Given a pair of right commas K,L ∶X ∗, a morphism Φ from K to L, written Φ ∶ K→ L ∶X ∗, is dened by a functor ⌈Φ⌉ ∶ ⌈K⌉→ ⌈L⌉ making the triangle

⌈K⌉K

vv ⌈Φ⌉

X

⌈L⌉L

hh

commute. Given a pair of left commas K,L ∶ ∗ A, a morphism Φ from K to L, written Φ ∶ K → L ∶

∗A, is dened by a functor ⌈Φ⌉ ∶ ⌈K⌉→ ⌈L⌉ making the triangle

⌈K⌉K

((⌈Φ⌉

A

⌈L⌉ L

66

commute.

Remark 3.2.4.(1) Given a category X, all locally small right commas over X and morphisms among them

dene the category [X ↓] with the obvious identities and the composition. Indeed, [X ↓]is fully embedded into the slice category over X. Dually, given a category A, all locallysmall left commas over A and morphisms among them dene the category [↓A].

(2) The image of k ∈ ⌈K⌉ under the functor ⌈Φ⌉ is denoted by Φ (k).Proposition 3.2.5.

A right comma morphism Φ ∶ K → L ∶ X ∗ sends each ⌈K⌉-arrow h ∶ s → t to the lift ofK (h) at Φ (t) as indicated in

Φ (s) Φ(h)=K(h)Φ(t)// Φ (t)

K (s)K(h)

// K (t).

56


A left comma morphism Φ ∶ K → L ∶ ∗ A sends each ⌈K⌉-arrow h ∶ s → t to the lift ofK (h) at Φ (s) as indicated in

Φ (s) Φ(h)=K(h)Φ(s)// Φ (t)

K (s)K(h)

// K (t)

.

Proof. By the denition of a comma morphism,

K (h) = L (Φ (h))

; that is, Φ (h) is a lift of K (h) along L. The assertion thus follows from the uniqueness of thelift.

Remark 3.2.6. A right (resp. left) comma morphism Φ is thus determined by the objectfunction of the functor ⌈Φ⌉.

Denition 3.2.7. A [two-sided] comma K from a category X to a category A, written K ∶X A, is given by a category denoted by ⌈K⌉ and a comma bration K ∶ ⌈K⌉ →X ×A, i.e. aspan

XK0←Ð ⌈K⌉ K1Ð→A

, satisfying the following conditions:(1) for every object k in ⌈K⌉ and every X-arrow g ∶ y → K0 (k), there is a unique ⌈K⌉-arrow

gk ∶ s→ k, the lift of g at k, such that K0 (gk) = g and K1 (gk) = 1K1(k);(2) for every object k in ⌈K⌉ and every A-arrow f ∶ K1 (k) → b, there is a unique ⌈K⌉-arrow

fk ∶ k→ t, the lift of f at k, such that K1 (fk) = f and K0 (fk) = 1K0(k);

(3) for every ⌈K⌉-arrow h ∶ s→ t, the domain of the lift K0 (h)tequals the codomain of the lift

K1 (h)sand the triangle

s

K1(h)s

h

ww

K1 (s)K1(h)

t K0(h)t

oo K1 (t)

K0 (t) K0 (s)K0(h)oo

commutes.

Remark 3.2.8.(1) The bres of a comma K ∶XA are dened as follows:

a) the bre of K at a ∈ ∥A∥, written ⟨K⟩a, is the subcategory of ⌈K⌉ consisting of allobjects k with K1 (k) = a and all arrows h with K1 (h) = 1a.

b) the bre of K at x ∈ ∥X∥, written x ⟨K⟩, is the subcategory of ⌈K⌉ consisting of allobjects k with K0 (k) = x and all arrows h with K0 (h) = 1x.

c) the bre of K at (x,a) ∈ ∥X ×A∥, written x ⟨K⟩a, is the subcategory of ⌈K⌉ consistingof all objects k with K (k) = (x,a) and all arrows h with K (h) = 1(x,a).

(2) With the notion of bres, the conditions (1) and (2) in Denition 3.2.7 are restated asfollows:

57


a) for every a ∈ ∥A∥, K0 restricted to ⟨K⟩a is a right comma bration;b) for every x ∈ ∥X∥, K1 restricted to x ⟨K⟩ is a left comma bration.

(3) A comma K ∶ X A is called locally small if X and A are locally small and the brex ⟨K⟩a is small for every (x,a) ∈ ∥X ×A∥.

Denition 3.2.9. Given a pair of commas K,L ∶XA, a morphism Φ from K to L, writtenΦ ∶ K→ L ∶XA, is dened by a functor ⌈Φ⌉ ∶ ⌈K⌉→ ⌈L⌉ such that the triangle

⌈K⌉

K &&

⌈Φ⌉ // ⌈L⌉

LyyX ×A

commutes or, equivalently, the two triangles

⌈K⌉

⌈Φ⌉

K0

vvK1

((X A

⌈L⌉L0

hh

L1

66

commute.

Remark 3.2.10.(1) Given a pair of categories X and A, all locally small commas X A and morphisms

among them dene the category [X ↓A] with the obvious identities and the composition.Indeed, [X ↓A] is fully embedded into the slice category over X ×A.

(2) There are obvious isomorphisms

[X ↓] ≅ [X ↓ ∗] [↓A] ≅ [∗ ↓A]

, by which a right comma over X is identied with a two-sided comma from X to the terminal

category. a left comma over A is identied with a two-sided comma from the terminal category

to A.(3) The bre of a comma morphism Φ ∶ K→ L ∶XA at (x,a) ∈ ∥X ×A∥, written as

x ⟨Φ⟩a ∶ x ⟨K⟩a→ x ⟨L⟩a

, is the restriction of the functor ⌈Φ⌉ ∶ ⌈K⌉→ ⌈L⌉ to the bre of K at (x,a).(4) The image of k ∈ ⌈K⌉ under the functor ⌈Φ⌉ is denoted by Φ (k).

Proposition 3.2.11. A comma morphism Φ ∶ K → L ∶ X A sends each ⌈K⌉-arrow h ∶ s → tto the composite of the lift of K1 (h) at Φ (s) and the lift of K0 (h) at Φ (t) as indicated in

Φ (s)K1(h)Φ(s)

Φ(h)

ww

K1 (s)K1(h)

Φ (t) K0(h)Φ(t)

oo K1 (t)

K0 (t) K0 (s)K0(h)oo

.

58


Proof. By the denition of a comma morphism,

K0 (h) = L0 (Φ (h)) K1 (h) = L1 (Φ (h))

. The assertion thus follows from the condition (3) in Denition 3.2.7.

Remark 3.2.12. A comma morphism Φ is thus determined by the object function of the functor⌈Φ⌉.

Denition 3.2.13. Given a pair of commas K ∶ X A and L ∶ Y B, and given a pair offunctors P ∶ X → Y and Q ∶ A → B, a comma cell Φ ∶ P Q ∶ K L is dened by a functor⌈Φ⌉ ∶ ⌈K⌉→ ⌈L⌉ making the diagram

X

P

⌈K⌉⌈Φ⌉

K0oo K1 //A

Q

Y ⌈L⌉L0

ooL1

// B

commute.

Remark 3.2.14.(1) All locally small commas and cells among them dene the category COM with the obvious

identities and the composition.(2) There is an obvious forgetful functor ⌈−⌉ ∶ COM → CAT, sending each comma K to the

category ⌈K⌉.(3) Given a pair of categories X and A, there is a canonical embedding [X ↓A] COM,

identical on objects, dened by the arrow function Φ ↦ (Φ ∶ 1X 1A). The embedding isnot, in general, full.

(4) The bre of a comma cell Φ ∶ P Q ∶ K L at (x,a) ∈ ∥X ×A∥, written as

x ⟨Φ⟩a ∶ x ⟨K⟩a→ (x ⋅ P) ⟨L⟩ (Q ⋅ a)

, is the restriction of the functor ⌈Φ⌉ ∶ ⌈K⌉→ ⌈L⌉ to the bre of K at (x,a).Note. The following construction of a comma category is borrowed from [Ell06].

Denition 3.2.15. The comma category ⌈M↓⌉ of a module M ∶ X A is dened in thefollowing way:(1) the objects of ⌈M↓⌉ are all M-arrows m ∶ x a, to be precise, all triples (x,m,a) with

x ∈ ∥X∥, a ∈ ∥A∥, and m ∈ x ⟨M⟩a;(2) an arrow of ⌈M↓⌉ from (m ∶ x a) to (n ∶ y b) is a pair (g, f) consisting of an X-arrow

g ∶ x→ y and an A-arrow f ∶ a→ b making the square

xm //

g

a

f

y n// b

commute;(3) the composition law of ⌈M↓⌉ is that induced by the composition laws of X and A.

Remark 3.2.16.

59


(1) In the literature, a comma category is dened for a pair of functors XFÐ→ C

G←Ð A andwritten (F ↓ G). Note that

(F ↓ G) = ⌈⟨F ⟨C⟩G⟩↓⌉and for a moduleM ∶XA,

⌈M↓⌉ = (M0 ↓M1).

(2) The comma category ⌈M↓⌉ is identied with the full subcategory of the functor category[2, ⌊M⌋] consisting of all sections of the functor ⌊∆M⌋ ∶ ⌊M⌋→ 2 (see Remark 3.1.2(4)).

Denition 3.2.17.

(1) Given a module M ∶ X A, the comma M↓ ∶ X A is dened by the comma category⌈M↓⌉ and the pair of functorsM↓

0 ∶ ⌈M↓⌉→X andM↓1 ∶ ⌈M↓⌉→A induced by the functor

[0, ⌊M⌋] (evaluation at 0) and the functor [1, ⌊M⌋] (evaluation at 1) as indicated in thefollowing commutative diagram

X

M0

⌈M↓⌉M↓

0oo

M↓1 //A

M1

⌊M⌋ [2, ⌊M⌋][0,⌊M⌋]oo

[1,⌊M⌋]// ⌊M⌋

, where ⌈M↓⌉ [2, ⌊M⌋] denotes the inclusion given by the identication in Remark 3.2.16(2).(2) Given a module morphism Φ ∶M → N ∶ X A, the comma morphism Φ↓ ∶M↓ → N ↓ ∶

X A is dened by the functor ⌈Φ↓⌉ ∶ ⌈M↓⌉ → ⌈N ↓⌉ induced by the functor [2, ⌊Φ⌋](postcomposition with the collage functor ⌊Φ⌋ ∶ ⌊M⌋ → ⌊N ⌋ (see Denition 3.1.9(2))) asindicated in the following commutative diagram:

⌈M↓⌉ //

⌈Φ↓⌉

[2, ⌊M⌋][2,⌊Φ⌋]

⌈N ↓⌉ // [2, ⌊N ⌋]

(3) Given a module cell

XM //

P

A

Q

Y N//

Φ

B

, the comma cell

X

P

⌈M↓⌉⌈Φ↓⌉

M↓0oo

M↓1 //A

Q

Y ⌈N ↓⌉N ↓0

ooN ↓1

// B

is dened by the functor ⌈Φ↓⌉ ∶ ⌈M↓⌉ → ⌈N ↓⌉ induced by the functor [2, ⌊Φ⌋] (postcompo-sition with the collage functor ⌊Φ⌋ ∶ ⌊M⌋ → ⌊N ⌋ (see Denition 3.1.9(3))) as indicated inthe following commutative diagram:

⌈M↓⌉ //

⌈Φ↓⌉

[2, ⌊M⌋][2,⌊Φ⌋]

⌈N ↓⌉ // [2, ⌊N ⌋]

60


Remark 3.2.18. Given an arrowx

m //

g

a

f

y n// b

of the comma category ⌈M↓⌉, the decomposition diagram

xm //

1

a

fx

m f = g n //

g

b

1

y n// b

illustrates the way the commaM↓ ∶X A satises the condition (3) in Denition 3.2.7; notethat the upper square forms the lift of the A-arrow f at m and the lower square forms the liftof the X-arrow g at n.

Proposition 3.2.19. The module-to-comma correspondence given in Denition 3.2.17 is func-torial and denes the following functors:

(1) [X ∶A] ↓Ð→ [X ↓A] for categories X and A;

(2) MOD↓Ð→COM;


Remark 3.2.20. Given a category X, the functor [X ∶] ↓Ð→ [X ↓] is dened such that the diagram

[X ∶] ↓ //

≅

[X ↓]≅

[X ∶ ∗] ↓// [X ↓ ∗]

commutes, where ≅ denotes the canonical isomorphisms. The functor sends each rightmoduleM ∶X ∗ to the right commaM↓ ∶X ∗.

Given a category A, the functor [∶A] ↓Ð→ [↓A] is dened such that the diagram

[∶A] ↓ //

≅

[↓A]≅

[∗ ∶A] ↓// [∗ ↓A]

commutes, where ≅ denotes the canonical isomorphisms. The functor sends each left moduleM ∶ ∗A to the left commaM↓ ∶ ∗A.2

Denition 3.2.21.

(1) Given a comma K ∶XA, the module K↑ ∶XA is dened in the following way:

2The comma category ⌈M↓⌉ of a left moduleM is called the category of elements in the literature.

61


a) for a pair of objects x ∈ ∥X∥ and a ∈ ∥A∥, the set x ⟨K↑⟩a is dened by

x ⟨K↑⟩a = ∥x ⟨K⟩a∥

, the set of objects of the bre of K at (x,a).b) for an X-arrow g ∶ y → x and an object a ∈ ∥A∥, the function g ⟨K↑⟩a ∶ x ⟨K↑⟩a →

y ⟨K↑⟩a is dened by the assignment

k↦ dom (gk)

; that is, g ⟨K↑⟩a is dened such that it maps each k ∈ ∥x ⟨K⟩a∥ to the domain of thelift gk.

c) for an object x ∈ ∥X∥ and an A-arrow f ∶ a → b, the function x ⟨K↑⟩ f ∶ x ⟨K↑⟩a →x ⟨K↑⟩b is dened by the assignment

k↦ cod (fk)

; that is, x ⟨K↑⟩ f is dened such that it maps each k ∈ ∥x ⟨K⟩a∥ to the codomain of thelift fk.

(2) Given a comma morphism Φ ∶ K→ L ∶XA, the module morphism Φ↑ ∶ K↑ → L↑ ∶XAis dened by

Φ↑ = (∥x ⟨Φ⟩a∥ ∶ ∥x ⟨K⟩a∥→ ∥x ⟨L⟩a∥)(x,a)∈∥X×A∥

, where ∥x ⟨Φ⟩a∥ is the object function of the bre of Φ at (x,a) ∈ ∥X ×A∥.(3) Given a comma cell

X

P

⌈K⌉⌈Φ⌉

K0oo K1 //A

Q

Y ⌈L⌉L0

ooL1

// B

, the module cell

XK↑ //

P

A

Q

YL↑

//Φ↑

B

is dened by the module morphism Φ↑ ∶ K↑ → P ⟨L↑⟩Q ∶XA given by

Φ↑ = (∥x ⟨Φ⟩a∥ ∶ ∥x ⟨K⟩a∥→ ∥(x ⋅ P) ⟨L⟩ (Q ⋅ a)∥)(x,a)∈∥X×A∥

, where ∥x ⟨Φ⟩a∥ is the object function of the bre of Φ at (x,a) ∈ ∥X ×A∥.Proposition 3.2.22.

(1) K↑ dened in Denition 3.2.21(1) is indeed a module.(2) Φ↑ dened in Denition 3.2.21(2) is indeed a module morphism.

Proof.(1) For each object a ∈ ∥A∥ (resp. x ∈ ∥X∥), the functoriality of the slice ⟨K↑⟩a ∶ X∼ → Set

(resp. x ⟨K↑⟩ ∶A → Set) follows from the functoriality of K0 (resp. K1). By the bifunctorlemma (see [ML98] p37 Proposition 1), the proof is complete if we show that the square

x ⟨K↑⟩ax⟨K↑⟩f

g⟨K↑⟩a// y ⟨K↑⟩a

y⟨K↑⟩f

x ⟨K↑⟩bg⟨K↑⟩b

// y ⟨K↑⟩b

62


commutes for any X-arrow g ∶ y → x and any A-arrow f ∶ a → b. Given an objectk ∈ ∥x ⟨K↑⟩a∥, consider the diagram

k

fk

s

fs

gk

oo

gk fk

yy

a

f

t gt

oo b

x ygoo

, where s is domain of the lift gk and t is the codomain of the lift fk. x ⟨K↑⟩ f g ⟨K↑⟩bmaps k to the domain of gt, and g ⟨K↑⟩a y ⟨K↑⟩ f maps k to the codomain of fs. But theyare equal by the condition (3) in Denition 3.2.7.

(2) By [ML98] p38 Proposition 2, it suces to show that x ⟨Φ↑⟩a is natural in x for eacha ∈ ∥A∥ and natural in a for each x ∈ ∥X∥; that is the squares

x ⟨K↑⟩ag⟨K↑⟩a

x⟨Φ↑⟩a// x ⟨L↑⟩a

g⟨L↑⟩a

x ⟨K↑⟩ax⟨K↑⟩f

x⟨Φ↑⟩a// x ⟨L↑⟩a

x⟨L↑⟩f

y ⟨K↑⟩ay⟨Φ↑⟩a

// y ⟨L↑⟩a x ⟨K↑⟩bx⟨Φ↑⟩b

// x ⟨L↑⟩b

commute for any X-arrow g ∶ y → x and any A-arrow f ∶ a → b. Let k ∈ ∥x ⟨K⟩a∥. Thecomposite g ⟨K↑⟩a y ⟨Φ↑⟩a sends k to Φ (dom (gk)) and the composite x ⟨Φ↑⟩a g ⟨L↑⟩asends k to dom (gΦ(k)); similarly, the composite x ⟨K↑⟩ f x ⟨Φ↑⟩b sends k to Φ (cod (fk))and the composite x ⟨Φ↑⟩a x ⟨L↑⟩ f sends k to cod (fΦ(k)). But since Φ (gk) = gΦ(k) andΦ (fk) = fΦ(k) by Proposition 3.2.5, we have

dom (gΦ(k)) = dom (Φ (gk)) = Φ (dom (gk))

andcod (fΦ(k)) = cod (Φ (fk)) = Φ (cod (fk))

.

Proposition 3.2.23. The comma-to-module correspondence given in Denition 3.2.21 is func-torial and denes the following functors:

(1) [X ↓A] ↑Ð→ [X ∶A] for categories X and A;

(2) COM↑Ð→MOD.


Remark 3.2.24. Given a category X, the functor [X ↓] ↑Ð→ [X ∶] is dened such that the diagram

[X ↓] ↑ //

≅

[X ∶]≅

[X ↓ ∗] ↑// [X ∶ ∗]

commutes, where ≅ denotes the canonical isomorphisms. The functor sends each rightcomma K ∶X ∗ to the right module K↑ ∶X ∗.

63


Given a category A, the functor [↓A] ↑Ð→ [∶A] is dened such that the diagram

[↓A] ↑ //

≅

[∶A]≅

[∗ ↓A] ↑// [∗ ∶A]

commutes, where ≅ denotes the canonical isomorphisms. The functor sends each left commaK ∶ ∗A to the left module K↑ ∶ ∗A.

Theorem 3.2.25.

(1) For each module M ∶XA, there is a canonical isomorphism

εM ∶ ⟨M↓⟩↑ ≅M

.(2) For each comma K ∶XA, there is a canonical isomorphism

ηK ∶ K ≅ ⟨K↑⟩↓

.

Proof. By Denition 3.2.17 and Denition 3.2.21,(1) the module ⟨M↓⟩↑ is obtained fromM by changing each element m ∈ x ⟨M⟩a to the triple

(x,m,a).(2) the comma ⟨K↑⟩↓ is obtained from K by changing each object k ∈ ∥x ⟨K⟩a∥ to the triple

(x,k,a) and changing each ⌈K⌉-arrow h ∶ s→ t to the pair (K0 (h) ,K1 (h)).Hence εM is dened by the assignment (x,m,a) ↦ m and ηK is given by the functor ⌈ηK⌉consisting of the object function k↦ (x,k,a) and the arrow function h↦ (K0 (h) ,K1 (h)). Thebijectivity of the arrow function of ⌈ηK⌉ follows from the condition (3) in Denition 3.2.7.

Remark 3.2.26.(1) In Theorem10.2.7 and Theorem10.3.7 we will see that the corresponding functors

[X ↓A]↑

// [X ∶A]↓oo

COM↑

//MOD↓oo

in Proposition 3.2.19 and Proposition 3.2.23 form adjoint equivalences with the isomor-phisms in Theorem3.2.25.

(2) Given categories X and A, the functors

[X ↓A]↑

// [X ↑A]↓oo

COM↑

// CLG↓oo

are dened so that the diagrams

[X ↓A]↑

//

1

[X ↑A]↓oo

COM↑

//

1

CLG↓oo

[X ↓A]↑

//

1

OO

[X ∶A]↓oo

♯OO

COM↑

//

1

OO

MOD↓oo

♯

OO

commute, where and ♯ are the isomorphisms in Theorem3.1.15.

64


Denition 3.2.27. LetM ∶XA be a module. The right comma exponential transpose ofM is the functor

[M] ∶A→ [X ↓]


AM // [X ∶] ↓ // [X ↓]

of the right exponential transpose ofM and the functor in Remark 3.2.20. The left comma exponential transpose ofM is the functor

[M] ∶X→ [↓A]∼


XM // [∶A]∼ ↓ // [↓A]∼

of the left exponential transpose ofM and the functor in Remark 3.2.20.

Remark 3.2.28. The right comma exponential transpose of a moduleM ∶XA sends each object a ∈ ∥A∥

to the comma ⟨⟨M⟩a⟩↓ ∶X ∗ of the right module ⟨M⟩a. This comma is called the rightcomma ofM at a ∈ ∥A∥ and written as

⟨M↓a⟩ ∶= ⟨⟨M⟩a⟩↓

. Each right comma ⟨M↓a⟩ ∶X ∗ ofM consists of the comma category ⌈M↓a⌉ and theright comma bration [M↓a] ∶ ⌈M↓a⌉→X.

The left comma exponential transpose of a module M ∶ X A sends each object x ∈ ∥X∥to the comma ⟨x ⟨M⟩⟩↓ ∶ ∗ A of the left module x ⟨M⟩. This comma is called the leftcomma ofM at x ∈ ∥X∥ and written as

⟨x↓M⟩ ∶= ⟨x ⟨M⟩⟩↓

. Each left comma ⟨x↓M⟩ ∶ ∗ A of M consists of the comma category ⌈x↓M⌉ and theleft comma bration [x↓M] ∶ ⌈x↓M⌉→A.

Note. The following denition is a special case of Denition 3.2.27 where M is given by thehom of a category.

Denition 3.2.29. LetM ∶XA be a module. Given a category X, the functor

[X] ∶X→ [X ↓]

is dened by the composition

XX // [X ∶] ↓ // [X ↓]

of the right Yoneda functor for X and the functor in Remark 3.2.20.

65


Given a category A, the functor

[A] ∶A→ [↓A]∼


AA // [∶A]∼ ↓ // [↓A]∼

of the left Yoneda functor for A and the functor in Remark 3.2.20.

Remark 3.2.30. The functor X sends each object x ∈ ∥X∥ to the comma ⟨⟨X⟩x⟩↓ ∶ X ∗ of the right

module ⟨X⟩x. This comma is called the right comma of X at x ∈ ∥X∥ and written as

⟨X↓x⟩ ∶= ⟨⟨X⟩x⟩↓

. Each right comma ⟨X↓x⟩ ∶X ∗ of X consists of the slice category ⌈X↓x⌉ and the rightcomma bration [X↓x] ∶ ⌈X↓x⌉→X.

The functorA sends each object a ∈ ∥A∥ to the comma ⟨a ⟨A⟩⟩↓ ∶ ∗A of the left modulea ⟨A⟩. This comma is called the left comma of A at a ∈ ∥A∥ and written as

⟨a↓A⟩ ∶= ⟨a ⟨A⟩⟩↓

. Each left comma ⟨a↓A⟩ ∶ ∗ A of A consists of the coslice category ⌈a↓A⌉ and the leftcomma bration [a↓A] ∶ ⌈a↓A⌉→A.

66

4. Frames

4.1. Cylindrical frames

Denition 4.1.1. A [cylindrical] frame α of an endomoduleM ∶ E E is a family ofM-arrowsαe ∶ e e, one for each object e ∈ ∥E∥, that is natural in the sense that the square

eαe //

h

e

he′ αe′

// e′

commutes for every E-arrow h ∶ e → e′. The M-arrow αe is called the component of α at e.The set of frames ofM is denoted by ∏EM.

Remark 4.1.2.(1) If E is small, so is ∏EM.(2) If E is discrete, then ∏EM is nothing but the cartesian product ∏e∈∥E∥ e ⟨M⟩e.(3) The naturality of a frame α of an endomoduleM ∶ E E is also expressed in the extraordi-

nary1 form by regardingM as a left (resp. right) module over E∼×E (see Remark 1.1.10(3)). IfM is regarded as a left moduleM ∶ ∗ E∼ ×E, then the naturality of α is expressed

by the commutativity of the square

∗αe′

αe // (e,e)(e,h)

(e′,e′)(h,e′)

// (e,e′)

for each E-arrow h ∶ e→ e′ (see Remark 1.1.14(4)). IfM is regarded as a right moduleM ∶ E∼×E ∗, then the naturality of α is expressed

by the commutativity of the square

(e′,e)(h,e)

(e′,h) // (e′,e′)αe′(e,e) αe// ∗

for each E-arrow h ∶ e→ e′ (see Remark 1.1.14(4)).(4) A frame of an endomodule M ∶ E E is the same thing as a frame of the opposite

endomoduleM∼ ∶ E∼ E∼; that is,

∏E

M =∏E∼M∼

.1The term extraordinary (natural) is borrowed from [Kel05]. An alternative term extranatural seems morecommon in the other literature.

67

4. Frames

Example 4.1.3. Let E and C be categories.(1) A natural transformation α ∶ S→ T ∶ E→C is the same thing as a frame α of the composite

endomodule S ⟨C⟩T ∶ E E.(2)

An extraordinary natural transformation α from an object c ∈ ∥C∥ to a bifunctor K ∶E∼ × E → C is the same thing as a cylindrical frame α of the composite left modulec ⟨C⟩K ∶ ∗ E∼ ×E.

An extraordinary natural transformation α from a bifunctor K ∶ E∼ × E → C to anobject c ∈ ∥C∥ is the same thing as a cylindrical frame α of the composite right moduleK ⟨C⟩c ∶ E∼ ×E ∗.

Theorem 4.1.4. Let M ∶ E ×D E ×D be an endomodule. A family of M-arrows α(e,d) ∶(e,d) (e,d) indexed by the objects of E × D is a frame of M if and only if for eache ∈ ∥E∥, (α(e,d))d∈∥D∥ is a frame of the endomodule [e ×D] ⟨M⟩ [e ×D] ∶ D D (see Exam-

ple 1.1.23(6)) and for each d ∈ ∥D∥, (α(e,d))e∈∥E∥ is a frame of the endomodule [E × d] ⟨M⟩ [E × d] ∶E E.

Proof. See [ML98] p38 Proposition 2.

Denition 4.1.5. If Φ ∶M → N ∶ E E is a module morphism and α is a frame ofM, thentheir composite α Φ = Φ α is the frame of N dened by

[α Φ]e = αe ⋅ e ⟨Φ⟩e

for e ∈ ∥E∥. The function∏E

Φ ∶∏E

M→∏E

N

is dened byα ⋅∏

E

Φ = α Φ

.

Remark 4.1.6. The composite α Φ so dened does form a frame of N . Indeed, for eachE-arrow h ∶ e→ e′, the commutativity of the square

e[α Φ]e //

h

e

he′

[α Φ]e′// e′

, i.e.

eαe ⋅Φ //

h

e

he′

αe′ ⋅Φ// e′

, follows from the commutativity of the square

eαe //

h

e

he′ αe′

// e′

by the naturality of Φ.

68

4. Frames

Proposition 4.1.7. The assignment Φ↦∏E Φ is functorial and denes the left module

∏E

∶ ∗ [E ∶ E]

.


Denition 4.1.8. If H ∶ D → E is a functor and α is a frame of an endomodule M ∶ E E,then their composite H α = α H is the frame of the endomodule H ⟨M⟩H ∶DD dened by

[H α]d = α(H ⋅ d)

for d ∈ ∥D∥. The function∏H

M ∶∏E

M→∏D

H ⟨M⟩H


H

M = H α.

Remark 4.1.9. The composite H α so dened does form a frame of H ⟨M⟩H. Indeed, by thenaturality of α, the square

d

h

d ⋅Hα(H ⋅d) //

h ⋅H

H ⋅ dH ⋅ h

d

h

d′ d′ ⋅H α(H ⋅d′)// H ⋅ d′ d′

commutes for every D-arrow h ∶ d→ d′.

Proposition 4.1.10. Given functors CGÐ→D

HÐ→ E and a frame α of an endomodule M ∶ EE, the associative law

[G H] α = G [H α]holds.

Proof. For any c ∈ ∥C∥,

[[G H] α]c = α(H ⋅G ⋅ c) = [H α](G ⋅ c) = [G [H α]]c

.

Proposition 4.1.11. The function ∏HM is natural inM; that is, for every module morphismΦ ∶M→ N ∶ E E, the square

∏EM∏HM //

∏E Φ

∏D H ⟨M⟩H

∏D H⟨Φ⟩H

∏EN ∏HN//∏D H ⟨N ⟩H

commutes.

69

4. Frames

Proof. For any frame α ofM,

α ⋅∏E

Φ ⋅∏H

N = H [α Φ]

andα ⋅∏

H

M ⋅∏D

H ⟨Φ⟩H = [H α] H ⟨Φ⟩H

. Hence we need to verify that

H [α Φ] = [H α] H ⟨Φ⟩H

. But for any d ∈ ∥D∥,

[H [α Φ]]d = [α Φ](H ⋅ d)

= α(H ⋅ d) ⋅ (d ⋅H) ⟨Φ⟩ (H ⋅ d)= [H α]d ⋅ d ⟨H ⟨Φ⟩H⟩d= [[H α] H ⟨Φ⟩H]d

.

Proposition 4.1.12. The family of functions ∏HM ∶ ∏EM → ∏D H ⟨M⟩H, one for eachendomodule M ∶ E E, denes a left module cell

∗1

∏E // [E ∶ E][H∶H]

∗∏D

//

∏H

[D ∶D]

. Moreover, the assignment H↦∏H is contravariant functorial.

Proof. The rst assertion is immediate from Proposition 4.1.11. The functoriality of the as-signment is easily veried using Proposition 4.1.10.

4.2. Conical frames

Denition 4.2.1.

A [conical] frame α of a left moduleM ∶ ∗ E is a family ofM-arrows αe ∶ ∗ e, one foreach object e ∈ ∥E∥, that is natural in the sense that the triangle

e

h

∗

αe55

αe′ ((e′

commutes for every E-arrow h ∶ e→ e′. TheM-arrow αe is called the component of α at e.The set of frames ofM is denoted by ∏∗EM.

70

4. Frames

A [conical] frame α of a right module M ∶ E ∗ is a family of M-arrows αe ∶ e ∗, onefor each object e ∈ ∥E∥, that is natural in the sense that the triangle

e αe

))h

∗

e′αe′

66

commutes for every E-arrow h ∶ e→ e′. TheM-arrow αe is called the component of α at e.The set of frames ofM is denoted by ∏E∗M.

Remark 4.2.2.(1) If E is small, so is ∏∗EM (resp. ∏E∗M).(2) If E is discrete, then ∏∗EM (resp. ∏E∗M) is just the cartesian product ∏e∈∥E∥ ⟨M⟩e

(resp. ∏e∈∥E∥ e ⟨M⟩).(3)

A frame of a left moduleM ∶ ∗ E is the same thing as a frame of the opposite rightmoduleM∼ ∶ E∼ ∗; that is,

∏∗EM =∏

E∼∗M∼

. A frame of a right moduleM ∶ E ∗ is the same thing as a frame of the opposite left

moduleM∼ ∶ ∗ E∼; that is,

∏E∗M =∏

∗E∼M∼

.

Example 4.2.3. Given a functor K ∶ E→C and an object c ∈ ∥C∥, a cone α from c to K is the same thing as a frame α of the composite left module c ⟨C⟩K ∶

∗ E. a cone α from K to c is the same thing as a frame α of the composite right module K ⟨C⟩c ∶

E ∗.

Proposition 4.2.4.

A frame of a left module M ∶ ∗ E is the same thing as a frame of the endomodule∆E ⟨M⟩ ∶ E E (see Example 1.1.23(7)); that is,

∏∗EM =∏

E

∆E ⟨M⟩

. A frame of a right module M ∶ E ∗ is the same thing as a frame of the endomodule

⟨M⟩∆E ∶ E E (see Example 1.1.23(7)); that is,

∏E∗M =∏

E

⟨M⟩∆E

.

Proof. Consider a family of M-arrows αe ∶ ∗ e indexed by the objects e ∈ ∥E∥. Since thefunctor ∆E ∶ E ∗ sends each E-arrow h ∶ e→ e′ to the identity ∗→ ∗, the triangle

e

h

∗

αe55

αe′ ((e′

71

4. Frames

commutes i the square

e

h

∗h ⋅∆E

αe // e

he′ ∗ αe′

// e′

in ∆E ⟨M⟩ commutes. Hence α forms a frame ofM i it forms a frame of ∆E ⟨M⟩.Proposition 4.2.5.

A frame of a left module M ∶ ∗ E is the same thing as a cylindrical frame of the leftmodule ⟨M⟩ [∆E∼ ×E] ∶ ∗ E∼ ×E (see Example 1.1.23(8)); that is,

∏∗EM =∏

E

⟨M⟩ [∆E∼ ×E]

. A frame of a right module M ∶ E ∗ is the same thing as a cylindrical frame of the right

module [∆E∼ ×E] ⟨M⟩ ∶ E∼ ×E ∗ (see Example 1.1.23(8)); that is,

∏E∗M =∏

E

[∆E∼ ×E] ⟨M⟩

.

Proof. Consider a family of M-arrows αe ∶ ∗ e indexed by the objects e ∈ ∥E∥. Since⟨M⟩ [∆E∼ ×E] is dummy in its rst variable, for each E-arrow h ∶ e→ e′, the triangle

e

h

∗

αe55

αe′ ((e′


∗αe′

αe // (e,e)(e,h)

(e′,e′)(h,e′)

// (e,e′)

in ⟨M⟩ [∆E∼ ×E] commutes. Hence α forms a frame of M i it forms a cylindrical frame of⟨M⟩ [∆E∼ ×E].

Remark 4.2.6. Since the left module ⟨M⟩ [∆E∼ ×E] ∶ ∗ E∼ × E (resp. the right module[∆E∼ ×E] ⟨M⟩ ∶ E∼ ×E ∗) is the same thing as the two-sided module ∆E ⟨M⟩ ∶ E E (resp.⟨M⟩∆E ∶ E E), Proposition 4.2.5 is in fact just a restatement of Proposition 4.2.4, and viceversa.

Denition 4.2.7.

If Φ ∶M→ N ∶ ∗ E is a left module morphism and α is a frame ofM, then their compositeα Φ = Φ α is the frame of N dened by

[α Φ]e = αe ⋅ ⟨Φ⟩e

for e ∈ ∥E∥. The function∏∗E

Φ ∶∏∗EM→∏

∗EN


∗EΦ = α Φ

.

72

4. Frames

If Φ ∶ M → N ∶ E ∗ is a right module morphism and α is a frame of M, then theircomposite α Φ = Φ α is the frame of N dened by

[α Φ]e = αe ⋅ e ⟨Φ⟩

for e ∈ ∥E∥. The function∏E∗

Φ ∶∏E∗M→∏

E∗N


E∗Φ = α Φ

.

Remark 4.2.8. The composite α Φ so dened does form a frame of N . In fact, we have thefollowing.

Proposition 4.2.9.

For a left module morphism Φ ∶M→ N ∶ ∗ E, the diagram

∏∗EM∏∗E Φ

∏E∆E ⟨M⟩∏E ∆E⟨Φ⟩

∏∗EN ∏E∆E ⟨N ⟩

commutes; that is,

∏∗E

Φ =∏E

∆E ⟨Φ⟩.

For a right module morphism Φ ∶M→ N ∶ E ∗, the diagram

∏E∗M∏E∗ Φ

∏E ⟨M⟩∆E

∏E⟨Φ⟩∆E

∏E∗N ∏E ⟨N ⟩∆E

commutes; that is,

∏E∗

Φ =∏E

⟨Φ⟩∆E

.

Proof. See Proposition 4.2.4 for the identities ∏∗EM =∏E∆E ⟨M⟩ and ∏∗EN =∏E∆E ⟨N ⟩.We now need to verify that α Φ = α ∆E ⟨Φ⟩ for any frame α ∈ ∏∗EM = ∏E∆E ⟨M⟩. Butsince ⟨Φ⟩e = e ⟨∆E ⟨Φ⟩⟩e for e ∈ ∥E∥, we have

[α Φ]e = αe ⋅ ⟨Φ⟩e = αe ⋅ e ⟨∆E ⟨Φ⟩⟩e = [α ∆E ⟨Φ⟩]e.

Proposition 4.2.10.

The assignment Φ↦∏∗E Φ is functorial and denes the left module

∏∗E

∶ ∗ [∶ E]

. In fact, ∏∗E is obtained from the left module ∏E in Proposition 4.1.7 by the composition

∗ ∏E // [E ∶ E] [∶ E][∆E∶E]oo

.

73

4. Frames

The assignment Φ↦∏E∗ Φ is functorial and denes the left module

∏E∗

∶ ∗ [E ∶]

. In fact, ∏E∗ is obtained from the left module ∏E in Proposition 4.1.7 by the composition

∗ ∏E // [E ∶ E] [E ∶][E∶∆E]oo

.

Proof. The second assertion is immediate from Proposition 4.2.4 and Proposition 4.2.9. Therst assertion follows from the second.

Denition 4.2.11.

If H ∶D→ E is a functor and α is a frame of a left moduleM ∶ ∗ E, then their compositeH α = α H is the frame of the left module ⟨M⟩H ∶ ∗D dened by

[H α]d = α(H ⋅ d)

for d ∈ ∥D∥. The function∏∗H

M ∶∏∗EM→∏

∗D⟨M⟩H


∗H

M = H α.

If H ∶D→ E is a functor and α is a frame of a right moduleM ∶ E ∗, then their compositeH α = α H is the frame of the right module H ⟨M⟩ ∶D ∗ dened by

[H α]d = α(H ⋅ d)

for d ∈ ∥D∥. The function∏H∗M ∶∏

E∗M→∏

D∗H ⟨M⟩


H∗M = H α

.

Remark 4.2.12. The composite H α so dened does form a frame of ⟨M⟩H (resp. H ⟨M⟩). Infact, we have the following.

Proposition 4.2.13. Let H ∶D→ E be a functor. For any left module M ∶ ∗ E, the diagram

∏∗EM∏∗HM

∏E∆E ⟨M⟩∏H ∆E⟨M⟩

∏∗D ⟨M⟩H ∏D∆D ⟨M⟩H ∏D H ⟨∆E ⟨M⟩⟩H

commutes; that is,

∏∗H

M =∏H

∆E ⟨M⟩.

74

4. Frames

For any right module M ∶ E ∗, the diagram

∏E∗M∏H∗M

∏E ⟨M⟩∆E

∏H⟨M⟩∆E

∏D∗ H ⟨M⟩ ∏D H ⟨M⟩∆D ∏D H ⟨⟨M⟩∆E⟩H

commutes; that is,

∏H∗M =∏

H

⟨M⟩∆E

.

Proof. See Proposition 4.2.4 for the identities∏∗EM =∏E∆E ⟨M⟩ and∏∗D ⟨M⟩H =∏D∆D ⟨M⟩H,and observe that the identity

H ⟨∆E ⟨M⟩⟩H = [H ∆E] ⟨M⟩H = ∆D ⟨M⟩H

holds. It is now clear that for any frame α ∈ ∏∗EM = ∏E∆E ⟨M⟩, the composition H α inDenition 4.2.11 and Denition 4.1.8 yields the same frame.

Proposition 4.2.14. Let H ∶D→ E be a functor. The family of functions ∏∗HM ∶∏∗EM→∏∗D ⟨M⟩H, one for each left moduleM ∶ ∗ E,

denes a left module cell

∗ ∏∗E //

1

[∶ E][∶H]

∗∏∗D

//

∏∗H

[∶D]

. In fact, the cell ∏∗H is obtained from the cell ∏H in Proposition 4.1.12 by the pastingcomposition

∗ ∏E //

1

[E ∶ E][H∶H]

[∶ E][∆E∶E]oo

[∶H]

∗∏D

//

∏H

[D ∶D] [∶D][∆D∶D]oo

. Moreover, the assignment H↦∏∗H is contravariant functorial. The family of functions ∏H∗M ∶∏E∗M→∏D∗ H ⟨M⟩, one for each right moduleM ∶ E

∗, denes a left module cell

∗ ∏E∗ //

1

[E ∶][H∶]

∗∏D∗

//

∏H∗

[D ∶]

. In fact, the cell ∏H∗ is obtained from the cell ∏H in Proposition 4.1.12 by the pastingcomposition

∗ ∏E //

1

[E ∶ E][H∶H]

[E ∶][E∶∆E]oo

[H∶]

∗∏D

//

∏H

[D ∶D] [D ∶][D∶∆D]oo

. Moreover, the assignment H↦∏H∗ is contravariant functorial.

Proof. The rst assertion follows from the second, which is immediate from Proposition 4.2.13.The functoriality of the assignment H↦∏∗H is now reduced to that of the assignment H↦∏H

(see Proposition 4.1.12) by virtue of Proposition 1.2.34.

75

4. Frames

4.3. Ordinary cylinders

Denition 4.3.1. LetM ∶XA be a module. Given a functor G ∶A→X, a right cylinder α from G toM, written

α ∶ GM

or diagrammatically as

XM

//α AGoo

, is dened by a frame α of the composite endomodule G ⟨M⟩ ∶AA. Given a functor F ∶X→A, a left cylinder α fromM to F, written

α ∶M F

or diagrammatically as

XM //

F//α A

, is dened by a frame α of the composite endomodule ⟨M⟩F ∶XX.

Remark 4.3.2.(1)

The naturality of a right cylinder α ∶ G M is expressed by the commutativity of thesquare

s ⋅G αs //

f ⋅G

s

f

t ⋅G αt// t

for each A-arrow f ∶ s→ t. The naturality of a left cylinder α ∶M F is expressed by the commutativity of the

square

sαs //

f

F ⋅ sF ⋅ f

t αt// F ⋅ t

for each X-arrow f ∶ s→ t.(2) By Remark 4.1.2(4) and Remark 1.1.28(4),

∏X

⟨M⟩F =∏X∼

F ⟨M∼⟩

. Hence a left cylinder α ∶M F is the same thing as a right cylinder α ∶ FM∼.(3)

If in Denition 4.3.1, A is the terminal category, then a right cylinder

XM

//α ∗Goo

is identied with an arrow of the right moduleM ∶X ∗.

76

4. Frames

If in Denition 4.3.1, X is the terminal category, then a left cylinder

∗M //

F//α A

is identied with an arrow of the left moduleM ∶ ∗A.

Denition 4.3.3. Let E be a category andM ∶XA be a module. Given a pair of functorsS ∶ E→X and T ∶ E→A, a [two-sided] cylinder α from S to T alongM, written

α ∶ S T ∶ EM

or diagrammatically asE

S

T

##X M

//

α

A

, is dened by a frame α of the composite endomodule S ⟨M⟩T ∶ E E.

Remark 4.3.4.(1) The naturality of a cylinder α ∶ S T ∶ E M is expressed by the commutativity of the

square

e ⋅ S αe //

h ⋅ S

T ⋅ eT ⋅ h

e′ ⋅ S αe′// T ⋅ e′

for each E-arrow h ∶ e→ e′.(2)

A right cylinder

XM

//α AGoo

from G toM can be depicted as a two-sided cylinder

AG

1

##X M

//

α

A

from G to the identity 1A alongM. A left cylinder

XM //

F//α A

fromM to F can be depicted as a two-sided cylinder

X1

F

##X M

//

α

A

from the identity 1X to F alongM.

77

4. Frames

(3) Since S ⟨⟨M⟩T⟩ = S ⟨M⟩T = ⟨S ⟨M⟩⟩T, the following right, two-sided, and left cylindersare the same thing:

ES

T

##X

⟨M⟩T//α E

SooX M

//

α

A ES⟨M⟩ //

T//α A

(4) By Example 4.1.3(1), a natural transformation α ∶ S → T ∶ E → C is the same thing as acylinder α ∶ S T ∶ E ⟨C⟩ along the hom of C. Conversely, sinceM = [M0] ⟨⌊M⌋⟩ [M1]for any moduleM ∶XA (see Remark 3.1.16(2)), a cylinder

ES

T

##X M

//

α

A

from S to T alongM is the same thing as a natural transformation

ES

zz

T

$$X M0

// ⌊M⌋α

AM1

oo

from S M0 toM1 T in the collage category ⌊M⌋.Example 4.3.5.

(1) Since [S F] ⟨D⟩T = S ⟨F ⟨D⟩⟩T, a natural transformation

CS

T

##E

F//

α

D

from S F to T is the same thing as a cylinder

CS

T

##E

F⟨D⟩//

α

D

from S to T along the representable module F ⟨D⟩. Since S ⟨D⟩ [F T] = S ⟨⟨D⟩F⟩T, a natural transformation

CS

T

##D

α

EF

oo

from S to F T is the same thing as a cylinder

CS

T

##D

⟨D⟩F//

α

E

from S to T along the corepresentable module ⟨D⟩F.

78

4. Frames

(2) As a special case of above, consider functors as in:

XF

//AGoo

Since [G F] ⟨A⟩ [1A] = G ⟨F ⟨A⟩⟩, a natural transformation ε ∶ G F → 1A ∶ A → A isthe same thing as a right cylinder

XF⟨A⟩

//ε AGoo

from G to the representable module F ⟨A⟩. Since [1X] ⟨X⟩ [G F] = ⟨⟨X⟩G⟩F, a natural transformation η ∶ 1X → G F ∶ X → X is

the same thing as a left cylinder

X⟨X⟩G //

F//η A

from the corepresentable module ⟨X⟩G to F.

Denition 4.3.6. Given a category E and a module M ∶ X A, the module of cylindersEM,

⟨E,M⟩ ∶ [E,X] [E,A], is dened by

(S) ⟨E,M⟩ (T ) =∏E

S ⟨M⟩T

for S ∈ [E,X] and T ∈ [E,X].

Remark 4.3.7.(1) For a pair of functors S ∶ E → X and T ∶ E → A, the set (S) ⟨E,M⟩ (T) consists of all

cylinders S T ∶ EM.(2) Given a natural transformation σ ∶ S′ → S, a cylinder α ∶ S T, and a second natural

transformation τ ∶ T→ T′ as in

ES′σ

S T ##

τT′

X M

//

α

A

, their composite is the cylinder

ES′

T′

##X M

//

σ α τ

A

dened byσ α τ = α σ ⟨M⟩ τ = α ⋅∏

E

σ ⟨M⟩ τ

, the image of a cylindrical frame α ∈∏E S ⟨M⟩T under the function

∏E

σ ⟨M⟩ τ ∶∏E

S ⟨M⟩T→∏E

S′ ⟨M⟩T′

79

4. Frames

. By Example 1.1.23(3), each component of the cylinder σ α τ is given by

[σ α τ]e = σe αe τe.

(3) If M is given by the hom of a category C, then the composition in (2) above is justthe usual composition of natural transformations. The module of cylinders E ⟨C⟩ (i.e.natural transformations E→C) is thus the same thing as the hom of the functor category[E,C]:

⟨E, ⟨C⟩⟩ = ⟨E,C⟩(4) If E is small andM is locally small, then the module ⟨E,M⟩ is locally small.

Proposition 4.3.8. Given a category E and a composite module P ⟨N ⟩Q as in

X

P

P⟨N ⟩Q //A

Q

Y N//

1

B

, the identity

[E,X][E,P]

⟨E,P⟨N ⟩Q⟩ // [E,A][E,Q]

[E,Y]⟨E,N ⟩

//

1

[E,B]

, i.e.⟨E,P ⟨N ⟩Q⟩ = [E,P] ⟨E,N ⟩ [E,Q]

, holds.

Proof. For any G ∈ [E,X] and F ∈ [E,A],

(G) ⟨E,P ⟨N ⟩Q⟩ (F ) =∏E

G ⟨P ⟨N ⟩Q⟩F

=∏E

[G P] ⟨N ⟩ [Q F ]

= (G P) ⟨E,N ⟩ (Q F )= (G ⋅ [E,P]) ⟨E,N ⟩ ([E,Q] ⋅ F )= (G) ⟨[E,P] ⟨E,N ⟩ [E,Q]⟩ (F )

.

Remark 4.3.9. The cell ⟨E,P ⟨N ⟩Q⟩ 1 ⟨E,N ⟩ above sends each cylinder

ES

T

##X

P⟨N ⟩Q//

α

A

to the cylinderE

S P

Q T

##Y N

//

α

B

dened by the same frame.

80

4. Frames

Denition 4.3.10. If α ∶ EM is a cylinder and Φ ∶M→ N is a module morphism as in

ES

T

##X

M //

NΦ

66

α

A

, then their composite α Φ = Φ α is the cylinder

ES

T

##X N

//

α Φ

A

dened byα Φ = α S ⟨Φ⟩T = α ⋅∏

E

S ⟨Φ⟩T


∏E

S ⟨Φ⟩T ∶∏E

S ⟨M⟩T→∏E

S ⟨N ⟩T

.

Remark 4.3.11. By Example 1.1.23(2), each component of the cylinder α Φ is given by

[α Φ]e = αe ⋅ (e ⋅ S) ⟨Φ⟩ (T ⋅ e)

.

Denition 4.3.12. Given a category E and a module morphism Φ ∶M → N ∶ X A, themodule morphism

⟨E,Φ⟩ ∶ ⟨E,M⟩→ ⟨E,N ⟩ ∶ [E,X] [E,A], postcomposition with Φ, is dened by

(S) ⟨E,Φ⟩ (T) =∏E

S ⟨Φ⟩T

for each pair of functors S ∶ E→X and T ∶ E→A.

Remark 4.3.13.(1) The module morphism ⟨E,Φ⟩ maps each cylinder α ∶ S T ∶ E M to the cylinder

α Φ ∶ S T ∶ E N dened in Denition 4.3.10.(2) The assignment Φ→ ⟨E,Φ⟩ is functorial; indeed the functor

⟨E,−⟩ ∶ [X ∶A]→ [[E,X] ∶ [E,A]]

is dened by(S) ⟨E,M⟩ (T ) =∏

E

S ⟨M⟩T

for S ∈ [E,X], T ∈ [E,A], and M ∈ [X ∶A].Note. By Remark 1.2.2(3), the following denition is regarded as a special case of Deni-tion 4.3.10, and vice versa.

81

4. Frames

Denition 4.3.14. Given a cylinder α and a cell Φ as in

ES

T

##X

P

M //

α

A

Q

Y N//

Φ

B

, their composite α Φ = Φ α is the cylinder

ES P

Q T

##Y N

//

α Φ

B

dened byα Φ = α S ⟨Φ⟩T = α ⋅∏

E

S ⟨Φ⟩T


∏E

S ⟨Φ⟩T ∶∏E

S ⟨M⟩T→∏E

S ⟨P ⟨N ⟩Q⟩T =∏E

[S P] ⟨N ⟩ [Q T]

.

Remark 4.3.15.(1) Each component of the cylinder α Φ is given by

[α Φ]e = αe ⋅ (e ⋅ S) ⟨Φ⟩ (T ⋅ e)

(cf. Remark 4.3.11).(2) If a cell is given by the hom of a functor H as in

ES

T

##C

H

⟨C⟩ //

α

C

H

B⟨B⟩

//

⟨H⟩

B

, then the composite α ⟨H⟩ is just the usual composite of a natural transformation and afunctor; that is,

α ⟨H⟩ = α H

.


Denition 4.3.16. Given a category E and a cell

X

P

M //A

Q

Y N//

Φ

B

82

4. Frames

, the cell

[E,X] ⟨E,M⟩ //

[E,P]

[E,A][E,Q]

[E,Y]⟨E,N ⟩

//

⟨E,Φ⟩

[E,B]


⟨E,M⟩ ⟨E,Φ⟩ // ⟨E,P ⟨N ⟩Q⟩ = [E,P] ⟨E,N ⟩ [E,Q]


Remark 4.3.17.(1) The cell ⟨E,Φ⟩ sends each cylinder α ∶ S T ∶ EM to the cylinder α Φ ∶ S P Q T ∶

E N dened in Denition 4.3.14.(2) If Φ is given by the hom of a functor H ∶C→ B, the postcomposition cell

[E,C] ⟨E,⟨C⟩⟩ //

[E,H]

[E,C][E,H]

[E,B]⟨E,⟨B⟩⟩

//

⟨E,⟨H⟩⟩

[E,B]

sends each cylinder α ∶ E ⟨C⟩ to the cylinder α ⟨H⟩ ∶ E ⟨B⟩; that is, sends eachnatural transformation α ∶ E → C to the natural transformation α H ∶ E → B, the usualcomposite of a natural transformation and a functor (see Remark 4.3.15(2)). Hence thepostcomposition cell ⟨E, ⟨H⟩⟩ is the same thing as the hom

[E,C] ⟨E,C⟩ //

[E,H]

[E,C][E,H]

[E,B]⟨E,B⟩

//

⟨E,H⟩

[E,B]

of the postcomposition functor [E,H]; that is,

⟨E, ⟨H⟩⟩ = ⟨E,H⟩

.

Note. Proposition 4.3.18 is analogous to Proposition 1.2.25. The proofs given for them arealmost identical.

Proposition 4.3.18. The assignment Φ↦ ⟨E,Φ⟩ is functorial.

Proof. Clearly, the assignment Φ ↦ ⟨E,Φ⟩ preserves the identities. To verify that it preservesthe composition, let Φ and Ψ be a composable pair of cells and consider the cells ⟨E,Φ⟩, ⟨E,Ψ⟩,

83

4. Frames

and ⟨E,Φ Ψ⟩ depicted in

XM //

P

A

Q

[E,X] ⟨E,M⟩ //

[E,P]

[E,A][E,Q]

Y N //

P′

Φ

B

Q′

[E,Y] ⟨E,N ⟩ //

[E,P′]

⟨E,Φ⟩

[E,B][E,Q′]

Z L//

Ψ

C [E,Z]⟨E,L⟩

//

⟨E,Ψ⟩

[E,C]

XM //

P P′

A

Q′ Q

[E,X] ⟨E,M⟩ //

[E,P P′]

[E,A][E,Q′ Q]

Z L//

Φ Ψ

C [E,Z]⟨E,L⟩

//

⟨E,Φ Ψ⟩

[E,C]

. We need to verify that the composition of the cells ⟨E,Φ⟩ and ⟨E,Ψ⟩ yields the cell ⟨E,Φ Ψ⟩.First note that [E,P P′] = [E,P] [E,P′] and [E,Q′ Q] = [E,Q′] [E,Q] by the func-toriality of the operation [E,−]. The cell ⟨E,Φ⟩ ⟨E,Ψ⟩ is dened by the module mor-phism ⟨E,Φ⟩ [E,P] ⟨E,Ψ⟩ [E,Q] and the cell ⟨E,Φ Ψ⟩ is dened by the module morphism⟨E,Φ P ⟨Ψ⟩Q⟩. But by the functoriality of the operation ⟨E,−⟩ (see Remark 4.3.13(2)) andProposition 4.3.8,

⟨E,Φ P ⟨Ψ⟩Q⟩ = ⟨E,Φ⟩ ⟨E,P ⟨Ψ⟩Q⟩ = ⟨E,Φ⟩ [E,P] ⟨E,Ψ⟩ [E,Q]

.

Remark 4.3.19.(1) Given a small category E, the functor

⟨E,−⟩ ∶MOD→MOD

is dened by the object functionM↦ ⟨E,M⟩ and the arrow function Φ↦ ⟨E,Φ⟩, extendingthe functor ⟨E,−⟩ in Remark 4.3.13(2) as shown in

[X ∶A]

⟨E,−⟩ // [[E,X] ∶ [E,A]]

MOD⟨E,−⟩

//MOD

, where denotes the canonical embedding in Remark 1.2.19(2).(2) For E small, the identity ⟨E, ⟨H⟩⟩ = ⟨E,H⟩ in Remark 4.3.17(2) is now expressed by the

commutativity of

CAT[E,−] //

⟨−⟩

CAT

⟨−⟩

MOD⟨E,−⟩

//MOD

.

Proposition 4.3.20. If a cell Φ is fully faithful, so is the cell ⟨E,Φ⟩.

Proof. Since the operation ⟨E,−⟩ is functorial, it preserves isomorphisms.

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4. Frames

Example 4.3.21. The postcomposition

[E,X][E,M0]

⟨E,M⟩ // [E,A][E,M1]

[E, ⌊M⌋]⟨E,⌊M⌋⟩

//

⟨E,IM⟩

[E, ⌊M⌋]

with the unit cell in Remark 3.1.16(2) sends each cylinder α ∶ S T ∶ E M to the naturaltransformation α ∶ S M0 →M1 T ∶ E→ ⌊M⌋ (see Remark 4.3.4(4)).

Denition 4.3.22. Given a category E and a cell morphisms τ ∶ Φ → Ψ ∶M N , the cellmorphism

⟨E, τ⟩ ∶ ⟨E,Φ⟩→ ⟨E,Ψ⟩ ∶ ⟨E,M⟩ ⟨E,N ⟩, postcomposition with τ , is dened by the pair of postcomposition natural transformations

[E, τ0] ∶ [E,Φ0]→ [E,Ψ0] [E, τ1] ∶ [E,Φ1]→ [E,Ψ1]

(see Preliminary 9), where τ0 ∶ Φ0 → Ψ0 and τ1 ∶ Φ1 → Ψ1 are the left and right components ofτ .

Remark 4.3.23.(1) ⟨E, τ⟩ so dened does form a cell morphism. Indeed, given a cylinder α ∶ S T ∶ E M,

the commutativity of

S ⋅ [E,Φ0]S ⋅ [E,τ0]

α ⋅ ⟨E,Φ⟩ // [E,Φ1] ⋅T[E,τ1] ⋅T

S ⋅ [E,Ψ0]α ⋅ ⟨E,Ψ⟩

// [E,Ψ1] ⋅T

, i.e.

S Φ0

S τ0

α Φ // Φ1 T

τ1 T

S Ψ0 α Ψ// Ψ1 T


e ⋅ S ⋅Φ0

e ⋅ S ⋅ τ0

αe ⋅Φ // Φ1 ⋅T ⋅ eτ1 ⋅T ⋅ e

e ⋅ S ⋅Ψ0 αe ⋅Ψ// Ψ1 ⋅T ⋅ e

for each object e ∈ ∥E∥.(2) The assignment τ ↦ ⟨E, τ⟩ denes the functor

⟨E,−⟩ ∶ [M ∶ N ]→ [⟨E,M⟩ ∶ ⟨E,N ⟩]

; indeed, by the denition of the cell morphism ⟨E, τ⟩, the functoriality of ⟨E,−⟩ is reducedto that of [E,−] ∶ [Mi,Ni]→ [[E,Mi] , [E,Ni]] for i = 0,1.

85

4. Frames

Denition 4.3.24. If H is a functor and α is a cylinder as in

D

HE

S

T

##X M

//

α

A

, then their composite H α = α H is the cylinder

DH S

T H

##X M

//

H α

A

dened byH α = α ⋅∏

H

S ⟨M⟩T


∏H

S ⟨M⟩T ∶∏E

S ⟨M⟩T→∏D

H ⟨S ⟨M⟩T⟩H =∏D

[H S] ⟨M⟩ [T H]

.

Remark 4.3.25.(1) Each component of the cylinder H α is given by

[H α]d = α(H ⋅ d)

(see Denition 4.1.8).

(2) If D is a subcategory of E, then the composition of the inclusion DDÐ→ E with α yields a

cylinder D α, the restriction of α to D.(3) IfM is given by the hom of a category, then the composite H α is just the usual composite

of a functor and a natural transformation.

Note. By Remark 4.3.4(2), the following denition is a special case of Denition 4.3.24 (andvice versa by Remark 4.3.4(3)).

Denition 4.3.26.

If K is a functor and α is a right cylinder as in

XM

//α AGoo

EKoo

, then their composite K α = α K is the two-sided cylinder

EK G

K

##X M

//

K α

A

dened byK α = α ⋅∏

K

G ⟨M⟩

86

4. Frames

, the image of a cylindrical frame α ∈∏A G ⟨M⟩ under the function

∏K

G ⟨M⟩ ∶∏A

G ⟨M⟩→∏E

K ⟨G ⟨M⟩⟩K =∏E

[K G] ⟨M⟩K

. If K is a functor and α is a left cylinder as in

EK //X

M //

F//α A

, then their composite K α = α K is the two-sided cylinder

EK

F K

##X M

//

K α

A

dened byK α = α ⋅∏

K

⟨M⟩F

, the image of a cylindrical frame α ∈∏X ⟨M⟩F under the function

∏K

⟨M⟩F ∶∏X

⟨M⟩F→∏E

K ⟨⟨M⟩F⟩K =∏E

K ⟨M⟩ [F K].

Remark 4.3.27. Each component of the cylinder K α is given by

[K α]e = α(K ⋅ e)


Denition 4.3.28. Given a functor H ∶D→ E and a moduleM ∶XA, the cell

[E,X] ⟨E,M⟩ //

[H,X]

[E,A][H,A]

[D,X]⟨D,M⟩

//

⟨H,M⟩

[D,A]

, precomposition with H, is dened by

(S) ⟨H,M⟩ (T) =∏H

S ⟨M⟩T

for each pair of functors S ∶ E→X and T ∶ E→A.

Remark 4.3.29.(1) The cell ⟨H,M⟩ sends each cylinder α ∶ S T ∶ EM to the cylinder H α ∶ H S T H ∶

DM dened in Denition 4.3.24.(2) The precomposition cell ⟨H,M⟩ is obtained from the left module cell in Proposition 4.1.12

by the pasting composition

∗1

∏E // [E ∶ E][H∶H]

[E,X]∼ × [E,A]−⟨M⟩−oo

[H,X]×[H,A]

∗∏D

//

∏H

[D ∶D] [D,X]∼ × [D,A]−⟨M⟩−oo

, where − ⟨M⟩− denotes the functor given by the assignment (S,T ) ↦ S ⟨M⟩T . (SeeRemark 1.2.5.)

87

4. Frames

(3) IfM is given by the hom of a category C, then the precomposition cell

[E,C] ⟨E,⟨C⟩⟩ //

[H,C]

[E,C][H,C]

[D,C]⟨D,⟨C⟩⟩

//

⟨H,⟨C⟩⟩

[D,C]

sends each cylinder α ∶ E ⟨C⟩ to the cylinder H α ∶D ⟨C⟩; that is, sends each naturaltransformation α ∶ E→C to the natural transformation H α ∶D→C, the usual compositeof a functor and a natural transformation. Hence the precomposition cell ⟨H, ⟨C⟩⟩ is thesame thing as the hom

[E,C] ⟨E,C⟩ //

[H,C]

[E,C][H,C]

[D,C]⟨D,C⟩

//

⟨H,C⟩

[D,C]

of the precomposition functor [H,C]; that is,

⟨H, ⟨C⟩⟩ = ⟨H,C⟩

.

Proposition 4.3.30. Given a moduleM, the assignment H↦ ⟨H,M⟩ denes the contravariantfunctor

⟨−,M⟩ ∶Cat∼ →MOD.

Proof. Since the cell ⟨H,M⟩ is obtained from the cell ∏H by the pasting composition in Re-mark 4.3.29(2), the functoriality of the assignment H ↦ ⟨H,M⟩ is reduced to that of the as-signment H↦∏H (see Proposition 4.1.12) by virtue of Proposition 1.2.34.

Example 4.3.31. Let E be a category andM ∶XA be a module. Given an object e ∈ ∥E∥,precomposition with the functor e ∶ ∗→ E yields the cell

[E,X][e,X]

⟨E,M⟩ // [E,A][e,A]

X M//

⟨e,M⟩

A

, evaluation at e, which sends each cylinder α ∶ S T ∶ EM to theM-arrow αe ∶ e⋅S T ⋅e,the component of α at e.

Remark 4.3.32. IfM in the example above is given by the hom of a category C, we have a cell

[E,C][e,C]

⟨E,⟨C⟩⟩ // [E,C][e,C]

C⟨C⟩

//

⟨e,⟨C⟩⟩

C

88

4. Frames

; by Remark 4.3.29(3), this cell is the same thing as the hom

[E,C][e,C]

⟨E,C⟩ // [E,C][e,C]

C⟨C⟩

//

⟨e,C⟩

C

of the precomposition functor [e,C] ∶ [E,C]→C, evaluation at e (see Preliminary 14).

Theorem 4.3.33. There is a functor

⟨−,−⟩ ∶Cat∼ ×MOD→MOD

such that(1) for each small category E, ⟨E,−⟩ ∶ MOD → MOD coincides with the functor in Re-

mark 4.3.19(1).(2) for each locally small module M, ⟨−,M⟩ ∶ Cat∼ → MOD coincides with the functor in

Proposition 4.3.30.

Proof. By the bifunctor lemma (see [ML98] p37 Proposition 1), it suces to show that thesquare

⟨E,M⟩ ⟨E,Φ⟩ //

⟨H,M⟩

⟨E,N ⟩⟨H,N ⟩

⟨D,M⟩⟨D,Φ⟩

// ⟨D,N ⟩

commutes for any functor H ∶D→ E and any cell Φ ∶M N ; that is,

α ⋅ ⟨E,Φ⟩ ⋅ ⟨H,N ⟩ = α ⋅ ⟨H,M⟩ ⋅ ⟨D,Φ⟩

for any cylinder α ∶ S T ∶ EM. But by Remark 4.3.17(1) and Remark 4.3.29(1),

α ⋅ ⟨E,Φ⟩ ⋅ ⟨H,N ⟩ = H (α Φ) = (H α) Φ = α ⋅ ⟨H,M⟩ ⋅ ⟨D,Φ⟩

Denition 4.3.34. Let D be a category andM ∶ E E be an endomodule. For any frame αofM, the frame [D, α] of the endomodule ⟨D,M⟩ ∶ [D,E] [D,E] is dened by

[D, α]H = H α ∶ H H ∶DM

for H a functor D→ E.

Remark 4.3.35.(1) [D, α] so dened does form a frame of ⟨D,M⟩. Indeed, given a natural transformation

τ ∶ H→ H′ ∶D→ E, the commutativity of the square

H[D,α]H=H α

//

τ

H

τ

H′[D,α]H′=H′ α

// H′

89

4. Frames

is reduced to the commutativity of the square

d ⋅Hd ⋅ τ

[H α]d=α(H ⋅d) // H ⋅ dτ ⋅ d

d ⋅H′[H′ α]d=α(H′ ⋅d)

// H′ ⋅ d

for each object d ∈ ∥D∥, i.e, the naturality of α.(2) If D is a category and α is a cylinder

ES

T

##X M

//

α

A

, i.e. a frame of the endomodule S ⟨M⟩T ∶ E E, then [D, α] is a frame of the endomodule⟨D,S ⟨M⟩T⟩ = [D,S] ⟨D,M⟩ [D,T] (see Proposition 4.3.8), i.e. a cylinder

[D,E][D,S]xx

[D,T]&&

[D,X]⟨D,M⟩

//

[D,α]

[D,A]

. The component of [D, α] at a functor H ∶ D → E is the cylinder H α dened inDenition 4.3.24. IfM is given by the hom of a category C, then [D, α] is the same thingas the postcomposition natural transformation

[D, α] ∶ [D,S]→ [D,T] ∶ [D,E]→ [D,C](see Preliminary 9).

(3) Let E be a category. If α is a right cylinder

XM

//α AGoo

, i.e. a frame of the endomodule G ⟨M⟩ ∶ A A, then [E, α] is a frame of the endo-module ⟨E,G ⟨M⟩⟩ = [E,G] ⟨E,M⟩ (see Proposition 4.3.8), i.e. a right cylinder

[E,X]⟨E,M⟩

//[E,α] [E,A][E,G]oo

. The component of [E, α] at a functor K ∶ E → A is the cylinder K α dened inDenition 4.3.26.

If α is a left cylinder

XM //

F//α A

, i.e. a frame of the endomodule ⟨M⟩F ∶XX, then [E, α] is a frame of the endomodule⟨E, ⟨M⟩F⟩ = ⟨E,M⟩ [E,F] (see Proposition 4.3.8), i.e. a left cylinder

[E,X]⟨E,M⟩ //

[E,F]//[E,α] [E,A]

. The component of [E, α] at a functor K ∶ E → X is the cylinder K α dened inDenition 4.3.26.

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4. Frames

4.4. Extraordinary cylinders

Note. This section is largely analogous to Section 4.3.

Denition 4.4.1. Let E be a category andM ∶XA be a module. Given an object x ∈ ∥X∥ and a bifunctor K ∶ E∼ ×E→A, an [extraordinary] cylinder α from

x to K alongM, writtenα ∶ x K ∶ EM

or diagrammatically as∗

x

E∼ ×E

K

X M//

α

A

, is dened by a cylindrical frame α of the composite left module x ⟨M⟩K ∶ ∗ E∼ ×E (seeRemark 4.1.2(3)).

Given an object a ∈ ∥A∥ and a bifunctor K ∶ E∼ ×E→X, an [extraordinary] cylinder α fromK to a alongM, written

α ∶ K a ∶ EMor diagrammatically as

E∼ ×E

K

∗a

X M//

α

A

, is dened by a cylindrical frame α of the composite right module K ⟨M⟩a ∶ E∼ × E ∗(see Remark 4.1.2(3)).

Remark 4.4.2.(1)

The naturality of an extraordinary cylinder α ∶ x K ∶ E M is expressed by thecommutativity of the square

x

αe′

αe // K (e,e)K(e,h)

K (e′,e′)K(h,e′)

// K (e,e′)

for each E-arrow h ∶ e→ e′. The naturality of an extraordinary cylinder α ∶ K a ∶ E M is expressed by the

commutativity of the square

K (e′,e)K(h,e)

K(e′,h) // K (e′,e′)αe′

K (e,e) αe// a

for each E-arrow h ∶ e→ e′.(2) Any of the cylinders we saw in Section 4.3 has the naturality expressed in the ordinary

form, and called ordinary to distinguish it from extraordinary cylinders. However, theadjectives ordinary and extraordinary are often omitted when the context makes itclear which type of cylinder is being talked about. In fact, some bicylinders (see Section4.7) have both ordinary and extraordinary naturalities.

91

4. Frames

Notation 4.4.3. Given a bifunctor K ∶ E∼ ×E → C and an object c ∈ ∥C∥, an extraordinarynatural transformation α from c to K (resp. K to c) is denoted by

α ∶ c K ∶ E→C α ∶ K c ∶ E→C

.

Remark 4.4.4. By Example 4.1.3(2), an extraordinary natural transformation in a category Cis the same thing as an extraordinary cylinder along the hom of C. Conversely, an extraor-dinary cylinder along a module M ∶ X A is the same thing as an extraordinary naturaltransformation in the collage category ⌊M⌋ (cf. Remark 4.3.4(4)).

Denition 4.4.5. Given a category E and a moduleM ∶XA, the module of extraordinary cylinders EM,

⟨E,M⟩ ∶X [E∼ ×E,A]

, is dened by(x) ⟨E,M⟩ (K) =∏

E

x ⟨M⟩K

for x ∈X and K ∈ [E∼ ×E,A]. the module of extraordinary cylinders EM,

⟨E,M⟩ ∶ [E∼ ×E,X]A

, is dened by(K) ⟨E,M⟩ (a) =∏

E

K ⟨M⟩a

for a ∈A and K ∈ [E∼ ×E,X].

Remark 4.4.6.(1) We use the same symbol ⟨E,M⟩ for the module of extraordinary cylinders and the module

of ordinary cylinders (see Denition 4.3.6), and let the context determine which is meant.(2)

For an object x ∈ ∥X∥ and a bifunctor K ∶ E∼ ×E → A, the set (x) ⟨E,M⟩ (K) consistsof all cylinders x K ∶ EM.

For an object a ∈ ∥A∥ and a bifunctor K ∶ E∼ ×E → X, the set (K) ⟨E,M⟩ (a) consistsof all cylinders K a ∶ EM.

(3) Given an X-arrow f ∶ x′ → x, a cylinder α ∶ x K, and a natural transformation

τ ∶ K→ K′ as in∗

x′ f

x

E∼ ×E

Kτ K′

X M//

α

A


∗x′

E∼ ×E

K′

X M//

f α τ

A

92

4. Frames

dened byf α τ = α f ⟨M⟩ τ = α ⋅∏

E

f ⟨M⟩ τ

, the image of a cylindrical frame α ∈∏E x ⟨M⟩K under the function

∏E

f ⟨M⟩ τ ∶∏E

x ⟨M⟩K→∏E

x′ ⟨M⟩K′

. Each component of the cylinder α τ is given by

[f α τ]e = f αe τ(e,e)

(cf. Remark 4.3.7(2)). Given an A-arrow f ∶ a → a′, a cylinder α ∶ K a, and a natural transformation

τ ∶ K′ → K as inE∼ ×E

K′ τ

K

∗af a′

X M//

α

A


E∼ ×E

K′

∗a′

X M//

τ α f

A

dened byτ α f = α τ ⟨M⟩ f = α ⋅∏

E

τ ⟨M⟩ f

, the image of a cylindrical frame α ∈∏E K ⟨M⟩a under the function

∏E

τ ⟨M⟩ f ∶∏E

K ⟨M⟩a→∏E

K′ ⟨M⟩a′

. Each component of the cylinder τ α is given by

[τ α f]e = τ(e,e) αe f

(cf. Remark 4.3.7(2)).

Note. The following denition is a special case of Denition 4.4.5 whereM is given by the homof a category.

Denition 4.4.7. Given categories E and C, the module of extraordinary natural transformations E→C,

⟨E,C⟩ ∶C [E∼ ×E,C]

, is dened by(c) ⟨E,C⟩ (K) =∏

E

c ⟨C⟩K

for c ∈C and K ∈ [E∼ ×E,C]; that is,

⟨E,C⟩ ∶= ⟨E, ⟨C⟩⟩

.

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4. Frames

the module of extraordinary natural transformations E→C,

⟨E,C⟩ ∶ [E∼ ×E,C]C

, is dened by(K) ⟨E,C⟩ (c) =∏

E

K ⟨C⟩ c

for c ∈C and K ∈ [E∼ ×E,C]; that is,

⟨E,C⟩ ∶= ⟨E, ⟨C⟩⟩

.

Remark 4.4.8. We use the same symbol ⟨E,C⟩ for the module of extraordinary natural trans-formations and the hom of the functor category [E,C] (see Remark 4.3.7(3)), and let thecontext determine which is meant.


X

P

P⟨N ⟩Q //A

Q

Y N//

1

B

the identity

X

P

⟨E,P⟨N ⟩Q⟩ // [E∼ ×E,A][E∼×E,Q]

Y⟨E,N ⟩

//

1

[E∼ ×E,B]

, i.e.⟨E,P ⟨N ⟩Q⟩ = P ⟨E,N ⟩ [E∼ ×E,Q]

, holds. the identity

[E∼ ×E,X][E∼×E,P]

⟨E,P⟨N ⟩Q⟩ //A

Q

[E∼ ×E,Y]⟨E,N ⟩

//

1

B

, i.e.⟨E,P ⟨N ⟩Q⟩ = [E∼ ×E,P] ⟨E,N ⟩Q

, holds.

Proof. For any x ∈X and K ∈ [E∼ ×E,A],

(x) ⟨E,P ⟨N ⟩Q⟩ (K) =∏E

x ⟨P ⟨N ⟩Q⟩K

=∏E

(x ⋅ P) ⟨N ⟩ [Q K]

= (x ⋅ P) ⟨E,N ⟩ (Q K)= (x ⋅ P) ⟨E,N ⟩ ([E∼ ×E,Q] ⋅K)= (x) ⟨P ⟨E,N ⟩ [E∼ ×E,Q]⟩ (K)

.

94

4. Frames

Remark 4.4.10. The cell

X

P

⟨E,P⟨N ⟩Q⟩ // [E∼ ×E,A][E∼×E,Q]

Y⟨E,N ⟩

//

1

[E∼ ×E,B]

resp. [E∼ ×E,X][E∼×E,P]

⟨E,P⟨N ⟩Q⟩ //A

Q

[E∼ ×E,Y]⟨E,N ⟩

//

1

B

sends each cylinder

∗x

E∼ ×E

K

XP⟨N ⟩Q

//

α

A

resp. E∼ ×E

K

∗a

XP⟨N ⟩Q

//

α

A

to the cylinder

∗x ⋅ P

E∼ ×E

Q K

Y N//

α

B

resp. E∼ ×E

K P

∗Q ⋅ a

Y N//

α

B


Denition 4.4.11.

If α ∶ EM is a cylinder and Φ ∶M→ N is a module morphism as in

∗x

E∼ ×E

K

XM //

NΦ 88

α

A


∗x

E∼ ×E

K

X N//

α Φ

A

dened byα Φ = α x ⟨Φ⟩K = α ⋅∏

E

x ⟨Φ⟩K


∏E

x ⟨Φ⟩K ∶∏E

x ⟨M⟩K→∏E

x ⟨N ⟩K

. If α ∶ EM is a cylinder and Φ ∶M→ N is a module morphism as in

E∼ ×E

K

∗a

XM //

NΦ 88

α

A

95

4. Frames


E∼ ×E

K

∗a

X N//

α Φ

A

dened byα Φ = α K ⟨Φ⟩a = α ⋅∏

E

K ⟨Φ⟩a


∏E

K ⟨Φ⟩a ∶∏E

K ⟨M⟩a→∏E

K ⟨N ⟩a

.

Remark 4.4.12. Each component of the cylinder α Φ is given by

[α Φ]e = αe ⋅ (x) ⟨Φ⟩ (K (e,e))

(cf. Remark 4.3.11). Each component of the cylinder α Φ is given by

[α Φ]e = αe ⋅ (K (e,e)) ⟨Φ⟩ (a)

(cf. Remark 4.3.11).

Denition 4.4.13. Given a category E and a module morphism Φ ∶M→ N ∶XA, the module morphism

⟨E,Φ⟩ ∶ ⟨E,M⟩→ ⟨E,N ⟩ ∶X [E∼ ×E,A]


(x) ⟨E,Φ⟩ (K) =∏E

x ⟨Φ⟩K

for each pair of an object x ∈ ∥X∥ and a bifunctor K ∶ E∼ ×E→A. the module morphism

⟨E,Φ⟩ ∶ ⟨E,M⟩→ ⟨E,N ⟩ ∶ [E∼ ×E,X]A


(K) ⟨E,Φ⟩ (a) =∏E

K ⟨Φ⟩a

for each pair of an object a ∈ ∥A∥ and a bifunctor K ∶ E∼ ×E→X.

Remark 4.4.14.(1) The module morphism ⟨E,Φ⟩ ∶ ⟨E,M⟩ → ⟨E,N ⟩ ∶ X [E∼ ×E,A] maps each cylinder

α ∶ x K ∶ E M to the cylinder α Φ ∶ x K ∶ E N dened in Denition 4.4.11,and dually the module morphism ⟨E,Φ⟩ ∶ ⟨E,M⟩ → ⟨E,N ⟩ ∶ [E∼ ×E,X] A maps eachcylinder α ∶ K a ∶ EM to the cylinder α Φ ∶ K a ∶ E N .

96

4. Frames

(2) The assignment Φ→ ⟨E,Φ⟩ is functorial; indeed the functor

⟨E,−⟩ ∶ [X ∶A]→ [X ∶ [E∼ ×E,A]]

is dened by(x) ⟨E,M⟩ (K) =∏

E

x ⟨M⟩K

for x ∈X, K ∈ [E∼ ×E,A], and M ∈ [X ∶A], and dually the functor

⟨E,−⟩ ∶ [X ∶A]→ [[E∼ ×E,X] ∶A]

is dened by(K) ⟨E,M⟩ (a) =∏

E

K ⟨M⟩a

for a ∈A, K ∈ [E∼ ×E,X], and M ∈ [X ∶A].Note. By Remark 1.2.2(3), the following denition is regarded as a special case of Deni-tion 4.4.11, and vice versa.

Denition 4.4.15.

Given a cylinder α and a cell Φ as in

∗x

E∼ ×E

K

X

P

M //

α

A

Q

Y N//

Φ

B


∗x ⋅ P

E∼ ×E

Q K

Y N//

α Φ

B


E

x ⟨Φ⟩K


∏E

x ⟨Φ⟩K ∶∏E

x ⟨M⟩K→∏E

x ⟨P ⟨N ⟩Q⟩K =∏E

(x ⋅ P) ⟨N ⟩ [Q K]

. Given a cylinder α and a cell Φ as in

E∼ ×E

K

∗a

X

P

M //

α

A

Q

Y N//

Φ

B

97

4. Frames


E∼ ×E

K P

∗Q ⋅ a

X N//

α Φ

A


E

K ⟨Φ⟩a


∏E

K ⟨Φ⟩a ∶∏E

K ⟨M⟩a→∏E

K ⟨P ⟨N ⟩Q⟩a =∏E

[K P] ⟨N ⟩ (Q ⋅ a)

.

Remark 4.4.16.(1)

Each component of the cylinder α Φ is given by

[α Φ]e = αe ⋅ (x) ⟨Φ⟩ (K (e,e))

(cf. Remark 4.4.12). Each component of the cylinder α Φ is given by

[α Φ]e = αe ⋅ (K (e,e)) ⟨Φ⟩ (a)

(cf. Remark 4.4.12).(2) If a cell is given by the hom of a functor H, then the composite α ⟨H⟩ is just the usual

composite of an extraordinary natural transformation and a functor; that is,

α ⟨H⟩ = α H

.



X

P

M //A

Q

Y N//

Φ

B

the cell

X⟨E,M⟩ //

P

[E∼ ×E,A][E∼×E,Q]

Y⟨E,N ⟩

//

⟨E,Φ⟩

[E∼ ×E,B]


⟨E,M⟩ ⟨E,Φ⟩ // ⟨E,P ⟨N ⟩Q⟩ = P ⟨E,N ⟩ [E∼ ×E,Q]


98

4. Frames

the cell

[E∼ ×E,X][E∼×E,P]

⟨E,M⟩ //A

Q

[E∼ ×E,Y]⟨E,N ⟩

//

⟨E,Φ⟩

B


⟨E,M⟩ ⟨E,Φ⟩ // ⟨E,P ⟨N ⟩Q⟩ = [E∼ ×E,P] ⟨E,N ⟩Q


Remark 4.4.18. The cell ⟨E,Φ⟩ ∶ P [E∼ ×E,Q] ∶ ⟨E,M⟩ ⟨E,N ⟩ sends each cylinder α ∶ xK ∶ EM to the cylinder α Φ ∶ x ⋅P Q K ∶ E N dened in Denition 4.4.15, and duallythe cell ⟨E,Φ⟩ ∶ [E∼ ×E,P] Q ∶ ⟨E,M⟩ ⟨E,N ⟩ sends each cylinder α ∶ K a ∶ E M tothe cylinder α Φ ∶ K P Q ⋅ a ∶ E N .

Proposition 4.4.19. The assignment Φ↦ ⟨E,Φ⟩ is functorial.

Proof. The functoriality is veried similarly to Proposition 4.3.18 using Remark 4.4.14(2) andProposition 4.4.9 in place of Remark 4.3.13(2) and Proposition 4.3.8.

Remark 4.4.20. Given a small category E, the functor

⟨E,−⟩ ∶MOD→MOD

is dened by the object function M ↦ ⟨E,M⟩ and the arrow function Φ ↦ ⟨E,Φ⟩, extendingthe functor ⟨E,−⟩ in Remark 4.4.14(2) as shown in

[X ∶A]

⟨E,−⟩ // [X ∶ [E∼ ×E,A]]

MOD⟨E,−⟩

//MOD

[X ∶A]

⟨E,−⟩ // [[E∼ ×E,X] ∶A]

MOD⟨E,−⟩

//MOD

, where denotes the canonical embedding in Remark 1.2.19(2).

4.5. Weighted cylinders

Denition 4.5.1.

A cylinder

E

S

D

T

Foo

X M//

α

A

from F S to T alongM is said to be right weighted by F (or right F-weighted). A cylinder

D

S

F // E

T

X M//

α

A

from S to T F alongM is said to be left weighted by F (or left F-weighted).

99

4. Frames

Remark 4.5.2.(1)

Since [F S] ⟨M⟩T = F ⟨S ⟨M⟩T⟩, a right F-weighted cylinder above is the same thingas a right cylinder

ES⟨M⟩T

//α DFoo

. Conversely, a right cylinder

XM

//α AGoo

may be depicted as a right weighted cylinder

X

1

A

1

Goo

X M//

α

A

. Since S ⟨M⟩ [T F] = ⟨S ⟨M⟩T⟩F, a left F-weighted cylinder above is the same thing as

a left cylinder

DS⟨M⟩T //

F//α E

. Conversely, a left cylinder

XM //

F//α A

may be depicted as a left weighted cylinder

X

1

F //A

1

X M//

α

A

.(2) A cylinder

ES

T

##X M

//

α

A

is regarded to be weighted by the identity E→ E and sometimes depicted as

E

S

E1oo

T

E1 //

S

E

T

X M//

α

A X M//

α

A

.

Denition 4.5.3. Given a functor F ∶D→ E and a moduleM ∶XA,

100

4. Frames

the module of right F-weighted cylinders alongM,

⟨F M⟩ ∶ [E,X] [D,A]

, is dened by the composition

[E,X] [F,X] // [D,X] ⟨D,M⟩ // [D,A]

. the module of left F-weighted cylinders alongM,

⟨F M⟩ ∶ [D,X] [E,A]


[D,X] ⟨D,M⟩ // [D,A] [E,A][F,A]oo

.

Remark 4.5.4.(1)

For any pair of functors S ∶ E→X and T ∶D→A,

(S) ⟨F M⟩ (T) = (S) ⟨[F,X] ⟨D,M⟩⟩ (T) = (F S) ⟨D,M⟩ (T)

; that is, an ⟨F M⟩-arrow α ∶ S T is a right F-weighted cylinder in Denition 4.5.1. For any pair of functors S ∶D→X and T ∶ E→A,

(S) ⟨F M⟩ (T) = (S) ⟨⟨D,M⟩ [F,A]⟩ (T) = (S) ⟨D,M⟩ (T F)

; that is, an ⟨F M⟩-arrow α ∶ S T is a left F-weighted cylinder in Denition 4.5.1.(2) If, as a special case, F is given by the identity E→ E, we have

⟨1E M⟩ = ⟨E,M⟩ = ⟨1E M⟩

immediately from the denition (cf. Remark 4.5.2(2)).(3) As another special case whereM is given by the hom of a category C,

the module⟨F C⟩ ∶ [E,C] [D,C]

is given by the composition

[E,C] [F,C] // [D,C] ⟨D,C⟩ // [D,C]

, i.e. by the representable module of the precomposition functor [F,C]. An ⟨F C⟩-arrow α ∶ S T is a right F-weighted natural transformation

E

S

D

T

Foo

C⟨C⟩

//

α

C

. The identity⟨F C⟩ = ⟨F ⟨C⟩⟩

follows from the identity ⟨D,C⟩ = ⟨D, ⟨C⟩⟩ (see Remark 4.3.7(3)).

101

4. Frames

the module⟨F C⟩ ∶ [D,C] [E,C]


[D,C] ⟨D,C⟩ // [D,C] [E,C][F,C]oo

, i.e. by the corepresentable module of the precomposition functor [F,C]. An ⟨F C⟩-arrow α ∶ S T is a left F-weighted natural transformation

D

S

F // E

T

C⟨C⟩

//

α

C

. The identity⟨F C⟩ = ⟨F ⟨C⟩⟩

follows from the identity ⟨D,C⟩ = ⟨D, ⟨C⟩⟩ (see Remark 4.3.7(3)).

Denition 4.5.5. Given a functor F ∶D→ E and a module morphism Φ ∶M→ N ∶XA, the module morphism

⟨F Φ⟩ ∶ ⟨F M⟩→ ⟨F N ⟩ ∶ [E,X] [D,A], postcomposition with Φ, is dened by the composition

[E,X] [F,X] // [D,X]⟨D,M⟩

,,

⟨D,N ⟩22⟨D,Φ⟩ [D,A]

. the module morphism

⟨F Φ⟩ ∶ ⟨F M⟩→ ⟨F N ⟩ ∶ [D,X] [E,A], postcomposition with Φ, is dened by the composition

[D,X]⟨D,M⟩

,,

⟨D,N ⟩22⟨D,Φ⟩ [D,A] [E,A][F,A]oo

.

Remark 4.5.6.(1) The module morphism ⟨F Φ⟩ maps each cylinder α ∶ F S T ∶ D M to the cylinder

α Φ ∶ F S T ∶D N (cf. Remark 4.3.13(1)), and dually the module morphism ⟨F Φ⟩maps each cylinder α ∶ S T F ∶DM to the cylinder α Φ ∶ S T F ∶D N .

(2) The assignment Φ→ ⟨F Φ⟩ is functorial; indeed the functor

⟨F −⟩ ∶ [X ∶A]→ [[E,X] ∶ [D,A]]is dened by

⟨F M⟩ = [F,X] ⟨D,M⟩. Dually, the functor

⟨F −⟩ ∶ [X ∶A]→ [[D,X] ∶ [E,A]]is dened by

⟨F M⟩ = ⟨D,M⟩ [F,A].

102

4. Frames

Denition 4.5.7. Given a functor F ∶D→ E and a cell

X

P

M //A

Q

Y N//

Φ

B

the cell

[E,X] ⟨FM⟩ //

[E,P]

[D,A][D,Q]

[E,Y]⟨FN ⟩

//

⟨FΦ⟩

[D,B]

, postcomposition with Φ, is dened by the pasting composition

[E,X] [F,X] //

[E,P]

[D,X][D,P]

⟨D,M⟩ // [D,A][D,Q]

[E,Y][F,Y]

// [D,Y]⟨D,N ⟩

//

⟨D,Φ⟩

[D,B].

the cell

[D,X] ⟨FM⟩ //

[D,P]

[E,A][E,Q]

[D,Y]⟨FN ⟩

//

⟨FΦ⟩

[E,B]

, postcomposition with Φ, is dened by the pasting composition

[D,X] ⟨D,M⟩ //

[D,P]

[D,A][D,Q]

[E,A][F,A]oo

[E,Q]

[D,Y]⟨D,N ⟩

//

⟨D,Φ⟩

[D,B] [E,B][F,B]oo

.

Remark 4.5.8.(1) The cell ⟨F Φ⟩ sends each cylinder α ∶ F S T ∶DM to the cylinder α Φ ∶ F S P

Q T ∶D N given by the composition

E

S

D

T

Foo

X M //

P

α

A

Q

Y N//

Φ

B

(cf. Remark 4.3.17(1)), and dually the cell ⟨F Φ⟩ sends each cylinder α ∶ S TF ∶DMto the cylinder α Φ ∶ S P Q T F ∶D N given by the composition

D

S

F // E

T

X M //

P

α

A

Q

Y N//

Φ

B

.

103

4. Frames

(2) If, as a special case, F is given by the identity E→ E, we have

⟨1E Φ⟩ = ⟨E,Φ⟩ = ⟨1E Φ⟩

immediately from the denition.(3) As another special case where Φ is given by the hom of a functor H ∶C→ B,

the cell

[E,C] ⟨FC⟩ //

[E,H]

[D,C][D,H]

[E,B]⟨FB⟩

//

⟨FH⟩

[D,B]

, postcomposition with H, is dened by the pasting composition

[E,C] [F,C] //

[E,H]

[D,C][D,H]

⟨D,C⟩ // [D,C][D,H]

[E,B][F,B]

// [D,B]⟨D,B⟩

//

⟨D,H⟩

[D,B]

. The cell ⟨F H⟩ sends each natural transformation α ∶ F S → T ∶ D → C to thenatural transformation α H ∶ F S H→ H T ∶D→ B, the usual composite of a naturaltransformation and a functor. The identity

⟨F H⟩ = ⟨F ⟨H⟩⟩

follows from the identity ⟨D,H⟩ = ⟨D, ⟨H⟩⟩ (see Remark 4.3.17(2)). the cell

[D,C] ⟨FC⟩ //

[D,H]

[E,C][E,H]

[D,B]⟨FB⟩

//

⟨FH⟩

[E,B]

, postcomposition with H, is dened by the pasting composition

[D,C] ⟨D,C⟩ //

[D,H]

[D,C][D,H]

[E,C][F,C]oo

[E,H]

[D,B]⟨D,B⟩

//

⟨D,H⟩

[D,B] [E,B][F,B]oo

. The cell ⟨F H⟩ sends each natural transformation α ∶ S → T F ∶ D → C to thenatural transformation α H ∶ S H→ H T F ∶D→ B, the usual composite of a naturaltransformation and a functor. The identity

⟨F H⟩ = ⟨F ⟨H⟩⟩

follows from the identity ⟨D,H⟩ = ⟨D, ⟨H⟩⟩ (see Remark 4.3.17(2)).

Proposition 4.5.9. The assignment Φ↦ ⟨F Φ⟩ (resp. Φ↦ ⟨F Φ⟩) is functorial.

Proof. Since the cell ⟨F Φ⟩ is obtained from the cell ⟨D,Φ⟩ by the pasting composition asin Denition 4.5.7, the functoriality of the assignment Φ ↦ ⟨F Φ⟩ is reduced to that of theassignment Φ↦ ⟨D,Φ⟩ (see Proposition 4.3.18) by virtue of Proposition 1.2.34.

104

4. Frames

Remark 4.5.10. Given a functor F ∶D→ E with D small, the functor

⟨F −⟩ ∶MOD→MOD

is dened by the object functionM↦ ⟨F M⟩ and the arrow function Φ↦ ⟨F Φ⟩, extendingthe functor ⟨F −⟩ in Remark 4.5.6(2) as shown in

[X ∶A]

⟨F−⟩ // [[E,X] ∶ [D,A]]

MOD⟨F−⟩

//MOD

, where denotes the canonical embedding in Remark 1.2.19(2). Dually, the functor

⟨F −⟩ ∶MOD→MOD

is dened by the object functionM↦ ⟨F M⟩ and the arrow function Φ↦ ⟨F Φ⟩, extendingthe functor ⟨F −⟩ in Remark 4.5.6(2) as shown in

[X ∶A]

⟨F−⟩ // [[D,X] ∶ [E,A]]

MOD⟨F−⟩

//MOD

.

4.6. Cones

Note. This section is largely analogous to Section 4.3 (resp. Section 4.4); in fact cones may beregarded as special sorts of ordinary (resp. extraordinary) cylinders.

Denition 4.6.1.

Let M ∶ ∗ A be a left module. Given a functor K ∶ E → A, a cone α to K along M,written

α ∶ ∗ K ∶ ∗EM, is dened by a frame α of the composite left module ⟨M⟩K ∶ ∗ E.

LetM ∶ X ∗ be a right module. Given a functor K ∶ E → X, a cone α from K alongM,written

α ∶ K ∗ ∶ E∗M, is dened by a frame α of the composite right module K ⟨M⟩ ∶ E ∗.

Remark 4.6.2. By Proposition 4.2.4, a frame α of the left module ⟨M⟩K is the same thing as a frame of the endomodule

∆E ⟨M⟩K ∶ E E. Hence a cone α ∶ ∗ K ∶ ∗E M is the same thing as a cylinderα ∶ ∆E K ∶ EM and depicted as

E∆

||

K

##∗ M

//

α

A

.

105

4. Frames

a frame α of the right module K ⟨M⟩ is the same thing as a frame of the endomoduleK ⟨M⟩∆E ∶ E E. Hence a cone α ∶ K ∗ ∶ E∗ M is the same thing as a cylinderα ∶ K ∆E ∶ EM and depicted as

EK

∆

""X M

//

α

∗.

Denition 4.6.3. Let E be a category andM ∶XA be a module. Given an object x ∈ ∥X∥ and a functor K ∶ E→A, a cone α from x to K alongM, written

α ∶ x K ∶ ∗EM

, is dened by a frame α of the composite left module x ⟨M⟩K ∶ ∗ E. Given an object a ∈ ∥A∥ and a functor K ∶ E→X, a cone α from K to a alongM, written

α ∶ K a ∶ E∗M

, is dened by a frame α of the composite right module K ⟨M⟩a ∶ E ∗.Remark 4.6.4.(1) By Proposition 4.2.4,

a frame α of the left module x ⟨M⟩K is the same thing as a frame of the endomodule∆E ⟨x ⟨M⟩K⟩ = [∆E x]K ⟨M⟩. Hence a cone α ∶ x K ∶ ∗EM is the same thing asa cylinder α ∶ ∆E x K ∶ EM right weighted by ∆E and depicted as

∗x

E∆oo

K

X M//

α

A

or, a bit more economically, as

E∆x

K

##X M

//

α

A

. a frame α of the right module K ⟨M⟩a is the same thing as a frame of the endomodule

⟨K ⟨M⟩a⟩∆E = K ⟨M⟩ [a ∆E]. Hence a cone α ∶ K a ∶ E∗ M is the same thing asa cylinder α ∶ K a ∆E ∶ EM left weighted by ∆E and depicted as

E

K

∆ // ∗a

X M//

α

A

or, a bit more economically, as

EK

∆a

##X M

//

α

A

.

106

4. Frames

(2) By Proposition 4.2.5, a frame α of the left module x ⟨M⟩K is the same thing as a cylindrical frame of the left

module ⟨x ⟨M⟩K⟩ [∆E∼ ×E] = x ⟨M⟩ [K [∆E∼ ×E]]. Hence a cone α ∶ x K ∶ ∗EMis the same thing as an extraordinary cylinder α ∶ x K [∆E∼ ×E] ∶ E M; a coneis thus regarded as an extraordinary cylinder α ∶ x K ∶ E M with K ∶ E∼ ×E → Adummy in the rst variable.

a frame α of the right module K ⟨M⟩a is the same thing as a cylindrical frame of the rightmodule [∆E∼ ×E] ⟨K ⟨M⟩a⟩ = [[∆E∼ ×E] K] ⟨M⟩a. Hence a cone α ∶ K a ∶ E∗Mis the same thing as an extraordinary cylinder α ∶ [∆E∼ ×E] K a ∶ E M; a coneis thus regarded as an extraordinary cylinder α ∶ K a ∶ E M with K ∶ E∼ ×E → Xdummy in the rst variable.

(3) By Remark 4.2.2(3) and Remark 1.1.28(4), the identity

∏∗E

x ⟨M⟩K =∏E∼∗

K ⟨M∼⟩x

holds; hence a cone α ∶ x K ∶ ∗E M is the same thing as a cone α ∶ K x ∶ E∼∗ M∼.

the identity

∏E∗

K ⟨M⟩a =∏∗E∼

a ⟨M∼⟩K

holds; hence a cone α ∶ K a ∶ E∗ M is the same thing as a cone α ∶ a K ∶ ∗E∼ M∼.

(4) A cone dened in Denition 4.6.1 is a special instance of a cone dened in Denition 4.6.3where X (resp. A) is the terminal category. Conversely, a cone α ∶ x K along a moduleM ∶XA is the same thing as a cone α ∶ ∗ K along

the left module x ⟨M⟩ ∶ ∗X. a cone α ∶ K a along a moduleM ∶XA is the same thing as a cone α ∶ K ∗ along

the right module ⟨M⟩a ∶X ∗.(5)

If E is discrete, a cone α ∶ x K ∶ ∗E M is called discrete. Given an object x ∈ ∥X∥and a family of objects ai ∈ ∥A∥ indexed by some set I, a discrete cone α from x to(ai)i∈I alongM is given by a family ofM-arrows αi ∶ x ai indexed by I. Conversely,a family ofM-arrows αi ∶ x ai indexed by I denes a discrete cone from x to (ai)i∈IalongM.

If E is discrete, a cone α ∶ K a ∶ E∗ M is called discrete. Given an object a ∈ ∥A∥and a family of objects xi ∈ ∥X∥ indexed by some set I, a discrete cone α from (xi)i∈Ito a along M is given by a family of M-arrows αi ∶ xi a indexed by I. Conversely,a family ofM-arrows αi ∶ xi a indexed by I denes a discrete cone from (xi)i∈I to aalongM.

Denition 4.6.5. Given a category E and a moduleM ∶XA, the module of cones ∗EM,

⟨∗E,M⟩ ∶X [E,A]

, is dened by(x) ⟨∗E,M⟩ (K) =∏

∗Ex ⟨M⟩K

for x ∈X and K ∈ [E,A].

107

4. Frames

the module of cones E∗M,

⟨E∗,M⟩ ∶ [E,X]A

, is dened by(K) ⟨E∗,M⟩ (a) =∏

E∗K ⟨M⟩a

for a ∈A and K ∈ [E,X].

Remark 4.6.6.(1)

For an object x ∈ ∥X∥ and a functor K ∶ E → A, the set (x) ⟨∗E,M⟩ (K) consists of allcones x K ∶ ∗EM.

For an object a ∈ ∥A∥ and a functor K ∶ E → X, the set (K) ⟨E∗,M⟩ (a) consists of allcones K a ∶ E∗M.

(2) Given an X-arrow f ∶ x′ → x, a cone α ∶ x K, and a natural transformation τ ∶ K → K′

as in

∗x′ f

x

E∆oo

Kτ K′

X M//

α

A

, their composite is the cone

∗x′

E∆oo

K′

X M//

f α τ

A

dened byf α τ = α f ⟨M⟩ τ = α ⋅∏

∗Ef ⟨M⟩ τ

, the image of a conical frame α ∈∏∗E x ⟨M⟩K under the function

∏∗E

f ⟨M⟩ τ ∶∏∗E

x ⟨M⟩K→∏∗E

x′ ⟨M⟩K′

. Given an A-arrow f ∶ a → a′, a cone α ∶ K a, and a natural transformation τ ∶ K′ → K

as in

E

K′ τ

K

∆ // ∗af a′

X M//

α

A

, their composite is the cone

E

K′

∆ // ∗a′

X M//

τ α f

A

dened byτ α f = α τ ⟨M⟩ f = α ⋅∏

E∗τ ⟨M⟩ f

108

4. Frames

, the image of a conical frame α ∈∏E∗ K ⟨M⟩a under the function

∏E∗τ ⟨M⟩ f ∶∏

E∗K ⟨M⟩a→∏

E∗K′ ⟨M⟩a′

.(3) If E is small andM is locally small, then the module ⟨∗E,M⟩ (resp. ⟨E∗,M⟩) is locally

small.(4) By Remark 4.6.4(3),

the identity

[E,A]∼

∼

⟨∗E,M⟩∼ //X∼

1

[E∼,A∼]⟨E∼∗,M∼⟩

//

1

X∼

holds, giving a canonical isomorphism

⟨∗E,M⟩∼ ≅ ⟨E∼∗,M∼⟩

. the identity

A∼

1

⟨E∗,M⟩∼ // [E,X]∼

∼

A∼⟨∗E∼,M∼⟩

//

1

[E∼,X∼]

holds, giving a canonical isomorphism

⟨E∗,M⟩∼ ≅ ⟨∗E∼,M∼⟩

.


X

P

P⟨N ⟩Q //A

Q

Y N//

1

B

the identity

X

P

⟨∗E,P⟨N ⟩Q⟩ // [E,A][E,Q]

Y⟨∗E,N ⟩

//

1

[E,B]

, i.e.⟨∗E,P ⟨N ⟩Q⟩ = P ⟨∗E,N ⟩ [E,Q]

, holds. the identity

[E,X][E,P]

⟨E∗,P⟨N ⟩Q⟩ //A

Q

[E,Y]⟨E∗,N ⟩

//

1

B

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4. Frames

, i.e.⟨E∗,P ⟨N ⟩Q⟩ = [E,P] ⟨E∗,N ⟩Q

, holds.

Proof. For any x ∈X and K ∈ [E,A],

(x) ⟨∗E,P ⟨N ⟩Q⟩ (K) =∏∗Ex ⟨P ⟨N ⟩Q⟩K

=∏∗E

(x ⋅ P) ⟨N ⟩ [Q K]

= (x ⋅ P) ⟨∗E,N ⟩ (Q K)= (x ⋅ P) ⟨∗E,N ⟩ ([E,Q] ⋅K)= (x) ⟨P ⟨∗E,N ⟩ [E,Q]⟩ (K)

.

Remark 4.6.8. The cell

X

P

⟨∗E,P⟨N ⟩Q⟩ // [E,A][E,Q]

Y⟨∗E,N ⟩

//

1

[E,B]

resp. [E,X][E,P]

⟨E∗,P⟨N ⟩Q⟩ //A

Q

[E,Y]⟨E∗,N ⟩

//

1

B

sends each cone

∗x

E∆oo

K

XP⟨N ⟩Q

//

α

A

resp. E

K

∆ // ∗a

XP⟨N ⟩Q

//

α

A

to the cone

∗x ⋅ P

E∆oo

Q K

Y N//

α

B

resp. E

K P

∆ // ∗Q ⋅ a

Y N//

α

B


Denition 4.6.9.

If α ∶ ∗EM is a cone and Φ ∶M→ N is a module morphism as in

∗x

E∆oo

K

XM //

NΦ 99

α

A

, then their composite α Φ = Φ α is the cone

∗x

E∆oo

K

X N//

α Φ

A

110

4. Frames


∗Ex ⟨Φ⟩K


∏∗E

x ⟨Φ⟩K ∶∏∗E


x ⟨N ⟩K

. If α ∶ E∗M is a cone and Φ ∶M→ N is a module morphism as in

E

K

∆ // ∗a

XM //

NΦ 99

α

A

, then their composite α Φ = Φ α is the cone

E

K

∆ // ∗a

X N//

α Φ

A


E∗K ⟨Φ⟩a


∏E∗

K ⟨Φ⟩a ∶∏E∗

K ⟨M⟩a→∏E∗

K ⟨N ⟩a

.

Remark 4.6.10. Each component of the cone α Φ is given by

[α Φ]e = αe ⋅ x ⟨Φ⟩ (K ⋅ e)

(cf. Remark 4.3.11). Each component of the cone α Φ is given by

[α Φ]e = αe ⋅ (e ⋅K) ⟨Φ⟩a


Denition 4.6.11. Given a category E and a module morphism Φ ∶M→ N ∶XA, the module morphism

⟨∗E,Φ⟩ ∶ ⟨∗E,M⟩→ ⟨∗E,N ⟩ ∶X [E,A]


(x) ⟨∗E,Φ⟩ (K) =∏∗E

x ⟨Φ⟩K

for each pair of an object x ∈ ∥X∥ and a functor K ∶ E→A.

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4. Frames

the module morphism

⟨E∗,Φ⟩ ∶ ⟨E∗,M⟩→ ⟨E∗,N ⟩ ∶ [E,X]A


(K) ⟨E∗,Φ⟩ (a) =∏E∗

K ⟨Φ⟩a

for each pair of an object a ∈ ∥A∥ and a functor K ∶ E→X.

Remark 4.6.12.(1) The module morphism ⟨∗E,Φ⟩ maps each cone α ∶ x K ∶ ∗E M to the cone α Φ ∶

x K ∶ ∗E N dened in Denition 4.6.9, and dually the module morphism ⟨E∗,Φ⟩ mapseach cone α ∶ K a ∶ E∗M to the cone α Φ ∶ K a ∶ E∗ N .

(2) The assignment Φ→ ⟨∗E,Φ⟩ is functorial; indeed the functor

⟨∗E,−⟩ ∶ [X ∶A]→ [X ∶ [E,A]]

is dened by(x) ⟨∗E,M⟩ (K) =∏

∗Ex ⟨M⟩K

for x ∈X, K ∈ [E,A], and M ∈ [X ∶A]. Dually, the functor

⟨E∗,−⟩ ∶ [X ∶A]→ [[E,X] ∶A]

is dened by(K) ⟨E∗,M⟩ (a) =∏

E∗K ⟨M⟩a

for a ∈A, K ∈ [E,X], and M ∈ [X ∶A].Note. By Remark 1.2.2(3), the following denition is regarded as a special case of Deni-tion 4.6.9, and vice versa.

Denition 4.6.13.

Given a cone α and cell Φ as in

∗x

E∆oo

K

X

P

M //

α

A

Q

Y N//

Φ

B

, their composite α Φ = Φ α is the cone

∗x ⋅ P

E∆oo

Q K

Y N//

α Φ

B


∗Ex ⟨Φ⟩K


∏∗E

x ⟨Φ⟩K ∶∏∗E


x ⟨P ⟨N ⟩Q⟩K =∏∗E

(x ⋅ P) ⟨N ⟩ [Q K]

.

112

4. Frames

Given a cone α and cell Φ as in

E

K

∆ // ∗a

X

P

M //

α

A

Q

Y N//

Φ

B

, their composite α Φ = Φ α is the cone

E

K P

∆ // ∗Q ⋅ a

X N//

α Φ

A


E∗K ⟨Φ⟩a


∏E∗

K ⟨Φ⟩a ∶∏E∗

K ⟨M⟩a→∏E∗

K ⟨P ⟨N ⟩Q⟩a =∏E∗

[K P] ⟨N ⟩ (Q ⋅ a)

.

Remark 4.6.14. Each component of the cone α Φ is given by

[α Φ]e = αe ⋅ x ⟨Φ⟩ (K ⋅ e)

(cf. Remark 4.6.10). Each component of the cone α Φ is given by

[α Φ]e = αe ⋅ (e ⋅K) ⟨Φ⟩a




X

P

M //A

Q

Y N//

Φ

B

the cell

X⟨∗E,M⟩ //

P

[E,A][E,Q]

Y⟨∗E,N ⟩

//

⟨∗E,Φ⟩

[E,B]


⟨∗E,M⟩ ⟨∗E,Φ⟩ // ⟨∗E,P ⟨N ⟩Q⟩ = P ⟨∗E,N ⟩ [E,Q]


113

4. Frames

the cell

[E,X] ⟨E∗,M⟩ //

[E,P]

A

Q

[E,Y]⟨E∗,N ⟩

//

⟨E∗,Φ⟩

B


⟨E∗,M⟩ ⟨E∗,Φ⟩ // ⟨E∗,P ⟨N ⟩Q⟩ = [E,P] ⟨∗E,N ⟩Q


Remark 4.6.16. The cell ⟨∗E,Φ⟩ sends each cone α ∶ x K ∶ ∗E M to the cone α Φ ∶x ⋅P Q K ∶ ∗E N dened in Denition 4.6.13, and dually the cell ⟨E∗,Φ⟩ sends each coneα ∶ K a ∶ E∗M to the cone α Φ ∶ K P Q ⋅ a ∶ E∗ N .

Proposition 4.6.17. The assignment Φ↦ ⟨∗E,Φ⟩ (resp. Φ↦ ⟨E∗,Φ⟩) is functorial.

Proof. The functoriality is veried similarly to Proposition 4.3.18 using Remark 4.6.12(2) andProposition 4.6.7 in place of Remark 4.3.13(2) and Proposition 4.3.8.

Remark 4.6.18.(1) In fact the functoriality of the assignment Φ ↦ ⟨∗E,Φ⟩ (resp. Φ ↦ ⟨E∗,Φ⟩) is reduced to

that of the assignment Φ↦ ⟨E,Φ⟩ by Corollary 4.6.20 below.(2) Given a small category E, the functor

⟨∗E,−⟩ ∶MOD→MOD

is dened by the object function M ↦ ⟨∗E,M⟩ and the arrow function Φ ↦ ⟨∗E,Φ⟩,extending the functor ⟨∗E,−⟩ in Remark 4.6.12(2) as shown in

[X ∶A]

⟨∗E,−⟩ // [X ∶ [E,A]]

MOD⟨∗E,−⟩

//MOD

, where denotes the canonical embedding in Remark 1.2.19(2). Dually, the functor

⟨E∗,−⟩ ∶MOD→MOD

is dened by the object function M ↦ ⟨E∗,M⟩ and the arrow function Φ ↦ ⟨E∗,Φ⟩,extending the functor ⟨E∗,−⟩ in Remark 4.6.12(2) as shown in

[X ∶A]

⟨E∗,−⟩ // [[E,X] ∶A]

MOD⟨E∗,−⟩

//MOD

.

Note. In Remark 4.6.4(1) we saw that a cone ∗E M is regarded as a cylinder E M rightweighted by the functor ∆E. This fact is restated below showing that a module of cones isgiven by a module of cylinders weighted by ∆E (see Denition 4.5.3).

114

4. Frames

Theorem 4.6.19. For a module M ∶XA, the module ⟨∗E,M⟩ ∶X [E,A] is given by the composition

X[∆E,X] // [E,X] ⟨E,M⟩ // [E,A]

; that is,⟨∗E,M⟩ = ⟨∆E M⟩

. the module ⟨E∗,M⟩ ∶ [E,X]A is given by the composition

[E,X] ⟨E,M⟩ // [E,A] A[∆E,A]oo

; that is,⟨E∗,M⟩ = ⟨∆E M⟩

.Similarly, for a module morphism Φ ∶M→ N ∶XA, the module morphism ⟨∗E,Φ⟩ ∶ ⟨∗E,M⟩→ ⟨∗E,N ⟩ ∶X [E,A] is given by the composition

X[∆E,X] // [E,X]

⟨E,M⟩,,

⟨E,N ⟩22⟨E,Φ⟩ [E,A]

; that is,⟨∗E,Φ⟩ = ⟨∆E Φ⟩

. the module morphism ⟨E∗,Φ⟩ ∶ ⟨E∗,M⟩→ ⟨E∗,N ⟩ ∶ [E,X]A is given by the composition

[E,X]⟨E,M⟩

,,

⟨E,N ⟩22⟨E,Φ⟩ [E,A] A

[∆E,A]oo

; that is,⟨E∗,Φ⟩ = ⟨∆E Φ⟩

.


(x) ⟨∗E,M⟩ (K) =∏∗Ex ⟨M⟩K

=∏E

∆E ⟨x ⟨M⟩K⟩ ∗1

=∏E

[∆E x] ⟨M⟩K

= [∆E x] ⟨E,M⟩ (K)= (x ⋅ [∆E,X]) ⟨E,M⟩ (K)= (x) ⟨[∆E,X] ⟨E,M⟩⟩ (K)

(∗1 by Proposition 4.2.4).

115

4. Frames

Corollary 4.6.20.

the postcomposition cell

X⟨∗E,M⟩ //

P

[E,A][E,Q]

Y⟨∗E,N ⟩

//

⟨∗E,Φ⟩

[E,B]

in Denition 4.6.15 is obtained from the postcomposition cell in Denition 4.3.16 by thepasting composition

X[∆E,X] //

P

[E,X][E,P]

⟨E,M⟩ // [E,A][E,Q]

Y[∆E,Y]

// [E,Y]⟨E,N ⟩

//

⟨E,Φ⟩

[E,B]

; that is,⟨∗E,Φ⟩ = ⟨∆E Φ⟩

(see Denition 4.5.7). the postcomposition cell

[E,X] ⟨E∗,M⟩ //

[E,P]

A

Q

[E,Y]⟨E∗,N ⟩

//

⟨E∗,Φ⟩

B


[E,X] ⟨E,M⟩ //

[E,P]

[E,A][E,Q]

A[∆E,A]oo

Q

[E,Y]⟨E,N ⟩

//

⟨E,Φ⟩

[E,B] B[∆E,B]oo

; that is,⟨E∗,Φ⟩ = ⟨∆E Φ⟩


Proof. Immediate from Theorem4.6.19, recalling the denition of the postcomposition cell⟨E,Φ⟩ and the denition of pasting composition.

Remark 4.6.21. By Theorem4.6.19 and Corollary 4.6.20, the compositions in Remark 4.6.6(2),Denition 4.6.9, and Denition 4.6.13 are in fact the same things as those in Remark 4.3.7(2),Denition 4.3.10, and Denition 4.3.14 with the cone α regarded as the cylinder weighted by∆E.

Denition 4.6.22. Given a category E and a cell morphisms τ ∶ Φ→ Ψ ∶M N , the cell morphism

⟨∗E, τ⟩ ∶ ⟨∗E,Φ⟩→ ⟨∗E,Ψ⟩ ∶ ⟨∗E,M⟩ ⟨∗E,N ⟩

, postcomposition with τ , is dened by the pair of natural transformations

τ0 ∶ Φ0 → Ψ0 [E, τ1] ∶ [E,Φ1]→ [E,Ψ1]

, the left component of τ and the postcomposition with the right component of τ (seePreliminary 9).

116

4. Frames

the cell morphism

⟨∗E, τ⟩ ∶ ⟨∗E,Φ⟩→ ⟨∗E,Ψ⟩ ∶ ⟨∗E,M⟩ ⟨∗E,N ⟩

, postcomposition with τ , is dened by the pair of natural transformations

[E, τ0] ∶ [E,Φ0]→ [E,Ψ0] τ1 ∶ Φ1 → Ψ1

, the postcomposition with the left component of τ (see Preliminary 9) and the right com-ponent of τ .

Remark 4.6.23.(1) ⟨∗E, τ⟩ (resp. ⟨∗E, τ⟩) so dened does form a cell morphism. Indeed, given a cone α ∶ x

K ∶ ∗EM (resp. α ∶ K a ∶ E∗M), the commutativity of

x ⋅Φ0

x ⋅ τ0

α ⋅ ⟨∗E,Φ⟩ // [E,Φ1] ⋅K[E,τ1] ⋅K

x ⋅Ψ0α ⋅ ⟨∗E,Ψ⟩

// [E,Ψ1] ⋅K

resp. K ⋅ [E,Φ0]K ⋅ [E,τ0]

α ⋅ ⟨E∗,Φ⟩ // Φ1 ⋅ aτ1 ⋅ a

K ⋅ [E,Ψ0]α ⋅ ⟨E∗,Ψ⟩

// Ψ1 ⋅ a

, i.e.

x ⋅Φ0

x ⋅ τ0

α ⋅Φ // Φ1 K

τ1 K

x ⋅Ψ0 α ⋅Ψ// Ψ1 K

resp. K Φ0

K τ0

α ⋅Φ // Φ1 ⋅ aτ1 ⋅ a

K Ψ0 α ⋅Ψ// Ψ1 ⋅ a


x ⋅Φ0

x ⋅ τ0

αe ⋅Φ // Φ1 ⋅K ⋅ eτ1 ⋅K ⋅ e

x ⋅Ψ0 αe ⋅Ψ// Ψ1 ⋅K ⋅ e

resp. e ⋅K ⋅Φ0

e ⋅K ⋅ τ0

αe ⋅Φ // Φ1 ⋅ aτ1 ⋅ a

e ⋅K ⋅Ψ0 αe ⋅Ψ// Ψ1 ⋅ a

for each object e ∈ ∥E∥.(2) The assignment τ ↦ ⟨∗E, τ⟩ denes the functor

⟨∗E,−⟩ ∶ [M ∶ N ]→ [⟨∗E,M⟩ ∶ ⟨∗E,N ⟩]

; indeed, by the denition of the cell morphism ⟨∗E, τ⟩, the functoriality of ⟨∗E,−⟩ isreduced to that of [E,−] ∶ [M1,N1] → [[E,M1] , [E,N1]]. Dually, the assignment τ ↦⟨E∗, τ⟩ denes the functor

⟨E∗,−⟩ ∶ [M ∶ N ]→ [⟨E∗,M⟩ ∶ ⟨E∗,N ⟩]

.

Denition 4.6.24.

If H is a functor and α is a cone as in

∗x

E∆oo

K

DHoo

X M//

α

A

117

4. Frames

, then their composite H α = α H is the cone

∗x

D∆oo

K H

X M//

H α

A


∗H

x ⟨M⟩K


∏∗H

x ⟨M⟩K ∶∏∗E

x ⟨M⟩K→∏∗D

⟨x ⟨M⟩K⟩H =∏∗D

x ⟨M⟩ [K H]

. If H is a functor and α is a cone as in

DH // E

K

∆ // ∗a

X M//

α

A

, then their composite H α = α H is the cone

D

H K

∆ // ∗a

X M//

H α

A


H∗K ⟨M⟩a


∏H∗

K ⟨M⟩a ∶∏E∗

K ⟨M⟩a→∏D∗

H ⟨K ⟨M⟩a⟩ =∏D∗

[H K] ⟨M⟩a

.

Remark 4.6.25. Each component of the cone H α is given by

[H α]d = α(H ⋅ d)


Denition 4.6.26. Given a functor H ∶D→ E and a moduleM ∶XA, the cell

X⟨∗E,M⟩ //

1

[E,A][H,A]

X⟨∗D,M⟩

//

⟨∗H,M⟩

[D,A]


(x) ⟨∗H,M⟩ (K) =∏∗H

x ⟨M⟩K

for x ∈ ∥X∥ and K a functor E→A.

118

4. Frames

the cell

[E,X] ⟨E∗,M⟩ //

[H,X]

A

1

[D,X]⟨D∗,M⟩

//

⟨H∗,M⟩

A


(K) ⟨H∗,M⟩ (a) =∏H∗

K ⟨M⟩a

for a ∈ ∥A∥ and K a functor E→X.

Remark 4.6.27. The cell ⟨∗H,M⟩ sends each cone α ∶ x K ∶ ∗E M to the cone H α ∶x K H ∶ ∗D M dened in Denition 4.6.24, and dually the cell ⟨H∗,M⟩ sends each coneα ∶ K a ∶ E∗M to the cone H α ∶ H K a ∶D∗M.

Example 4.6.28.

(1) LetM ∶ X A be a module and E be a category. Given a subcategory D of E, precom-

position with the inclusion DDÐ→ E yields

the cell

X⟨∗E,M⟩ //

1

[E,A][D,A]

X⟨∗D,M⟩

//

⟨∗D,M⟩

[D,A]

, restriction to D, which sends each cone α ∶ x K ∶ ∗E M to the cone D α ∶ x K D ∶ ∗DM, α restricted to D.

the cell

[E,X] ⟨E∗,M⟩ //

[D,X]

A

1

[D,X]⟨D∗,M⟩

//

⟨D∗,M⟩

A

, restriction to D, which sends each cone α ∶ K a ∶ E∗ M to the cone D α ∶D K a ∶D∗M, α restricted to D.

(2) LetM ∶XA be a module and E be a category. Given an object e ∈ ∥E∥, precompositionwith the functor e ∶ ∗→ E yields the cell

X⟨∗E,M⟩ //

1

[E,A][e,A]

X M//

⟨∗e,M⟩

A

, evaluation at e, which sends each cone α ∶ x K ∶ ∗E M to the M-arrowαe ∶ x K ⋅ e, the component of α at e.

the cell

[E,X] ⟨E∗,M⟩ //

[e,X]

A

1

X M//

⟨e∗,M⟩

A

, evaluation at e, which sends each cone α ∶ K a ∶ E∗ M to the M-arrowαe ∶ e ⋅K a, the component of α at e.

119

4. Frames

Proposition 4.6.29.

The cell ⟨∗H,M⟩ is obtained from the cell ⟨H,M⟩ in Denition 4.3.28 by the pasting com-position

X[∆E,X] //

1

[E,X] ⟨E,M⟩ //

[H,X]

[E,A][H,A]

X[∆D,X]

// [D,X]⟨D,M⟩

//

⟨H,M⟩

[D,A].

The cell ⟨H∗,M⟩ is obtained from the cell ⟨H,M⟩ in Denition 4.3.28 by the pasting com-position

[E,X] ⟨E,M⟩ //

[H,X]

[E,A][H,A]

A[∆E,A]oo

1

[D,X]⟨D,M⟩

//

⟨H,M⟩

[D,A] A[∆D,A]oo

.

Proof. We need to verify that ⟨∗H,M⟩ = [∆E,X] ⟨H,M⟩. But by Proposition 4.2.13, for anyobject x ∈ ∥X∥ and any functor K ∶ E→A,

(x) ⟨∗H,M⟩ (K) =∏∗H

x ⟨M⟩K

=∏H

∆E ⟨x ⟨M⟩K⟩ ∗1

=∏H

[∆E x] ⟨M⟩K

= [∆E x] ⟨H,M⟩ (K)= (x ⋅ [∆E,X]) ⟨H,M⟩ (K)= (x) ⟨[∆E,X] ⟨H,M⟩⟩ (K)


Proposition 4.6.30. If M is a locally small module, then the assignment H↦ ⟨∗H,M⟩ denes the contravariant functor

⟨∗−,M⟩ ∶Cat∼ →MOD

. the assignment H↦ ⟨H∗,M⟩ denes the contravariant functor

⟨−∗,M⟩ ∶Cat∼ →MOD

.

Proof. Since the cell ⟨∗H,M⟩ is obtained from the cell ⟨H,M⟩ by the pasting composition inProposition 4.6.29, the functoriality of the assignment H ↦ ⟨∗H,M⟩ is reduced to that of theassignment H↦ ⟨H,M⟩ (see Proposition 4.3.30) by virtue of Proposition 1.2.34.

Theorem 4.6.31.

There is a functor⟨∗−,−⟩ ∶Cat∼ ×MOD→MOD

such that

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4. Frames

(1) for each small category E, ⟨∗E,−⟩ ∶MOD →MOD coincides with the functor in Re-mark 4.6.18(2).

(2) for each locally small moduleM, ⟨∗−,M⟩ ∶Cat∼ →MOD coincides with the functor inProposition 4.6.30.

There is a functor⟨−∗,−⟩ ∶Cat∼ ×MOD→MOD

such that(1) for each small category E, ⟨E∗,−⟩ ∶MOD →MOD coincides with the functor in Re-

mark 4.6.18(2).(2) for each locally small moduleM, ⟨−∗,M⟩ ∶Cat∼ →MOD coincides with the functor in

Proposition 4.6.30.

Proof. Similar to the proof of Theorem4.3.33.

Note. In Remark 4.6.4(2) we saw that a cone is a special instance of an extraordinary cylinder.This fact is restated below showing that a module of cones is given by a module of extraordinarycylinders composed with a diagonal functor.

Theorem 4.6.32. For a module M ∶XA, the module ⟨∗E,M⟩ ∶X [E,A] is given by the composition

X⟨E,M⟩ // [E∼ ×E,A] [E,A][∆E∼×E,A]oo

; that is,⟨∗E,M⟩ = ⟨E,M⟩ [∆E∼ ×E,A]

. the module ⟨E∗,M⟩ ∶ [E,X]A is given by the composition

[E,X] [∆E∼×E,X] // [E∼ ×E,X] ⟨E,M⟩ //A

; that is,⟨E∗,M⟩ = [∆E∼ ×E,X] ⟨E,M⟩

.Similarly, for a module morphism Φ ∶M→ N ∶XA, the module morphism ⟨∗E,Φ⟩ ∶ ⟨∗E,M⟩→ ⟨∗E,N ⟩ ∶X [E,A] is given by the composition

X

⟨E,M⟩--

⟨E,N ⟩11⟨E,Φ⟩ [E∼ ×E,A] [E,A][∆E∼×E,A]oo

; that is,⟨∗E,Φ⟩ = ⟨E,Φ⟩ [∆E∼ ×E,A]

. the module morphism ⟨E∗,Φ⟩ ∶ ⟨E∗,M⟩→ ⟨E∗,N ⟩ ∶ [E,X]A is given by the composition

[E,X] [∆E∼×E,X] // [E∼ ×E,X]⟨E,M⟩

++

⟨E,N ⟩

33⟨E,Φ⟩ A

; that is,⟨E∗,Φ⟩ = [∆E∼ ×E,X] ⟨E,Φ⟩

.

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4. Frames


(x) ⟨∗E,M⟩ (K) =∏∗Ex ⟨M⟩K

=∏E

⟨x ⟨M⟩K⟩ [∆E∼ ×E] ∗1

=∏E

x ⟨M⟩ [K [∆E∼ ×E]]

= (x) ⟨E,M⟩ (K [∆E∼ ×E])= (x) ⟨E,M⟩ ([∆E∼ ×E,A] ⋅K)= (x) ⟨⟨E,M⟩ [∆E∼ ×E,A]⟩ (K)


Corollary 4.6.33.

the postcomposition cell

X⟨∗E,M⟩ //

P

[E,A][E,Q]

Y⟨∗E,N ⟩

//

⟨∗E,Φ⟩

[E,B]


X⟨E,M⟩ //

P

[E∼ ×E,A][E∼×E,Q]

[E,A][∆E∼×E,A]oo

[E,Q]

Y⟨E,N ⟩

//

⟨E,Φ⟩

[E∼ ×E,B] [E,B][∆E∼×E,B]oo

. the postcomposition cell

[E,X] ⟨E∗,M⟩ //

[E,P]

A

Q

[E,Y]⟨E∗,N ⟩

//

⟨E∗,Φ⟩

B


[E,X] [∆E∼×E,X] //

[E,P]

[E∼ ×E,X][E∼×E,P]

⟨E,M⟩ //A

Q

[E,Y][∆E∼×E,Y]

// [E∼ ×E,Y]⟨E,N ⟩

//

⟨E,Φ⟩

B

.

Proof. Immediate from Theorem4.6.32, recalling the denition of the postcomposition cell⟨E,Φ⟩ and the denition of pasting composition.

122

4. Frames

4.7. Bicylinders

Denition 4.7.1. A cylinder from a product category is called a bicylinder. Given a pair ofcategories E and D, and a moduleM ∶XA, the following types of bicylinders E ×DMare considered: ordinary

E ×DS

yy

T

%%X M

//

α

A

, extraordinary

∗x

[E ×D]∼ × [E ×D]K

[E ×D]∼ × [E ×D]K

∗a

X M//

α

A X M//

α

A

, and complexE

S

E ×D∼ ×D

K

E ×D∼ ×D

K

E

T

X M//

α

A X M//

α

A

.

Remark 4.7.2.(1)

A complex bicylinder α along a module M ∶ X A from a functor S ∶ E → X to abifunctor K ∶ E×D∼×D→A is dened by a cylindrical frame α of the composite moduleS ⟨M⟩K ∶ E E×D∼ ×D (regarded as an endomodule E×D E×D). The module ofcomplex bicylinder E ×DM,

⟨E ×D,M⟩ ∶ [E,X] [E ×D∼ ×D,A]

, is dened by(S) ⟨E ×D,M⟩ (K) = ∏

E×DS ⟨M⟩K

for S ∈ [E,X] and K ∈ [E ×D∼ ×D,A]. A complex bicylinder α along a moduleM ∶XA from a bifunctor K ∶ E×D∼×D→X

to a functor T ∶ E → A is dened by a cylindrical frame α of the composite moduleK ⟨M⟩T ∶ E ×D∼ ×D E (regarded as an endomodule E ×D E ×D). The moduleof complex bicylinder E ×DM,

⟨E ×D,M⟩ ∶ [E ×D∼ ×D,X] [E,A]

, is dened by(K) ⟨E ×D,M⟩ (T ) = ∏

E×DK ⟨M⟩T

for K ∈ [E ×D∼ ×D,X] and T ∈ [E,A].(2) All bicylinders in the denition above is thus given by a cylindrical frame of an endomodule

E×D E×D. The same notation ⟨E ×D,M⟩ is used to denote the following modules ofbicylinders:

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4. Frames

Module Bicylinder⟨E ×D,M⟩ ∶ [E ×D,X] [E ×D,A] Ordinary⟨E ×D,M⟩ ∶X [[E ×D]∼ × [E ×D] ,A] Extraordinary (left)⟨E ×D,M⟩ ∶ [[E ×D]∼ × [E ×D] ,X]A Extraordinary (right)⟨E ×D,M⟩ ∶ [E,X] [E ×D∼ ×D,A] Complex (left)⟨E ×D,M⟩ ∶ [E ×D∼ ×D,X] [E,A] Complex (right)

(3) An ordinary bicylinder along the hom of a category is just a natural transformation betweenbifunctors.

Denition 4.7.3. The right and left exponential transposes of a bicylinder

E ×DS

yy

T

%%X M

//

α

A

are the cylinders

DS⊣

yyT⊣

%%

ES⊢

yyT⊢

%%[E,X]

⟨E,M⟩//

α⊣

[E,A] [D,X]⟨D,M⟩

//

α⊢

[D,A]

dened by[α⊣d]e = α(e,d) = [α⊢e]d

for e ∈ ∥E∥ and d ∈ ∥D∥.Remark 4.7.4.(1) By Theorem4.1.4, α⊣ and α⊢ and all of their components do form cylinders, and the right

and left exponential transpositions form the iso cells

[E ×D,X] ⟨E×D,M⟩ //

⊣

[E ×D,A]⊣

[E ×D,X] ⟨E×D,M⟩ //

⊢

[E ×D,A]⊢

[D, [E,X]]⟨D,⟨E,M⟩⟩

//

⊣

[D, [E,A]] [E, [D,X]]⟨E,⟨D,M⟩⟩

//

⊢

[E, [D,A]]

, natural in E, D, andM.(2) The right and left exponential transposes of an extraordinary bicylinder is dened in the

same way. The right and left exponential transposition for extraordinary bicylinders E ×DM form the iso cells

X⟨E×D,M⟩ //

1

[[E ×D]∼ × [E ×D] ,A]⊣

[[E ×D]∼ × [E ×D] ,X]⊣

⟨E×D,M⟩ //A

1

X⟨D,⟨E,M⟩⟩

//

⊣

[D∼ ×D, [E∼ ×E,A]] [D∼ ×D, [E∼ ×E,X]]⟨D,⟨E,M⟩⟩

//

⊣

A

and

X⟨E×D,M⟩ //

1

[[E ×D]∼ × [E ×D] ,A]⊣

[[E ×D]∼ × [E ×D] ,X]⊣

⟨E×D,M⟩ //A

1

X⟨E,⟨D,M⟩⟩

//

⊣

[E∼ ×E, [D∼ ×D,A]] [E∼ ×E, [D∼ ×D,X]]⟨E,⟨D,M⟩⟩

//

⊣

A

.

124

4. Frames

(3) The right and left exponential transposes of a complex bicylinder is also dened in thesame way. Since they play an important role in Section 12.2, we treat them separatelybelow.

Denition 4.7.5.

The right and left exponential transposes of a bicylinder

E

S

E ×D∼ ×D

K

X M//

α

A

are the cylinders

∗S

D∼ ×D

K⊣

E

S

E

K⊢

Eoo

[E,X]⟨E,M⟩

//

α⊣

[E,A] X⟨D,M⟩

//

α⊢

[D∼ ×D,A]


for e ∈ ∥E∥ and d ∈ ∥D∥. The left slice of α at e ∈ ∥E∥ is the cylinder

∗S(e)

D∼ ×D

K⊢(e)

X M//

[α⊢]e

A

given by the component of the cylinder α⊢ at e. The right and left exponential transposes of a bicylinder

E ×D∼ ×D

K

E

T

X M//

α

A

are the cylinders

D∼ ×D

K⊣

∗T

E

K⊢

E // E

T

[E,X]⟨E,M⟩

//

α⊣

[E,A] [D∼ ×D,X]⟨D,M⟩

//

α⊢

A


for e ∈ ∥E∥ and d ∈ ∥D∥. The left slice of α at e ∈ ∥E∥ is the cylinder

D∼ ×D

K⊢(e)

∗T(e)

X M//

[α⊢]e

A

given by the component of the cylinder α⊢ at e.

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4. Frames

Remark 4.7.6. The right and left exponential transpositions form the iso cells

[E,X] ⟨E×D,M⟩ //

1

[E ×D∼ ×D,A]⊣

[E ×D∼ ×D,X] ⟨E×D,M⟩ //

⊣

[E,A]1

[E,X]⟨D,⟨E,M⟩⟩

//

⊣

[D∼ ×D, [E,A]] [D∼ ×D, [E,X]]⟨D,⟨E,M⟩⟩

//

⊣

[E,A]

and

[E,X] ⟨E×D,M⟩ //

1

[E ×D∼ ×D,A]⊢

[E ×D∼ ×D,X] ⟨E×D,M⟩ //

⊢

[E,A]1

[E,X]⟨E,⟨D,M⟩⟩

//

⊢

[E, [D∼ ×D,A]] [E, [D∼ ×D,X]]⟨E,⟨D,M⟩⟩

//

⊢

[E,A]

, natural in E, D, andM.

Denition 4.7.7.

The transpose of a cylinder∗

S

D∼ ×D

K

[E,X]⟨E,M⟩

//

α

[E,A]

is the cylinder

E

S

E

K⊺

Eoo

X⟨D,M⟩

//

α⊺

[D∼ ×D,A]

dened by[α⊺e]d = [αd]e

for e ∈ ∥E∥ and d ∈ ∥D∥. The slice of α at e ∈ ∥E∥ is the cylinder

∗S(e)

D∼ ×D

K⊺(e)

X M//

[α⊺]e

A

given by the component of the cylinder α⊺ at e. The transpose of a cylinder

D∼ ×D

K

∗T

[E,X]⟨E,M⟩

//

α

[E,A]

is the cylinder

E

K⊺

E // E

T

[D∼ ×D,X]⟨D,M⟩

//

α⊺

A


126

4. Frames

for e ∈ ∥E∥ and d ∈ ∥D∥. The slice of α at e ∈ ∥E∥ is the cylinder

D∼ ×D

K⊺(e)

∗T(e)

X M//

[α⊺]e

A

given by the component of the cylinder α⊺ at e.

Remark 4.7.8.(1) The transposition α ↦ α⊺ forms the iso cells

[E,X] ⟨D,⟨E,M⟩⟩ //

1

[D∼ ×D, [E,A]]⊺

[D∼ ×D, [E,X]] ⟨D,⟨E,M⟩⟩ //

⊺

[E,A]1

[E,X]⟨E,⟨D,M⟩⟩

//

⊺

[E, [D∼ ×D,A]] [E, [D∼ ×D,X]]⟨E,⟨D,M⟩⟩

//

⊺

[E,A]

, natural in E, D andM, making the diagram

⟨E ×D,M⟩⊣

ww⊢''

⟨D, ⟨E,M⟩⟩ ⊺// ⟨E, ⟨D,M⟩⟩

commute, where ⊣ and ⊢ are the cells in Remark 4.7.6.(2) The diagram

⟨D, ⟨E,M⟩⟩

⟨D,⟨e,M⟩⟩ &&

⊺ // ⟨E, ⟨D,M⟩⟩

⟨e,⟨D,M⟩⟩xx⟨D,M⟩

commutes, where ⟨e,M⟩ and ⟨e, ⟨D,M⟩⟩ are evaluations at e ∈ ∥E∥ (see Example 4.3.31).Hence the postcomposition

∗S

D∼ ×D

K

D∼ ×D

K

∗T

[E,X][e,X]

⟨E,M⟩ //

α

[E,A][e,A]

[E,X][e,X]

⟨E,M⟩ //

α

[E,A][e,A]

X M//

⟨e,M⟩

A X M//

⟨e,M⟩

A

of α with the evaluation ⟨e,M⟩ yields the same cylinder as the one given by the componentof α⊺ at e, i.e. the slice of α at e (cf. Preliminary 16).

4.8. Wedges

Note. A combination of a cone and a cylinder is called a wedge2. Dened below are represen-tative instances of wedges.

2This terminology diers from the literature, which uses the term wedge to refer to what this book calls anextraordinary cylinder.

127

4. Frames

Denition 4.8.1. Let E and D be categories andM ∶XA be a module. Given a functor S ∶ E→X and a bifunctor K ∶ E×D→A, a wedge α from S to K alongM,

writtenα ∶ S K ∶ E × ∗DM

, is a cylinder

E

S

E ×D

K

E×∆Doo

X M//

α

A

right weighted by the projection E ×∆D. Given a functor T ∶ E → A and a bifunctor K ∶ E ×D → X, a wedge α from K to T alongM, written

α ∶ K T ∶ E ×D∗M, is a cylinder

E ×D

K

E×∆D // E

T

X M//

α

A

left weighted by the projection E ×∆D.

Remark 4.8.2.(1) Just like a cone is a special instance of a cylinder, a wedge is a special instance of a

bicylinder.(2) A wedge α ∶ S K ∶ E×∗DM is also dened by a bicylinder α ∶ S K [E ×∆D∼ ×D] ∶

E ×DM, i.e. by a bicylinder

E

S

E ×D∼ ×D

K

X M//

α

A

with a dummy variable varying over D∼ introduced to K.(3) A wedge also arises as a combination of a cone and an extraordinary cylinder. For example,

given an object x ∈ ∥X∥ and a bifunctor K ∶ E∼ ×E ×D→A, an extraordinary wedge

α ∶ x K ∶ E × ∗DM

is dened by one of the following bicylinders:

D

∆D

E∼ ×E ×D

K

∗

x

E∼ ×D∼ ×E ×D

E×∆D∼×E×D

∗x

E∼ ×E ×D

K

X M//

α

A X M//

α

A

Denition 4.8.3. Let E and D be categories andM ∶XA be a module.

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4. Frames

The module of wedges E × ∗DM,

⟨E × ∗D,M⟩ ∶ [E,X] [E ×D,A]


[E,X] [E×∆D,X] // [E ×D,X] ⟨E×D,M⟩ // [E ×D,A]

; that is,⟨E × ∗D,M⟩ ∶= ⟨E ×∆D M⟩

(see Denition 4.5.3). The module of wedges E ×D∗M,

⟨E ×D∗,M⟩ ∶ [E ×D,X] [E,A]


[E ×D,X] ⟨E×D,M⟩ // [E ×D,A] [E,A][E×∆D,A]oo

; that is,⟨E ×D∗,M⟩ ∶= ⟨E ×∆D M⟩


Remark 4.8.4.(1)

For a functor S ∶ E → X and a bifunctor K ∶ E ×D → A, the set (S) ⟨E × ∗D,M⟩ (K)consists of all wedges S K ∶ E × ∗DM.

For a functor T ∶ E → A and a bifunctor K ∶ E ×D → X, the set (K) ⟨E ×D∗,M⟩ (T)consists of all wedges K T ∶ E ×D∗M.

(2) The module ⟨E × ∗D,M⟩ is alternatively dened by the composition

[E,X] ⟨E×D,M⟩ // [E ×D∼ ×D,A] [E ×D,A][E×∆D∼×D,A]oo

(see Remark 4.8.2(2)).(3) The module of extraordinary wedges

⟨E × ∗D,M⟩ ∶X [E∼ ×E ×D,A]

(see Remark 4.8.2(3)) is dened by the composition

X[∆D,X] // [D,X] ⟨E×D,M⟩ // [E∼ ×E ×D,A]

, or by the composition

X⟨E×D,M⟩ // [E∼ ×D∼ ×E ×D,A] [E∼ ×E ×D,A][E∼×∆D∼×E×D,A]oo

.

Note. The following denition is analogous to Denition 4.7.5.

Denition 4.8.5.

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4. Frames

The right and left exponential transposes of a wedge

E

S

E ×D

K

E×∆Doo

X M//

α

A

are the cone and the cylinder

∗S

D

K⊣

∆Doo E

S

E

K⊢

Eoo

[E,X]⟨E,M⟩

//

α⊣

[E,A] X⟨∗D,M⟩

//

α⊢

[D,A]


for e ∈ ∥E∥ and d ∈ ∥D∥. The left slice of α at e ∈ ∥E∥ is the cone

∗S(e)

D

K⊢(e)

∆Doo

X M//

[α⊢]e

A

given by the component of the cylinder α⊢ at e. The right and left exponential transposes of a wedge

E ×D

K

E×∆D // E

T

X M//

α

A

are the cone and the cylinder

D

K⊣

∆D // ∗T

E

K⊢

E // E

T

[E,X]⟨E,M⟩

//

α⊣

[E,A] [D,X]⟨D∗,M⟩

//

α⊢

A


for e ∈ ∥E∥ and d ∈ ∥D∥. The left slice of α at e ∈ ∥E∥ is the cone

D

K⊢(e)

∆D // ∗T(e)

X M//

[α⊢]e

A

given by the component of the cylinder α⊢ at e.

Remark 4.8.6.

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4. Frames

The right and left exponential transpositions of wedges E × ∗DM form the iso cells

[E,X] ⟨E×∗D,M⟩ //

1

[E ×D,A]⊣

[E,X] ⟨E×∗D,M⟩ //

1

[E ×D,A]⊢

[E,X]⟨∗D,⟨E,M⟩⟩

//

⊣

[D, [E,A]] [E,X]⟨E,⟨∗D,M⟩⟩

//

⊢

[E, [D,A]]

, natural in E, D, and M. In fact, these iso cells are obtained from the iso cells in Re-mark 4.7.4(1) by pasting a commutative diagram of diagonal functors (see Preliminary 18)as shown in

[E,X] [E×∆D,X] //

1

[E ×D,X] ⟨E×D,M⟩ //

⊣

[E ×D,A]⊣

[E,X][∆D,[E,X]]

// [D, [E,X]]⟨D,⟨E,M⟩⟩

//

⊣

[D, [E,A]]

[E,X] [E×∆D,X] //

1

[E ×D,X] ⟨E×D,M⟩ //

⊢

[E ×D,A]⊢

[E,X][E,[∆D,X]]

// [E, [D,X]]⟨E,⟨D,M⟩⟩

//

⊢

[E, [D,A]].

The right and left exponential transpositions of wedges E ×D∗M form the iso cells

[E ×D,X] ⟨E×D∗,M⟩ //

⊣

[E,A]1

[E ×D,X] ⟨E×D∗,M⟩ //

⊢

[E,A]1

[D, [E,X]]⟨D∗,⟨E,M⟩⟩

//

⊣

[E,A] [E, [D,X]]⟨E,⟨D∗,M⟩⟩

//

⊢

[E,A]

, natural in E, D, and M. In fact, these iso cells are obtained from the iso cells in Re-mark 4.7.4(1) by pasting a commutative diagram of diagonal functors (see Preliminary 18)as shown in

[E ×D,X] ⟨E×D,M⟩ //

⊣

[E ×D,A]⊣

[E,A][E×∆D,A]oo

1

[D, [E,X]]⟨D,⟨E,M⟩⟩

//

⊣

[D, [E,A]] [E,A][∆D,[E,A]]oo

[E ×D,X] ⟨E×D,M⟩ //

⊢

[E ×D,A]⊢

[E,A][E×∆D,A]oo

1

[E, [D,X]]⟨E,⟨D,M⟩⟩

//

⊢

[E, [D,A]] [E,A][E,[∆D,A]]oo

.

Note. The following denition is analogous to Denition 4.7.7.

Denition 4.8.7.

The transpose of a cone

∗S

D

K

∆Doo

[E,X]⟨E,M⟩

//

α

[E,A]

131

4. Frames

is the cylinder

E

S

E

K⊺

Eoo

X⟨∗D,M⟩

//

α⊺

[D,A]


for e ∈ ∥E∥ and d ∈ ∥D∥. The slice of α at e ∈ ∥E∥ is the cone

∗S(e)

D

K⊺(e)

∆Doo

X M//

[α⊺]e

A

given by the component of the cylinder α⊺ at e. The transpose of a cone

D

K

∆D // ∗T

[E,X]⟨E,M⟩

//

α

[E,A]

is the cylinder

E

K⊺

E // E

T

[D,X]⟨D∗,M⟩

//

α⊺

A


for e ∈ ∥E∥ and d ∈ ∥D∥. The slice of α at e ∈ ∥E∥ is the cone

D

K⊺(e)

∆D // ∗T(e)

X M//

[α⊺]e

A

given by the component of the cylinder α⊺ at e.

Remark 4.8.8.(1) The transposition α ↦ α⊺ forms

the iso cell

[E,X] ⟨∗D,⟨E,M⟩⟩ //

1

[D, [E,A]]⊺

[E,X]⟨E,⟨∗D,M⟩⟩

//

⊺

[E, [D,A]]

, natural in E, D, andM, making the diagram

⟨E × ∗D,M⟩⊣

vv⊢

((⟨∗D, ⟨E,M⟩⟩ ⊺

// ⟨E, ⟨∗D,M⟩⟩

commute, where ⊣ and ⊢ are the cells in Remark 4.8.6.

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4. Frames

the iso cell

[D, [E,X]] ⟨D∗,⟨E,M⟩⟩ //

⊺

[E,A]1

[E, [D,X]]⟨E,⟨D∗,M⟩⟩

//

⊺

[E,A]

, natural in E, D, andM, making the diagram

⟨E ×D∗,M⟩⊣

vv⊢

((⟨D∗, ⟨E,M⟩⟩ ⊺

// ⟨E, ⟨D∗,M⟩⟩

commute, where ⊣ and ⊢ are the cells in Remark 4.8.6.(2)

The diagram

⟨∗D, ⟨E,M⟩⟩

⟨∗D,⟨e,M⟩⟩ ''

⊺ // ⟨E, ⟨∗D,M⟩⟩

⟨e,⟨∗D,M⟩⟩ww⟨∗D,M⟩

commutes, where ⟨e,M⟩ and ⟨e, ⟨∗D,M⟩⟩ are evaluations at e ∈ ∥E∥ (see Exam-ple 4.3.31); the postcomposition

∗S

D

K⊺

∆Doo

[E,X][e,X]

⟨E,M⟩ //

α

[E,A][e,A]

X M//

⟨e,M⟩

A

of α with the evaluation ⟨e,M⟩ thus yields the same cone as the one given by thecomponent of α⊺ at e, i.e. the slice of α at e (cf. Preliminary 16).

The diagram

⟨D∗, ⟨E,M⟩⟩

⟨D∗,⟨e,M⟩⟩ ''

⊺ // ⟨E, ⟨D∗,M⟩⟩

⟨e,⟨D∗,M⟩⟩ww⟨D∗,M⟩

commutes, where ⟨e,M⟩ and ⟨e, ⟨D∗,M⟩⟩ are evaluations at e ∈ ∥E∥ (see Exam-ple 4.3.31); the postcomposition

D

K⊺

∆D // ∗T

[E,X][e,X]

⟨E,M⟩ //

α

[E,A][e,A]

X M//

⟨e,M⟩

A

of α with the evaluation ⟨e,M⟩ thus yields the same cone as the one given by thecomponent of α⊺ at e, i.e. the slice of α at e (cf. Preliminary 16).

133

4. Frames

4.9. Cones and wedges in a category

Notation 4.9.1. Given a functor K ∶ E→C and an object c ∈ ∥C∥, a cone α from c to K is denoted by

α ∶ c K ∶ ∗E→C

. a cone α from K to c is denoted by

α ∶ K c ∶ E∗→C

.

Remark 4.9.2.(1) By Example 4.2.3, a cone in a category C is just a special instance of a cone in Deni-

tion 4.6.3 where M is the hom of C. Conversely, a cone along a module M is the samething as a cone in the collage category ⌊M⌋ (cf. Remark 4.3.4(4)).

(2) By Remark 4.6.4(1), A cone α ∶ c K ∶ ∗E→C is the same thing as a natural transformation

∗c

E∆oo

K

C⟨C⟩

//

α

C

right weighted by ∆E. A cone α ∶ K c ∶ E∗→C is the same thing as a natural transformation

E

K

∆ // ∗c

C⟨C⟩

//

α

C

left weighted by ∆E.

Note. The module of cones in a category is dened as below as a special case of Denition 4.6.5whereM is given by the hom of a category.

Denition 4.9.3. Given categories E and C, the module of cones ∗E→C,

⟨∗E,C⟩ ∶C [E,C], is dened by

(c) ⟨∗E,C⟩ (K) =∏∗Ec ⟨C⟩K

for c ∈C and K ∈ [E,C]; that is,

⟨∗E,C⟩ ∶= ⟨∗E, ⟨C⟩⟩

.

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4. Frames

the module of cones E∗→C,⟨E∗,C⟩ ∶ [E,C]C

, is dened by(K) ⟨E∗,C⟩ (c) =∏

E∗K ⟨C⟩ c

for c ∈C and K ∈ [E,C]; that is,

⟨E∗,C⟩ ∶= ⟨E∗, ⟨C⟩⟩

.

Remark 4.9.4.(1) For an object c ∈ ∥C∥ and a functor K ∶ E→C,

the set (c) ⟨∗E,C⟩ (K) consists of all cones c K ∶ ∗E→C. the set (K) ⟨E∗,C⟩ (c) consists of all cones K c ∶ E∗→C.

(2) By applying Theorem4.6.19 to the hom endomodule ⟨C⟩ and noting Remark 4.5.4(3), wehave

⟨∗E,C⟩ = ⟨∆E C⟩ ⟨E∗,C⟩ = ⟨∆E C⟩; hence the module ⟨∗E,C⟩ ∶C [E,C] is given by the composition

C[∆E,C] // [E,C] ⟨E,C⟩ // [E,C]

, i.e. by the representable module of the diagonal functor [∆E,C]. the module ⟨E∗,C⟩ ∶ [E,C]C is given by the composition

[E,C] ⟨E,C⟩ // [E,C] C[∆E,C]oo

, i.e. by the corepresentable module of the diagonal functor [∆E,C].Note. The following denition is a special case of Denition 4.6.15 where Φ is given by the homof a functor.

Denition 4.9.5. Given a category E and a functor H ∶C→ B, the cell

C⟨∗E,C⟩ //

H

[E,C][E,H]

B⟨∗E,B⟩

//

⟨∗E,H⟩

[E,B]

, postcomposition with H, is dened by the module morphism

⟨∗E, ⟨C⟩⟩ ⟨∗E,⟨H⟩⟩ // ⟨∗E,H ⟨B⟩H⟩ = [H] ⟨∗E, ⟨B⟩⟩ [E,H]

, postcomposition (see Denition 4.6.11) with the hom of H; that is,

⟨∗E,H⟩ ∶= ⟨∗E, ⟨H⟩⟩

.

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4. Frames

the cell

[E,C] ⟨E∗,C⟩ //

[E,H]

C

H

[E,B]⟨E∗,B⟩

//

⟨E∗,H⟩

B

, postcomposition with H, is dened by the module morphism

⟨E∗, ⟨C⟩⟩ ⟨E∗,⟨H⟩⟩ // ⟨E∗,H ⟨B⟩H⟩ = [E,H] ⟨E∗, ⟨B⟩⟩ [H]

, postcomposition (see Denition 4.6.11) with the hom of H; that is,

⟨E∗,H⟩ ∶= ⟨E∗, ⟨H⟩⟩

.

Remark 4.9.6.(1) The cell ⟨∗E,H⟩ sends each cone α ∶ c K ∶ ∗E→C to the cone αH ∶ c⋅H HK ∶ ∗E→ B,

the usual composite of a cone and a functor, and dually the cell ⟨E∗,H⟩ sends each coneα ∶ K c ∶ E∗→C to the cone α H ∶ K H H ⋅ c ∶ E∗→ B.

(2) By replacing Φ in Corollary 4.6.20 with the hom cell ⟨H⟩ and noting Remark 4.5.8(3), wehave

⟨∗E,H⟩ = ⟨∆E H⟩ ⟨E∗,H⟩ = ⟨∆E H⟩; hence ⟨∗E,H⟩ and ⟨E∗,H⟩ are obtained from the hom of the postcomposition functor[E,C] by the pasting compositions

C[∆E,C] //

H

[E,C][E,H]

⟨E,C⟩ // [E,C][E,H]

[E,C] ⟨E,C⟩ //

[E,H]

[E,C][E,H]

C[∆E,C]oo

H

B[∆E,B]

// [E,B]⟨E,B⟩

//

⟨E,H⟩

[E,B] [E,B]⟨E,B⟩

//

⟨E,H⟩

[E,B] B[∆E,B]oo

.


Denition 4.9.7. Given a functor H ∶D→ E and a category C, the cell

C⟨∗E,C⟩ //

1

[E,C][H,C]

C⟨∗D,C⟩

//

⟨∗H,C⟩

[D,C]


(c) ⟨∗H,C⟩ (K) =∏∗H

c ⟨C⟩K

for c ∈ ∥C∥ and K a functor E→C; that is,

⟨∗H,C⟩ ∶= ⟨∗H, ⟨C⟩⟩

.

136

4. Frames

the cell

[E,C] ⟨E∗,C⟩ //

[H,C]

C

1

[D,C]⟨D∗,C⟩

//

⟨H∗,C⟩

C


(K) ⟨H∗,C⟩ (c) =∏H∗

K ⟨C⟩c

for c ∈ ∥C∥ and K a functor E→C; that is,

⟨H∗,C⟩ ∶= ⟨H∗, ⟨C⟩⟩

.

Remark 4.9.8.(1) The cell ⟨∗H,C⟩ sends each cone α ∶ x K ∶ ∗E→C to the cone H α ∶ x K H ∶ ∗D→C,

the usual composite of a functor and a cone, and dually the cell ⟨H∗,C⟩ sends each coneα ∶ K a ∶ E∗→C to the cone H α ∶ H K a ∶D∗→C.

(2) ReplacingM in Proposition 4.6.29 with the hom endomodule ⟨C⟩ and noting that ⟨H, ⟨C⟩⟩ =⟨H,C⟩ (see Remark 4.3.29(3)), we see that ⟨∗H,C⟩ and ⟨H∗,C⟩ are obtained from the homof the precomposition functor [H,C] by the pasting compositions

C[∆E,C] //

1

[E,C] ⟨E,C⟩ //

[H,C]

[E,C][H,C]

[E,C] ⟨E,C⟩ //

[H,C]

[E,C][H,C]

C[∆E,C]oo

1

C[∆D,C]

// [D,C]⟨D,C⟩

//

⟨H,C⟩

[D,C] [D,C]⟨D,C⟩

//

⟨H,C⟩

[D,C] C[∆D,C]oo

.

Example 4.9.9.

(1) Let C and E be categories. Given a subcategory D of E, precomposition with the inclusion

DDÐ→ E yields

the cell

C⟨∗E,C⟩ //

1

[E,C][D,C]

C⟨∗D,C⟩

//

⟨∗D,C⟩

[D,C]

, restriction to D, which sends each cone α ∶ c K ∶ ∗E → C to the cone D α ∶ c K D ∶ ∗D→C, α restricted to D.

the cell

[E,C] ⟨E∗,C⟩ //

[D,C]

C

1

[D,C]⟨D∗,C⟩

//

⟨D∗,C⟩

C

, restriction to D, which sends each cone α ∶ K c ∶ E∗→C to the cone D α ∶D Kc ∶D∗→C, α restricted to D.

137

4. Frames

(2) Let C and E be categories. Given an object e ∈ ∥E∥, precomposition with the functore ∶ ∗→ E yields the cell

C⟨∗E,C⟩ //

1

[E,C][e,C]

C⟨C⟩

//

⟨∗e,C⟩

C

, evaluation at e, which sends each cone α ∶ c K ∶ ∗E→C to theC-arrow αe ∶ c→ K⋅e,the component of α at e.

the cell

[E,C] ⟨E∗,C⟩ //

[e,C]

C

1

C⟨C⟩

//

⟨e∗,C⟩

C

, evaluation at e, which sends each cone α ∶ K c ∶ E∗→C to theC-arrow αe ∶ e⋅K→ c,the component of α at e.

Note. A wedge in a category is dened as below as a special case of Denition 4.8.1 whereMis given by the hom of a category.

Denition 4.9.10. Let E, D, and C be categories. Given a functor S ∶ E→C and a bifunctor K ∶ E ×D→C, a wedge α from S to K, written

α ∶ S K ∶ E × ∗D→C

, is a natural transformation

E

S

E ×D

K

E×∆Doo

C⟨C⟩

//

α

C

right weighted by the projection E ×∆D. Given a functor T ∶ E→C and a bifunctor K ∶ E ×D→C, a wedge α from K to T, written

α ∶ K T ∶ E ×D∗→C

, is a natural transformation

E ×D

K

E×∆D // E

T

C⟨C⟩

//

α

C

left weighted by the projection E ×∆D.

Remark 4.9.11. A wedge in a category C dened above is just a special instance of a wedge inDenition 4.8.1 whereM is the hom of C. Conversely, a wedge along a moduleM is the samething as a wedge in the collage category ⌊M⌋ (cf. Remark 4.3.4(4)).

Note. The module of wedges in a category is dened as below as a special case of Denition 4.8.3whereM is given by the hom of a category.

Denition 4.9.12. Let E, D, and C be categories.

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4. Frames

The module of wedges E × ∗D→C,

⟨E × ∗D,C⟩ ∶ [E,C] [E ×D,C]

, is dened by⟨E × ∗D,C⟩ = ⟨E ×∆D C⟩ = ⟨E ×∆D ⟨C⟩⟩

(see Remark 4.5.4(3)); that is,

⟨E × ∗D,C⟩ ∶= ⟨E × ∗D, ⟨C⟩⟩

. The module of wedges E ×D∗→C,

⟨E ×D∗,C⟩ ∶ [E ×D,C] [E,C]

, is dened by⟨E ×D∗,C⟩ = ⟨E ×∆D C⟩ = ⟨E ×∆D ⟨C⟩⟩

(see Remark 4.5.4(3)); that is,

⟨E ×D∗,C⟩ ∶= ⟨E ×D∗, ⟨C⟩⟩

.

Remark 4.9.13.(1)

For a functor S ∶ E → C and a bifunctor K ∶ E ×D → C, the set (S) ⟨E × ∗D,C⟩ (K)consists of all wedges S K ∶ E × ∗D→C.

For a functor T ∶ E → C and a bifunctor K ∶ E ×D → C, the set (K) ⟨E ×D∗,C⟩ (T)consists of all wedges K T ∶ E ×D∗→C.

(2) By Remark 4.5.4(3), the module ⟨E × ∗D,C⟩ ∶ [E,C] [E ×D,C] is given by the composition

[E,C] [E×∆D,C] // [E ×D,C] ⟨E×D,C⟩ // [E ×D,C]

; that is, ⟨E × ∗D,C⟩ is the representable module of the diagonal functor [E ×∆D,C]. the module ⟨E ×D∗,C⟩ ∶ [E ×D,C] [E,C] is given by the composition

[E ×D,C] ⟨E×D,C⟩ // [E ×D,C] [E,C][E×∆D,C]oo

; that is, ⟨E ×D∗,C⟩ is the corepresentable module of the diagonal functor [E ×∆D,C].(3) The exponential transpositions of a wedge in a category is dened as a special case of

Denition 4.8.5 whereM is given by the hom of a category. The right and left exponential transpositions of wedges E × ∗D→C form the iso cells

[E,C] ⟨E×∗D,C⟩//

1

[E ×D,C]⊣

[E,C] ⟨E×∗D,C⟩//

1

[E ×D,C]⊢

[E,C]⟨∗D,⟨E,C⟩⟩

//

⊣

[D, [E,C]] [E,C]⟨E,⟨∗D,C⟩⟩

//

⊢

[E, [D,C]]

139

4. Frames

, natural in E, D, and C. In fact, these iso cells are obtained from the homs of the func-

tors [E ×D,C] ⊣Ð→ [D, [E,C]] and [E ×D,C] ⊢Ð→ [E, [D,C]] by pasting a commutativediagram of diagonal functors (see Preliminary 18) as shown in

[E,C] [E×∆D,C] //

1

[E ×D,C] ⟨E×D,C⟩ //

⊣

[E ×D,C]⊣

[E,C][∆D,[E,C]]

// [D, [E,C]]⟨D,⟨E,C⟩⟩

//

⟨⊣⟩

[D, [E,C]]

[E,C] [E×∆D,C] //

1

[E ×D,C] ⟨E×D,C⟩ //

⊢

[E ×D,C]⊢

[E,C][E,[∆D,C]]

// [E, [D,C]]⟨E,⟨D,C⟩⟩

//

⟨⊢⟩

[E, [D,C]].

The right and left exponential transpositions of wedges E ×D∗→C form the iso cells

[E ×D,C] ⟨E×D∗,C⟩ //

⊣

[E,C]1

[E ×D,C] ⟨E×D∗,C⟩ //

⊢

[E,C]1

[D, [E,C]]⟨D∗,⟨E,C⟩⟩

//

⊣

[E,C] [E, [D,C]]⟨E,⟨D∗,C⟩⟩

//

⊢

[E,C]

, natural in E, D, and C. In fact, these iso cells are obtained from the homs of the func-

tors [E ×D,C] ⊣Ð→ [D, [E,C]] and [E ×D,C] ⊢Ð→ [E, [D,C]] by pasting a commutativediagram of diagonal functors (see Preliminary 18) as shown in

[E ×D,C] ⟨E×D,C⟩ //

⊣

[E ×D,C]⊣

[E,C][E×∆D,C]oo

1

[D, [E,C]]⟨D,⟨E,C⟩⟩

//

⟨⊣⟩

[D, [E,C]] [E,C][∆D,[E,C]]oo

[E ×D,C] ⟨E×D,C⟩ //

⊢

[E ×D,C]⊢

[E,C][E×∆D,C]oo

1

[E, [D,C]]⟨E,⟨D,C⟩⟩

//

⟨⊢⟩

[E, [D,C]] [E,C][E,[∆D,C]]oo

.

4.10. Cones and wedges in Set

Notation 4.10.1.(1)

A cone α from a set S to a left module M ∶ ∗ E (i.e. a functor M ∶ E → Set) isdenoted by

α ∶ S M ∶ ∗ ∗Erather than α ∶ S M ∶ ∗E → Set, and the component at e ∈ ∥E∥ is written as⟨α⟩e ∶ S → ⟨M⟩e. The module of cones ∗E→ Set is denoted by

⟨∶ ∗E⟩ ∶ Set [∶ E]

rather than ⟨∗E,Set⟩ ∶ Set [E,Set].

140

4. Frames

A cone α from a set S to a right module M ∶ E ∗ (i.e. a functor M ∶ E∼ → Set) isdenoted by

α ∶ S M ∶ E∗ ∗rather than α ∶ S M ∶ ∗E∼ → Set, and the component at e ∈ ∥E∥ is written ase ⟨α⟩ ∶ S → e ⟨M⟩. The module of cones ∗E∼ → Set is denoted by

⟨E∗ ∶⟩ ∶ Set [E ∶]

rather than ⟨∗E∼,Set⟩ ∶ Set [E∼,Set].(2)

A wedge α from a right module L ∶ X ∗ (i.e. a functor L ∶ X∼ → Set) to a moduleM ∶X E (i.e. a functorM ∶X∼ ×E→ Set) is denoted by

α ∶ LM ∶X ∗E

rather than α ∶ L M ∶ X∼ × ∗E → Set, and the component at (x,e) ∈ ∥X ×E∥ iswritten as x ⟨α⟩e ∶ x ⟨L⟩ → x ⟨M⟩e. The module of wedges X∼ × ∗E → Set is denotedby

⟨X ∶ ∗E⟩ ∶ [X ∶] [X ∶ E]rather than ⟨X∼ × ∗E,Set⟩ ∶ [X∼,Set] [X∼ ×E,Set].

A wedge α from a left module L ∶ ∗ A (i.e. a functor L ∶ A → Set) to a moduleM ∶ EA (i.e. a functorM ∶ E∼ ×A→ Set) is denoted by

α ∶ LM ∶ E∗A

rather than α ∶ L M ∶ ∗E∼ ×A → Set, and the component at (e,a) ∈ ∥E ×A∥ iswritten as e ⟨α⟩a ∶ ⟨L⟩a → e ⟨M⟩a. The module of wedges ∗E∼ ×A → Set is denotedby

⟨E∗ ∶A⟩ ∶ [∶A] [E ∶A]rather than ⟨∗E∼ ×A,Set⟩ ∶ [A,Set] [E∼ ×A,Set].

Remark 4.10.2.(1) By Remark 4.9.4(2),

⟨∶ ∗E⟩ ∶ Set [∶ E] is the representable module of the diagonal functor [∶ ∆E] ∶ Set →[∶ E] (see Example 1.1.23(7)); that is, ⟨∶ ∗E⟩ is given by the composition

Set[∶∆E] // [∶ E] ⟨∶E⟩ // [∶ E]

, and a cone α ∶ S M ∶ ∗ ∗E is the same thing as a left module morphismα ∶ ⟨S⟩∆E M ∶ ∗ E.

⟨E∗ ∶⟩ ∶ Set [E ∶] is the representable module of the diagonal functor [∆E ∶] ∶ Set →[E ∶] (see Example 1.1.23(7)); that is, ⟨E∗ ∶⟩ is given by the composition

Set[∆E∶] // [E ∶] ⟨E∶⟩ // [E ∶]

, and a cone α ∶ S M ∶ E∗ ∗ is the same thing as a right module morphismα ∶ ∆E ⟨S⟩M ∶ E ∗.

(2) By Remark 4.9.13(2),

141

4. Frames

⟨X ∶ ∗E⟩ ∶ [X ∶] [X ∶ E] is the representable module of the diagonal functor [X ∶ ∆E] ∶[X ∶]→ [X ∶ E] (see Example 1.1.23(7)); that is, ⟨X ∶ ∗E⟩ is given by the composition

[X ∶] [X∶∆E] // [X ∶ E] ⟨X∶E⟩ // [X ∶ E]

, and a wedge α ∶ L M ∶ X ∗E is the same thing as a module morphism α ∶⟨L⟩∆E →M ∶X E.

⟨E∗ ∶A⟩ ∶ [∶A] [E ∶A] is the representable module of the diagonal functor [∆E ∶A] ∶[∶A]→ [E ∶A] (see Example 1.1.23(7)); that is, ⟨E∗ ∶A⟩ is given by the composition

[∶A] [∆E∶A] // [E ∶A] ⟨E∶A⟩ // [E ∶A]

, and a wedge α ∶ L M ∶ E∗ A is the same thing as a module morphism α ∶∆E ⟨L⟩→M ∶ EA.

Notation 4.10.3. The right and left exponential transpositions of wedges X∼ × ∗E→ Set are denoted by

[X ∶]1

⟨X∶∗E⟩ // [X ∶ E]

[X ∶]1

⟨X∶∗E⟩ // [X ∶ E]

[X ∶]⟨∗E,⟨X∶⟩⟩

//

[E, [X ∶]] [X ∶]⟨X∼,⟨∶∗E⟩⟩

//

[X∼, [∶ E]]

rather than

[X∼,Set] ⟨X∼×∗E,Set⟩//

1

[X∼ ×E,Set]⊣

[X∼,Set] ⟨X∼×∗E,Set⟩//

1

[X∼ ×E,Set]⊢

[X∼,Set]⟨∗E,⟨X∼,Set⟩⟩

//

⊣

[E, [X∼,Set]] [X∼,Set]⟨X∼,⟨∗E,Set⟩⟩

//

⊢

[X∼, [E,Set]].

The right and left exponential transpositions of wedges ∗E∼ ×A→ Set are denoted by

[∶A]1

⟨E∗∶A⟩ // [E ∶A]

[∶A] ⟨E∗∶A⟩ //

1

[E ∶A]

[∶A]⟨A,⟨E∗∶⟩⟩

//

[A, [E ∶]] [∶A]⟨∗E∼,⟨∶A⟩⟩

//

[E∼, [∶A]]

rather than

[A,Set] ⟨∗E∼×A,Set⟩//

1

[E∼ ×A,Set]⊣

[A,Set] ⟨∗E∼×A,Set⟩//

1

[E∼ ×A,Set]⊢

[A,Set]⟨A,⟨∗E∼,Set⟩⟩

//

⊣

[A, [E∼,Set]] [A,Set]⟨∗E∼,⟨A,Set⟩⟩

//

⊢

[E∼, [A,Set]]

. The cell ⟨E∗ ∶A⟩ // ⟨∗E∼, ⟨∶A⟩⟩ often appears in the opposite form

[E ∶A]∼ ⟨E∗∶A⟩∼ //

[∶A]∼

1

[E, [∶A]∼]⟨E∗,⟨∶A⟩∼⟩

//

[∶A]∼

(note that ⟨∗E∼, ⟨∶A⟩⟩∼ ≅ ⟨E∗, ⟨∶A⟩∼⟩ by Remark 4.6.6(4)).

142

4. Frames

Remark 4.10.4. By Remark 4.9.13(3),

the iso cell ⟨X ∶ ∗E⟩ // ⟨∗E, ⟨X ∶⟩⟩ is obtained from the hom of the functor [X ∶ E] Ð→[E, [X ∶]] by pasting a commutative diagram of diagonal functors in Remark 2.1.2(5) asshown in

[X ∶] [X∶∆E] //

1

[X ∶ E]

⟨X∶E⟩ // [X ∶ E]

[X ∶][∆E,[X∶]]

// [E, [X ∶]]⟨E,[X∶]⟩

//

⟨⟩

[E, [X ∶]]

. Likewise, the iso cell ⟨X ∶ ∗E⟩ // ⟨X∼, ⟨∶ ∗E⟩⟩ is obtained from the hom of the functor

[X ∶ E] Ð→ [X∼, [∶ E]] by the pasting composition

[X ∶] [X∶∆E] //

1

[X ∶ E]

⟨X∶E⟩ // [X ∶ E]

[X ∶][X∼,[∶∆E]]

// [X∼, [∶ E]]⟨X∼,[∶E]⟩

//

⟨⟩

[X∼, [∶ E]]

.

the iso cell ⟨E∗ ∶A⟩ // ⟨A, ⟨E∗ ∶⟩⟩ is obtained from the hom of the functor [E ∶A] Ð→[A, [E ∶]] by pasting a commutative diagram of diagonal functors in Remark 2.1.2(5) asshown in

[∶A]1

[∆E∶A] // [E ∶A]

⟨E∶A⟩ // [E ∶A]

[∶A][A,[∆E∶]]

// [A, [E ∶]]⟨A,[E∶]⟩

//

⟨⟩

[A, [E ∶]]

. Likewise, the iso cell ⟨E∗ ∶A⟩∼ // ⟨E∗, ⟨∶A⟩∼⟩ is obtained from the hom of the functor

[E ∶A]∼ Ð→ [E, [∶A]∼] by the pasting composition

[E ∶A]∼ ⟨E∶A⟩∼ //

[E ∶A]∼

[∶A]∼[∆E∶A]oo

1

[E, [∶A]∼]⟨E,[∶A]∼⟩

//

⟨⟩

[E, [∶A]∼] [∶A]∼[∆E,[∶A]∼]oo

.

143

5. Yoneda Lemma

5.1. Yoneda modules

Denition 5.1.1.

The right Yoneda module for a category X is the module

⟨Xd⟩ ∶X [X ∶]

given by the evaluation(x,M)↦ x ⟨M⟩ ∶X∼ × [X ∶]→ Set

; that is,(x) ⟨Xd⟩ (M) ∶= x ⟨M⟩

for x ∈X and M ∈ [X ∶]. The left Yoneda module for a category A is the module

⟨eA⟩ ∶ [∶A]∼ A

given by the evaluation(M,a)↦ ⟨M⟩a ∶ [∶A] ×A→ Set

; that is,(M) ⟨eA⟩ (a) ∶= ⟨M⟩a

for a ∈A and M ∈ [∶A].Remark 5.1.2.(1) The module Xd (resp. eA) is called the Yoneda module just because it is represented

(resp. corepresented) by the Yoneda functor X (resp. A) (see Theorem5.2.15).(2) For a right module M ∶ X ∗ and an object x ∈ ∥X∥, the set (x) ⟨Xd⟩ (M) consists of

all M-arrows x ∗, and for a left module M ∶ ∗ A and an object a ∈ ∥A∥, the set(M) ⟨eA⟩ (a) consists of allM-arrows ∗ a.

Proposition 5.1.3.

The right exponential transpose of the right Yoneda module ⟨Xd⟩ ∶X [X ∶] is the identity[X ∶]→ [X ∶]; that is,

[X ∶] ⟨Xd⟩ = 1 // [X ∶]; the right slice of Xd at a right module M ∶X ∗ is M itself:

⟨Xd⟩ (M) =M

The left exponential transpose of the left Yoneda module ⟨eA⟩ ∶ [∶A]∼ A is the identity[∶A]→ [∶A]; that is,

[∶A] ⟨eA⟩ = 1 // [∶A]; the left slice of eA at a left module M ∶ ∗A is M itself:

(M) ⟨eA⟩ =M

144

5. Yoneda Lemma

Proof. Immediate from the denition.

Proposition 5.1.4. Given a module (resp. module morphism) M ∶XA, the identity

M = ⟨Xd⟩ [M]holds; that is, M is recovered from its right exponential transpose by the composition

XXd // [X ∶] A

Moo

. Hence the right action of the right Yoneda module Xd on the functor category [A, [X ∶]]yields the inverse of the right exponential transposition [X ∶A] Ð→ [A, [X ∶]]; that is,

[X ∶A] [A, [X ∶]]⟨Xd⟩A =oo

. the identity

M = [M] ⟨eA⟩holds; that is, M is recovered from its left exponential transpose by the composition

XM // [∶A]∼ eA //A

. Hence the left action of the left Yoneda module eA on the functor category [X, [∶A]∼]yields the inverse of the left exponential transposition [X ∶A]∼ Ð→ [X, [∶A]∼]; that is,

[X ∶A]∼ [X, [∶A]∼]X⟨eA⟩ =oo

.

Proof. By the bijectiveness of exponential transposition, it suces to show that

M = ⟨⟨Xd⟩ [M]⟩

. But by Proposition 2.1.6 and Proposition 5.1.3,

⟨⟨Xd⟩ [M]⟩ = [⟨Xd⟩] [M] =M

.

Remark 5.1.5. For a moduleM ∶XA, the identities in Proposition 5.1.4 are expressed by

X

1

M //A

M

X

M

M //A

1

XXd

//

1

[X ∶] [∶A]∼ eA//

1

A

. These fully faithful cells identify each M-arrow m ∶ x a with the ⟨M⟩a-arrow m ∶ x ∗and with the x ⟨M⟩-arrow m ∶ ∗ a respectively (see Remark 5.1.2(2)).

Denition 5.1.6. Given categories X and A,

145

5. Yoneda Lemma

the right general Yoneda module for [A,X],

⟨XdA⟩ ∶ [A,X] [X ∶A]

, is dened by(G) ⟨XdA⟩ (M) =∏

A

G ⟨M⟩

for G ∈ [A,X] and M ∈ [X ∶A]. the left general Yoneda module for [X,A],

⟨XeA⟩ ∶ [X ∶A]∼ [X,A]

, is dened by(M) ⟨XeA⟩ (F ) =∏

X

⟨M⟩F

for M ∈ [X ∶A] and F ∈ [X,A].

Remark 5.1.7.(1) The module XdA (resp. XeA) is called the general Yoneda module just because it is

represented (resp. corepresented) by the general Yoneda functor XA (resp. XA) (seeTheorem5.3.16).

(2) For a module M ∶ X A and a functor G ∶ A → X, the set (G) ⟨XdA⟩ (M) consists ofall right cylinders G M, and for a moduleM ∶X A and a functor F ∶X →A, the set(M) ⟨XeA⟩ (F) consists of all left cylindersM F.

(3) The identication in Remark 4.3.2(3) yields canonical isomorphisms

⟨Xd⟩ ≅ ⟨Xd∗⟩

and⟨eA⟩ ≅ ⟨∗eA⟩

. Yoneda modules are thus special instances of general Yoneda modules.

Theorem 5.1.8. Given a category E and a module M ∶XA, the identities

[E,X]1

⟨E,M⟩ // [E,A]ME

[E,X]EM

⟨E,M⟩ // [E,A]1

[E,X]XdE

//

1

[X ∶ E] [E ∶A]∼EeA

//

1

[E,A]

hold.

Proof. Immediate from Denition 4.3.6 and Denition 5.1.6; indeed, for any S ∈ [E,X] and anyT ∈ [E,A],

S ⟨E,M⟩T =∏E

S ⟨M⟩T

=∏E

S ⟨⟨M⟩T ⟩ =∏E

⟨S ⟨M⟩⟩T

= S ⟨XdE⟩ (⟨M⟩T ) = (S ⟨M⟩) ⟨EeA⟩T= S ⟨XdE⟩ ([ME] ⋅ T ) = (S ⋅ [EM]) ⟨EeA⟩T= S ⟨⟨XdE⟩ [ME]⟩T = S ⟨[EM] ⟨EeA⟩⟩T

.

146

5. Yoneda Lemma

Remark 5.1.9.(1) This is the identity stated in Remark 4.3.4(3); the fully faithful cells in Theorem5.1.8

identify each two-sided cylinder

ES

T

##X M

//

α

A

with the right cylinder

X⟨M⟩Tα // ESoo

and with the left cylinder

ES⟨M⟩ //

T//α A

respectively.(2) If we replace E with the terminal category, then we obtain the cells in Remark 5.1.5.

Corollary 5.1.10. Given categories X and A, the identities

[A,X]1

XdA // [X ∶A]

[X ∶A]∼

XeA //

1

[X,A]1

[A,X]⟨A,⟨Xd⟩⟩

//

1

[A, [X ∶]] [X, [∶A]∼]⟨X,⟨eA⟩⟩

// [X,A]

hold.

Proof. ReplacingM in Theorem5.1.8 with the right and left Yoneda modules, we have

[A,X]1

⟨A,⟨Xd⟩⟩ // [A, [X ∶]]⟨Xd⟩A

[X, [∶A]∼]X⟨eA⟩

⟨X,⟨eA⟩⟩ // [X,A]1

[A,X]⟨XdA⟩

//

1

[X ∶A] [X ∶A]∼XeA

//

1

[X,A]

. Since ⟨Xd⟩A = and X⟨eA⟩ = by Proposition 5.1.4, the assertion follows by takingthe inverses.

Remark 5.1.11. The iso cell ⟨XdA⟩ ⟨A, ⟨Xd⟩⟩ identies each right cylinder

XM

//α AGoo

with the two-sided cylinderA

G

||

M$$

X⟨Xd⟩

//

α

[X ∶].

147

5. Yoneda Lemma

The iso cell ⟨XeA⟩ ⟨X, ⟨eA⟩⟩ identies each left cylinder

XM //

F//α A

with the two-sided cylinderX

Myy

F

""[∶A]∼

⟨eA⟩//

α

A

.

5.2. Yoneda morphisms

Denition 5.2.1. LetM ∶XA be a module. The right hom ofM is the module

⟨XM⟩ ∶X ⌊M⌋, from X to the collage category ofM, given by the composition

XM0 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋

; that is, ⟨XM⟩ is the representable module of the inclusionM0 ∶X→ ⌊M⌋; in short,

⟨XM⟩ ∶=M0 ⟨⌊M⌋⟩.

The left hom ofM is the module

⟨MA⟩ ∶ ⌊M⌋A

, from the collage category ofM to A, given by the composition

⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ AM1oo

; that is, ⟨MA⟩ is the corepresentable module of the inclusionM1 ∶A→ ⌊M⌋; in short,

⟨MA⟩ ∶= ⟨⌊M⌋⟩M1

.

Remark 5.2.2. The composition

X⟨XM⟩ // ⌊M⌋ X

M0oo

, i.e.

XM0 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ X

M0oo

, yields ⟨X⟩, the hom of X, and the composition

X⟨XM⟩ // ⌊M⌋ A

M1oo

, i.e.

XM0 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ A

M1oo

, yieldsM (see Remark 3.1.16(2)).

148

5. Yoneda Lemma

The composition

AM1 // ⌊M⌋ ⟨MA⟩ //A

, i.e.

AM1 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ A

M1oo

, yields ⟨A⟩, the hom of A, and the composition

XM0 // ⌊M⌋ ⟨MA⟩ //A

, i.e.

XM0 // ⌊M⌋ ⟨⌊M⌋⟩ // ⌊M⌋ A

M1oo

, yieldsM (see Remark 3.1.16(2)).

Denition 5.2.3. LetM ∶XA be a module. The right exponential transpose

[⟨XM⟩] ∶ ⌊M⌋→ [X ∶]

of the right hom ofM is called the right Yoneda functor forM. The left exponential transpose

[⟨MA⟩] ∶ ⌊M⌋→ [∶A]∼

of the left hom ofM is called the left Yoneda functor forM.

Remark 5.2.4. By Remark 5.2.2, the right slices of ⟨XM⟩ at s ∈ ∥X∥ and t ∈ ∥A∥ are given by

⟨XM⟩ s = ⟨XM⟩ (M0 ⋅ s) = ⟨⟨XM⟩M0⟩ s = ⟨X⟩ s

and⟨XM⟩ t = ⟨XM⟩ (M1 ⋅ t) = ⟨⟨XM⟩M1⟩ t = ⟨M⟩ t

; hence the right Yoneda functor ⟨XM⟩ sends each M-arrow m ∶ s t to the rightmodule morphism

⟨XM⟩m ∶ ⟨X⟩ s→ ⟨M⟩ t ∶X ∗which maps each X-arrow h ∶ x→ s to theM-arrow h m ∶ x t as indicated in

x

h

h ⋅ ⟨XM⟩m

%%s m

// t

(cf. Remark 2.1.2(1)). WhenM is understood, ⟨XM⟩m is also written as

Xm ∶ ⟨X⟩ s→ ⟨M⟩ t ∶X ∗

and called the right module morphism generated by X direct along m.

149

5. Yoneda Lemma

By Remark 5.2.2, the left slices of ⟨MA⟩ at s ∈ ∥X∥ and t ∈ ∥A∥ are given by

s ⟨MA⟩ = (s ⋅M0) ⟨MA⟩ = s ⟨M0 ⟨MA⟩⟩ = s ⟨M⟩

andt ⟨MA⟩ = (t ⋅M1) ⟨MA⟩ = t ⟨M1 ⟨MA⟩⟩ = t ⟨A⟩

; hence the left Yoneda functor ⟨MA⟩ sends eachM-arrow m ∶ s t to the left modulemorphism

m ⟨MA⟩ ∶ t ⟨A⟩→ s ⟨M⟩ ∶ ∗A

which maps each A-arrow h ∶ t→ a to theM-arrow m h ∶ s a as indicated in

sm //

m⟨MA⟩ ⋅ h %%

t

ha

(cf. Remark 2.1.2(1)). WhenM is understood, m ⟨MA⟩ is also written as

mA ∶ t ⟨A⟩→ s ⟨M⟩ ∶ ∗A

and called the left module morphism generated by A inverse along m.

Denition 5.2.5. LetM ∶XA be a module. The right Yoneda morphism forM is the cell

X

X

M //A

M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

sending eachM-arrow m ∶ s t to the right module morphism

Xm = ⟨XM⟩m ∶ ⟨X⟩ s→ ⟨M⟩ t ∶X ∗

. The left Yoneda morphism forM is the cell

X

M

M //A

A

[∶A]∼⟨∶A⟩∼

//

⟨MA⟩

[∶A]∼

sending eachM-arrow m ∶ s t to the left module morphism

mA = m ⟨MA⟩ ∶ t ⟨A⟩→ s ⟨M⟩ ∶ ∗A

.

Remark 5.2.6.(1)

150

5. Yoneda Lemma

The cell ⟨XM⟩ is formally given by the adjunct (see Theorem3.1.17) of the rightYoneda functor forM, i.e. by the composition

X

M0

M //A

M1⌊M⌋

⟨XM⟩

⟨⌊M⌋⟩ //

IM

⌊M⌋⟨XM⟩

[X ∶]⟨X∶⟩

//

⟨⟨XM⟩⟩

[X ∶]

of the unit cell ofM and the hom of the right Yoneda functor forM. The cell ⟨MA⟩ is formally given by the adjunct (see Theorem3.1.17) of the left

Yoneda functor forM, i.e. by the composition

X

M0

M //A

M1⌊M⌋

⟨MA⟩

⟨⌊M⌋⟩ //

IM

⌊M⌋⟨MA⟩

[∶A]∼⟨∶A⟩∼

//

⟨⟨MA⟩⟩

[∶A]∼

of the unit cell ofM and the hom of the left Yoneda functor forM.(2)

The Yoneda morphism

X

X

M // ∗M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

for a right module M ∶ X ∗ is dened as a special case of Denition 5.2.5 where Ais the terminal category under the identication [X ∶] ≅ [X ∶ ∗]. This conical cell sendseachM-arrow m ∶ r ∗ to the right module morphism

Xm ∶ ⟨X⟩ r→M ∶X ∗

which maps each X-arrow h ∶ x→ r to theM-arrow h m ∶ x ∗ as indicated in

x

h

h ⋅ ⟨Xm⟩

%%r m

// ∗

. Conversely, given an arrow m ∶ s t of a two-sided module M ∶ X A, the rightmodule morphism

Xm ∶ ⟨X⟩ s→ ⟨M⟩ t ∶X ∗coincides with that generated by X direct along the arrow m ∶ s ∗ of the right module⟨M⟩ t ∶X ∗.

The Yoneda morphism

∗M

M //A

A

[∶A]∼⟨∶A⟩∼

//

⟨MA⟩

[∶A]∼

151

5. Yoneda Lemma

for a left module M ∶ ∗ A is dened as a special case of Denition 5.2.5 where X isthe terminal category under the identication [∶A] ≅ [∗ ∶A]. This conical cell sendseachM-arrow m ∶ ∗ r to the left module morphism

mA ∶ r ⟨A⟩→M ∶ ∗A

which maps each A-arrow h ∶ r→ a to theM-arrow m h ∶ ∗ a as indicated in

∗ m //

⟨mA⟩ ⋅ h %%

r

ha

. Conversely, given an arrow m ∶ s t of a two-sided module M ∶ X A, the leftmodule morphism

mA ∶ t ⟨A⟩→ s ⟨M⟩ ∶ ∗A

coincides with that generated by A inverse along the arrow m ∶ ∗ t of the left modules ⟨M⟩ ∶ ∗A.

Proposition 5.2.7. Let M ∶XA be a module. The right slice (see Denition 2.1.8)

X

X

⟨M⟩t // ∗⟨M⟩t

[X ∶]⟨X∶⟩

//

⟨XM⟩t

[X ∶]

at t ∈ ∥A∥ of the right Yoneda morphism for M is given by the Yoneda morphism

X

X

⟨M⟩t // ∗⟨M⟩t

[X ∶]⟨X∶⟩

//

⟨X⟨⟨M⟩t⟩⟩

[X ∶]

for the right module ⟨M⟩ t, the right slice of M at t. The left slice (see Denition 2.1.8)

∗s⟨M⟩

s⟨M⟩ //A

A

[∶A]∼⟨∶A⟩∼

//

s⟨MA⟩

[∶A]∼

at s ∈ ∥X∥ of the left Yoneda morphism for M is given by the Yoneda morphism

∗s⟨M⟩

s⟨M⟩ //A

A

[∶A]∼⟨∶A⟩∼

//

⟨⟨s⟨M⟩⟩A⟩

[∶A]∼

for the left module s ⟨M⟩, the left slice of M at s.

152

5. Yoneda Lemma

Proof. Immediate from Remark 5.2.6(2).

Proposition 5.2.8.

Given a category X, the right Yoneda morphism

X

X

⟨X⟩ //X

X

[X ∶]⟨X∶⟩

//

⟨X⟨X⟩⟩

[X ∶]

for the hom of X is the same thing as the hom

X

X

⟨X⟩ //X

X

[X ∶]⟨X∶⟩

//

⟨X⟩

[X ∶]

of the right Yoneda functor for X; that is, for any X-arrow f ∶ s → t, the right modulemorphism

Xf ∶ ⟨X⟩ s→ ⟨X⟩ t ∶X ∗coincides with

⟨X⟩ f ∶ ⟨X⟩ s→ ⟨X⟩ t ∶X ∗.

Given a category A, the left Yoneda morphism

A

A

⟨A⟩ //A

A

[∶A]∼⟨∶A⟩∼

//

⟨⟨A⟩A⟩

[∶A]∼

for the hom of A is the same thing as the hom

A

A

⟨A⟩ //A

A

[∶A]∼⟨∶A⟩∼

//

⟨A⟩

[∶A]∼

of the left Yoneda functor for A; that is, for any A-arrow f ∶ s→ t, the left module morphism

fA ∶ t ⟨A⟩→ s ⟨A⟩ ∶ ∗A

coincides withf ⟨A⟩ ∶ t ⟨A⟩→ s ⟨A⟩ ∶ ∗A

.

Proof. Both Xf and ⟨X⟩ f map each X-arrow h ∶ x → s to the X-arrow h f ∶ x → t (seeRemark 2.3.2).

Example 5.2.9.

153

5. Yoneda Lemma

(1) Given a module and functors as in

ES //X

M //A DToo

the right Yoneda morphism for the composite module S ⟨M⟩T ∶ E D sends eachS ⟨M⟩T-arrow m ∶ s t to the right module morphism

Em ∶ ⟨E⟩ s→ ⟨S ⟨M⟩T⟩ t ∶ E ∗

which maps each E-arrow h ∶ e→ s to the S ⟨M⟩T-arrow h m ∶ e t as indicated in

e

h

e ⋅ Sh ⋅ S

h ⋅ ⟨Em⟩

''s s ⋅ S m

// T ⋅ t

(cf. Example 1.1.23(1)). The commutativity of the triangle above for every h ∈ ⟨E⟩ stranslates into the commutativity of

⟨E⟩ s Em //

⟨S⟩s

⟨S ⟨M⟩T⟩ t = S ⟨M⟩ (T ⋅ t)

⟨S ⟨X⟩S⟩ s = S ⟨X⟩ (S ⋅ s)S⟨Xm⟩

33

. the left Yoneda morphism for the composite module S ⟨M⟩T ∶ E D sends each

S ⟨M⟩T-arrow m ∶ s t to the left module morphism

mD ∶ t ⟨D⟩→ s ⟨S ⟨M⟩T⟩ ∶ ∗D

which maps each D-arrow h ∶ t→ d to the S ⟨M⟩T-arrow m h ∶ s d as indicated in

s ⋅ S m //

⟨mD⟩ ⋅ h ''

T ⋅ tT ⋅ h

t

h

T ⋅ d d

(cf. Example 1.1.23(1)). The commutativity of the triangle above for every h ∈ t ⟨D⟩translates into the commutativity of

t ⟨D⟩ mD //

t⟨T⟩

s ⟨S ⟨M⟩T⟩ = (s ⋅ S) ⟨M⟩T

t ⟨T ⟨A⟩T⟩ = (t ⋅T) ⟨A⟩T⟨mA⟩T

33

.(2) As a special case of (1) above,

consider a representable module F ⟨E⟩ ∶D E given by

DF // E

⟨E⟩ // E

. The right Yoneda morphism for F ⟨E⟩ sends each F ⟨E⟩-arrow f ∶ s t to the rightmodule morphism

Df ∶ ⟨D⟩ s→ F ⟨E⟩ t ∶D ∗

154

5. Yoneda Lemma

which maps each D-arrow h ∶ d→ s to the F ⟨E⟩-arrow h f ∶ d t as indicated in

d

h

d ⋅ Fh ⋅ F

h ⋅ ⟨Df⟩

&&s s ⋅ F

f// t

. The commutativity of the triangle above for every h ∈ ⟨D⟩ s translates into the com-mutativity of

⟨D⟩ s Df //

⟨F⟩s

F ⟨E⟩ t

⟨F ⟨E⟩F⟩ s = F ⟨E⟩ (F ⋅ s)F⟨E⟩f

55

. consider a corepresentable module ⟨E⟩F ∶ ED given by

E⟨E⟩ // E D

Foo

. The left Yoneda morphism for ⟨E⟩F sends each ⟨E⟩F-arrow f ∶ s t to the left modulemorphism

fD ∶ t ⟨D⟩→ s ⟨E⟩F ∶ ∗D

which maps each D-arrow h ∶ t→ d to the ⟨E⟩F-arrow f h ∶ s d as indicated in

sf //

⟨fD⟩ ⋅ h &&

F ⋅ tF ⋅ h

t

h

F ⋅ d d

. The commutativity of the triangle above for every h ∈ t ⟨D⟩ translates into the com-mutativity of

t ⟨D⟩ fD //

t⟨F⟩

s ⟨E⟩F

t ⟨F ⟨E⟩F⟩ = (t ⋅ F) ⟨E⟩Ff⟨E⟩F

55

.

Theorem 5.2.10. (Yoneda Lemma : Part one). Given a right module M ∶ X ∗ and an object r ∈ ∥X∥, the assignment m ↦ Xm yields a

bijectionr ⟨M⟩ ≅ (⟨X⟩ r) ⟨X ∶⟩ (M)

from the set of M-arrows r ∗ to the set of right module morphisms ⟨X⟩ r →M ∶ X ∗,whose inverse sends each right module morphism Φ ∶ ⟨X⟩ r→M to theM-arrow 1r⋅Φ ∶ r ∗,the image of the identity r→ r under the function r ⟨Φ⟩ ∶ r ⟨X⟩ r→ r ⟨M⟩.

Given a left module M ∶ ∗ A and an object r ∈ ∥A∥, the assignment m ↦ mA yields abijection

⟨M⟩ r ≅ (r ⟨A⟩) ⟨∶A⟩ (M)from the set of M-arrows ∗ r to the set of left module morphisms r ⟨A⟩ →M ∶ ∗ A,whose inverse sends each left module morphism Φ ∶ r ⟨A⟩→M to theM-arrow Φ ⋅1r ∶ ∗ r,the image of the identity r→ r under the function ⟨Φ⟩ r ∶ r ⟨A⟩ r→ ⟨M⟩ r.

155

5. Yoneda Lemma

Proof. Let m ∶ r ∗ be an M-arrow and Φ ∶ ⟨X⟩ r →M be a right module morphism. Weneed to show that m = 1r ⋅ ⟨Xm⟩ and Φ = X(1r ⋅Φ). Replacing h ∶ x → r with 1r ∶ r → r inthe triangle in Remark 5.2.6(2), we have

r1r

1r ⋅ ⟨Xm⟩

%%r m

// ∗

, i.e.m = 1r ⋅ ⟨Xm⟩

. For any object x ∈ ∥X∥ and any arrow h ∶ x→ r, the commutative triangle

xh

h

%%r

1r// r

yields a commutative trianglex

h

h ⋅Φ

%%r

1r ⋅Φ// ∗

by the naturality of Φ. Comparing this triangle with that in Remark 5.2.6(2), we have

Φ =X(1r ⋅Φ)

.

Theorem 5.2.11. Let M ∶XA be a module. The right Yoneda morphism

X

X

M //A

M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

for M is fully faithful. Specically, for each pair of objects s ∈ ∥X∥ and t ∈ ∥A∥, theassignment m↦Xm yields a bijection

s ⟨M⟩ t ≅ (⟨X⟩ s) ⟨X ∶⟩ (⟨M⟩ t)

from the set ofM-arrows s t to the set of right module morphisms ⟨X⟩ s→ ⟨M⟩ t ∶X ∗,whose inverse sends each right module morphism Φ ∶ ⟨X⟩ s → ⟨M⟩ t to the M-arrow 1s ⋅Φ ∶s t, the image of the identity s→ s under the function s ⟨Φ⟩ ∶ s ⟨X⟩ s→ s ⟨M⟩ t.

The left Yoneda morphism

X

M

M //A

A

[∶A]∼⟨∶A⟩

//

⟨MA⟩

[∶A]∼

for M is fully faithful. Specically, for each pair of objects s ∈ ∥X∥ and t ∈ ∥A∥, theassignment m↦ mA yields a bijection

s ⟨M⟩ t ≅ (t ⟨A⟩) ⟨∶A⟩ (s ⟨M⟩)

156

5. Yoneda Lemma

from the set ofM-arrows s t to the set of left module morphisms t ⟨A⟩→ s ⟨M⟩ ∶ ∗A,whose inverse sends each left module morphism Φ ∶ t ⟨A⟩ → s ⟨M⟩ to the M-arrow Φ ⋅ 1t ∶s t, the image of the identity t→ t under the function ⟨Φ⟩ t ∶ t ⟨A⟩ t→ s ⟨M⟩ t.

Proof. See Remark 5.2.12.

Remark 5.2.12. Theorem5.2.11 follows from Theorem5.2.10 (Yoneda lemma) by virtue ofProposition 5.2.7 and Proposition 2.1.9. Conversely, the Yoneda lemma is a special case ofTheorem5.2.11 where A (resp. X) is the terminal category.

Corollary 5.2.13. (Yoneda Embedding). For any category X, the right Yoneda functor [X] ∶X→ [X ∶] is fully faithful. Specically,

for each pair of objects s, t ∈ ∥X∥, the assignment f ↦ ⟨X⟩ f yields a bijection

s ⟨X⟩ t ≅ (⟨X⟩ s) ⟨X ∶⟩ (⟨X⟩ t)

from the set of X-arrows s→ t to the set of right module morphisms ⟨X⟩ s→ ⟨X⟩ t ∶X ∗,whose inverse sends each module morphism Φ ∶ ⟨X⟩ s → ⟨X⟩ t to the X-arrow 1s ⋅Φ ∶ s → t,the image of the identity s→ s under the function s ⟨Φ⟩ ∶ s ⟨X⟩ s→ s ⟨X⟩ t.

For any category A, the left Yoneda functor [A] ∶A→ [∶A]∼ is fully faithful. Specically,for each pair of objects s, t ∈ ∥A∥, the assignment f ↦ f ⟨A⟩ yields a bijection

s ⟨A⟩ t ≅ (t ⟨A⟩) ⟨∶A⟩ (s ⟨A⟩)

from the set of A-arrows s → t to the set of left module morphisms t ⟨A⟩ → s ⟨A⟩ ∶ ∗ A,whose inverse sends each module morphism Φ ∶ t ⟨A⟩ → s ⟨A⟩ to the A-arrow Φ ⋅ 1t ∶ s → t,the image of the identity t→ t under the function ⟨Φ⟩ t ∶ t ⟨A⟩ t→ s ⟨A⟩ t.

Proof. See Remark 5.2.14.

Remark 5.2.14. By Proposition 5.2.8, Corollary 5.2.13 (Yoneda embedding) is a special caseof Theorem5.2.11; to put it the other way round, Theorem5.2.11 generalizes the Yonedaembedding for general modules.

Theorem 5.2.15.

Given a category X, the right Yoneda morphism

X

X

Xd // [X ∶]⟨Xd⟩ = 1

[X ∶]⟨X∶⟩

//

⟨X⟨Xd⟩⟩

[X ∶]

for the right Yoneda module Xd yields a representation

⟨Xd⟩ ≅ [X] ⟨X ∶⟩ ∶X [X ∶]

of the right Yoneda module Xd by the right Yoneda functor X. The right slice of therepresentation cell ⟨X⟨Xd⟩⟩ at a right module M ∶ X ∗ is nothing but the Yonedamorphism for M (see Remark 5.2.6(2)); the cell thus sends each ⟨Xd⟩-arrow m ∶ r M(i.e. M-arrow m ∶ r ∗) to the right module morphism Xm ∶ ⟨X⟩ r→M.

157

5. Yoneda Lemma

Given a category A, the left Yoneda morphism

[∶A]∼

⟨eA⟩ = 1

eA //A

A

[∶A]∼⟨∶A⟩∼

//

⟨⟨eA⟩A⟩

[∶A]∼

for the left Yoneda module eA yields a corepresentation

⟨eA⟩ ≅ ⟨∶A⟩∼ [A] ∶ [∶A]∼ A

of the left Yoneda module eA by the left Yoneda functor A. The left slice of the corepre-sentation cell ⟨⟨eA⟩A⟩ at a left moduleM ∶ ∗A is nothing but the Yoneda morphismforM (see Remark 5.2.6(2)); the cell thus sends each ⟨eA⟩-arrow m ∶M r (i.e. M-arrowm ∶ ∗ r) to the left module morphism mA ∶ r ⟨A⟩→M.

Proof. See Proposition 5.1.3 for the identity ⟨Xd⟩ = 1. The rst assertion now follows fromthe fully faithfulness of the Yoneda morphism (Theorem5.2.11). By Proposition 5.2.7 andProposition 5.1.3, the right slice of ⟨X⟨Xd⟩⟩ atM is given by

⟨X⟨Xd⟩⟩M = ⟨X⟨⟨Xd⟩ (M)⟩⟩= ⟨XM⟩

, i.e. by the Yoneda morphism forM.

Corollary 5.2.16. (Yoneda Lemma : Part two). The bijection in Theorem5.2.10 is naturalin r and M.

Proof. This is a restatement of Theorem5.2.15.

Remark 5.2.17. The representation (resp. corepresentation) in Theorem5.2.15 is called theYoneda representation (resp. corepresentation) and denoted by X (resp. A) as in

X

X

Xd // [X ∶]1

[∶A]∼

1

eA //A

A

[X ∶]⟨X∶⟩

//

X

[X ∶] [∶A]∼⟨∶A⟩∼

//

A

[∶A]∼

; with this notation, the bijections of the Yoneda lemma are written as

(r) ⟨X⟩ (M) ∶ (r) ⟨Xd⟩ (M) ≅ (⟨X⟩ r) ⟨X ∶⟩ (M)

and(M) ⟨A⟩ (r) ∶ (M) ⟨eA⟩ (r) ≅ (r ⟨A⟩) ⟨∶A⟩ (M)

.

5.3. Yoneda morphisms for cylinders

Denition 5.3.1. Let E be a category andM ∶XA be a module. The right action

⟨XM⟩E ∶ [E, ⌊M⌋]→ [X ∶ E]of the right hom ofM on the functor category [E, ⌊M⌋] is called the right general Yonedafunctor for ⟨E,M⟩.

158

5. Yoneda Lemma

The left actionE⟨MA⟩ ∶ [E, ⌊M⌋]→ [E ∶A]∼

of the left hom of M on the functor category [E, ⌊M⌋] is called the left general Yonedafunctor for ⟨E,M⟩.

Remark 5.3.2. Consider a cylinder

ES

T

##X M

//

α

A

. i.e. a natural transformation α ∶ S M0 →M1 T ∶ E→ ⌊M⌋ (see Remark 4.3.4(4)). By Remark 5.2.2, the module ⟨XM⟩ acts onM0 S andM1 T and yields

⟨XM⟩ [M0 S] = ⟨⟨XM⟩M0⟩S = ⟨X⟩S

and⟨XM⟩ [M1 T] = ⟨⟨XM⟩M1⟩T = ⟨M⟩T

; hence ⟨XM⟩ acts on α and gives a module morphism

⟨XM⟩α ∶ ⟨X⟩S→ ⟨M⟩T ∶X E

which maps each ⟨X⟩S-arrow h ∶ x e to the ⟨M⟩T-arrow h αe ∶ x e as indicated in

x

h h ⋅ ⟨XM⟩α''

e ⋅ S αe// T ⋅ e

(cf. Remark 2.2.2(2)). WhenM is understood, ⟨XM⟩α is also written as

Xα ∶ ⟨X⟩S→ ⟨M⟩T ∶X E

and called the module morphism generated by X direct along α. By Remark 5.2.2, the module ⟨MA⟩ acts on S M0 and T M1 and yields

[S M0] ⟨MA⟩ = S ⟨M0 ⟨MA⟩⟩ = S ⟨M⟩

and[T M1] ⟨MA⟩ = T ⟨M1 ⟨MA⟩⟩ = T ⟨A⟩

; hence ⟨MA⟩ acts on α and gives a module morphism

α ⟨MA⟩ ∶ T ⟨A⟩→ S ⟨M⟩ ∶ EA

which maps each T ⟨A⟩-arrow h ∶ e a to the S ⟨M⟩-arrow αe h ∶ e a as indicated in

e ⋅ S αe //

α⟨MA⟩ ⋅ h ''

T ⋅ eha

(cf. Remark 2.2.2(2)). WhenM is understood, α ⟨MA⟩ is also written as

αA ∶ T ⟨A⟩→ S ⟨M⟩ ∶ EA

and called the module morphism generated by A inverse along α.

159

5. Yoneda Lemma

Denition 5.3.3. Let E be a category andM ∶XA be a module. The right general Yoneda morphism for ⟨E,M⟩ (see Denition 4.3.6) is the cell

[E,X]XE

⟨E,M⟩ // [E,A]ME

[X ∶ E]⟨X∶E⟩

//

⟨XM⟩E

[X ∶ E]

sending each cylinder α ∶ S T ∶ EM to the module morphism

Xα = ⟨XM⟩α ∶ ⟨X⟩S→ ⟨M⟩T ∶X E

. The left general Yoneda morphism for ⟨E,M⟩ (see Denition 4.3.6) is the cell

[E,X]EM

⟨E,M⟩ // [E,A]EA

[E ∶A]∼⟨E∶A⟩∼

//

E⟨MA⟩

[E ∶A]∼

sending each cylinder α ∶ S T ∶ EM to the module morphism

αA = α ⟨MA⟩ ∶ T ⟨A⟩→ S ⟨M⟩ ∶ EA

.

Remark 5.3.4.(1)

The cell ⟨XM⟩E is formally given by the composition

[E,X][E,M0]

⟨E,M⟩ // [E,A][E,M1]

[E, ⌊M⌋]⟨XM⟩E

⟨E,⌊M⌋⟩ //

⟨E,IM⟩

[E, ⌊M⌋]⟨XM⟩E

[X ∶ E]⟨X∶E⟩

//

⟨⟨XM⟩E⟩

[X ∶ E]

of the postcomposition cell in Example 4.3.21 and the hom of the right general Yonedafunctor.

The cell E⟨MA⟩ is formally given by the composition

[E,X][E,M0]

⟨E,M⟩ // [E,A][E,M1]

[E, ⌊M⌋]E⟨MA⟩

⟨E,⌊M⌋⟩ //

⟨E,IM⟩

[E, ⌊M⌋]E⟨MA⟩

[E ∶A]∼⟨E∶A⟩∼

//

⟨E⟨MA⟩⟩

[E ∶A]∼

of the postcomposition cell in Example 4.3.21 and the hom of the left general Yonedafunctor.

160

5. Yoneda Lemma

(2) The Yoneda morphism forM is identied with the special instance of the general Yonedamorphism for ⟨E,M⟩ where E is the terminal category.

Example 5.3.5.

(1) Given a right cylinder

XM

//α AGoo

, i.e. a two-sided cylinderA

G

1

##X M

//

α

A

, the category X acts on α and generates a module morphism

Xα = ⟨XM⟩α ∶ ⟨X⟩G→M ∶XA

direct along α, mapping each ⟨X⟩G-arrow h ∶ x a to the M-arrow h αa ∶ x a asindicated in

x

h h ⋅ ⟨Xα⟩

&&a ⋅G αa

// a

, and the category A acts on α and generates a module morphism

αA = α ⟨MA⟩ ∶ ⟨A⟩→ G ⟨M⟩ ∶AA

inverse along α, mapping each A-arrow h ∶ a → b to the G ⟨M⟩-arrow αa h ∶ a b asindicated in

a ⋅G αa //

⟨αA⟩ ⋅ h &&

a

hb

. Given a left cylinder

XM //

F//α A

, i.e. a two-sided cylinderX

1

F

##X M

//

α

A

, the category A acts on α and generates a module morphism

αA = α ⟨MA⟩ ∶ F ⟨A⟩→M ∶XA

inverse along α, mapping each F ⟨A⟩-arrow h ∶ x a to the M-arrow αx h ∶ x a asindicated in

xαx //

⟨αA⟩ ⋅ h &&

F ⋅ xha

161

5. Yoneda Lemma

, and the category X acts on α and generates a module morphism

Xα = ⟨XM⟩α ∶ ⟨X⟩→ ⟨M⟩F ∶XX

direct along α, mapping each X-arrow h ∶ y → x to the ⟨M⟩F-arrow h αx ∶ y x asindicated in

y

h

h ⋅ ⟨Xα⟩&&

x αx// F ⋅ x

.(2) Consider a pair of functors as in:

XF

//AGoo

Given a natural transformation ε ∶ G F→ 1A ∶A→A, i.e. a right cylinder

XF⟨A⟩

//ε AGoo

(see Example 4.3.5(2)), the category X acts on ε and generates a module morphism

Xε = ⟨X⟨F ⟨A⟩⟩⟩ ε ∶ ⟨X⟩G→ F ⟨A⟩ ∶XA

direct along ε, mapping each ⟨X⟩G-arrow h ∶ x a to the F ⟨A⟩-arrow h εa ∶ x a asindicated in

x

h

x ⋅ Fh ⋅ F

h ⋅ ⟨Xε⟩

''a ⋅G a ⋅G ⋅ F εa

// a

. Given a natural transformation η ∶ 1X → G F ∶X→X, i.e. a left cylinder

X⟨X⟩G //

F//η A

(see Example 4.3.5(2)), the category A acts on η and generates a module morphism

ηA = η ⟨⟨⟨X⟩G⟩A⟩ ∶ F ⟨A⟩→ ⟨X⟩G ∶XA

inverse along η, mapping each F ⟨A⟩-arrow h ∶ x a to the ⟨X⟩G-arrow ηx h ∶ x aas indicated in

xηx //

⟨ηA⟩ ⋅ h ''

G ⋅ F ⋅ xG ⋅ h

F ⋅ xh

G ⋅ a a.

(3)

162

5. Yoneda Lemma

Given a right F-weighted cylinder

E

S

DFoo

T

X M//

α

A

, i.e. a right cylinder

ES⟨M⟩T

//α DFoo

(see Remark 4.5.2(1)), the category E acts on α and generates a module morphism

Eα = ⟨E⟨S ⟨M⟩T⟩⟩α ∶ ⟨E⟩F→ S ⟨M⟩T ∶ ED

, i.e. a cell

E⟨E⟩F //

S

D

T

X M//

Eα

A

direct along α, mapping each ⟨E⟩F-arrow h ∶ e d to theM-arrow (h ⋅ S) αd ∶ e ⋅ ST ⋅ d as indicated in

e

h

e ⋅ Sh ⋅ S

h ⋅ ⟨Eα⟩

((d ⋅ F d ⋅ F ⋅ S αd

// T ⋅ d.

Given a left F-weighted cylinder

DF //

S

E

T

X M//

α

A

, i.e. a left cylinder

DS⟨M⟩T //

F//α E

(see Remark 4.5.2(1)), the category E acts on α and generates a module morphism

αE = α ⟨⟨S ⟨M⟩T⟩E⟩ ∶ F ⟨E⟩→ S ⟨M⟩T ∶D E

, i.e. a cell

DF⟨E⟩ //

S

E

T

X M//

αE

A

inverse along α, mapping each F ⟨E⟩-arrow h ∶ d e to theM-arrow αd (T ⋅ h) ∶ d ⋅ST ⋅ e as indicated in

d ⋅ S αd //

⟨αE⟩ ⋅ h ((

T ⋅ F ⋅ dT ⋅ h

F ⋅ dh

T ⋅ e e.

163

5. Yoneda Lemma

(4) Given a cone α ∶ r K ∶ ∗EM, i.e. a cylinder

∗r

E∆oo

K

X M//

α

A

, the category X acts on α and generates a wedge

Xα = ⟨XM⟩α ∶ ⟨X⟩ r ⟨M⟩K ∶X ∗E

direct along α, mapping each X-arrow h ∶ x → r to the ⟨M⟩K-arrow h αe ∶ x e asindicated in

x

h

h ⋅ ⟨Xα⟩&&

r αe// K ⋅ e

for each e ∈ ∥E∥. Given a cone α ∶ K r ∶ E∗M, i.e. a cylinder

E

K

∆ // ∗r

X M//

α

A

, the category A acts on α and generates a wedge

αA = α ⟨MA⟩ ∶ r ⟨A⟩ K ⟨M⟩ ∶ E∗A

inverse along α, mapping each A-arrow r → a to the K ⟨M⟩-arrow αe h ∶ e a asindicated in

e ⋅K αe //

⟨αA⟩ ⋅ h &&

r

ha

for each e ∈ ∥E∥.

Proposition 5.3.6.

Given a right cylinder

XM

//α AGoo

, the triangle

G ⟨X⟩G

G⟨Xα⟩

⟨A⟩⟨G⟩oo

αAwwG ⟨M⟩

commutes.

164

5. Yoneda Lemma

Given a left cylinder

XM //

F//α A

, the triangle

⟨X⟩ ⟨F⟩ //

Xα ''

F ⟨A⟩F

⟨αA⟩F

⟨M⟩F

commutes.

Proof. We need to show that the identity

h ⋅G ⋅ ⟨Xα⟩ = ⟨αA⟩ ⋅ h

holds for any A-arrow h ∶ a→ b. But since the triangles

a ⋅Gh ⋅G

h ⋅G ⋅ ⟨Xα⟩

&&

a ⋅G αa //

⟨αA⟩ ⋅ h &&

a

h

b ⋅G αa// a b

commute (see Example 5.3.5(1)), h ⋅ G ⋅ ⟨Xα⟩ and ⟨αA⟩ ⋅ h are identical, being the diagonalof the naturality square

a ⋅Gh ⋅G

αa // a

h

b ⋅G αb

// b

.

Note. A two-sided cylinderE

S

T

##X M

//

α

A

is also depicted as a right cylinder

X⟨M⟩T

//α ESoo

or a left cylinder

ES⟨M⟩ //

T//α A

(see Remark 4.3.4(3)). Proposition 5.3.7 says that X (resp. A) generates the same modulemorphism direct (resp. inverse) along α irrespective of the way α is depicted.

Proposition 5.3.7. Consider a cylinder as in Note above. Then

165

5. Yoneda Lemma

the module morphism

Xα = ⟨XM⟩α ∶ ⟨X⟩S→ ⟨M⟩T ∶X E

coincides withXα = ⟨X⟨M⟩T⟩α ∶ ⟨X⟩S→ ⟨M⟩T ∶X E

. the module morphism

αA = α ⟨MA⟩ ∶ T ⟨A⟩→ S ⟨M⟩ ∶ EA

coincides withαA = α ⟨S ⟨M⟩A⟩ ∶ T ⟨A⟩→ S ⟨M⟩ ∶ EA

.

Proof. Both map each ⟨X⟩S-arrow h ∶ x e to the ⟨M⟩T-arrow h αe ∶ x e (see Exam-ple 5.3.5(1)).

Note. Given a pair of categoriesX andA, we saw in Remark 4.3.7(3) that the hom of the functorcategory [A,X] is the same thing as the module ⟨A, ⟨X⟩⟩, and will see in Proposition 5.3.8that the hom of the right general Yoneda functor for [A,X] is the same thing as the rightgeneral Yoneda morphism for ⟨A, ⟨X⟩⟩.

Proposition 5.3.8. Given a pair of categories X and A, the right general Yoneda morphism

[A,X]XA

⟨A,⟨X⟩⟩ // [A,X]XA

[X ∶A]⟨X∶A⟩

//

⟨X⟨X⟩⟩A

[X ∶A]

for ⟨A, ⟨X⟩⟩ is the same thing as the hom

[A,X]XA

⟨A,X⟩ // [A,X]XA

[X ∶A]⟨X∶A⟩

//

⟨XA⟩

[X ∶A]

of the right general Yoneda functor for [A,X]; that is, for any natural transformationτ ∶ S→ T ∶A→X, the module morphism

Xτ ∶ ⟨X⟩S→ ⟨X⟩T ∶XA

coincides with⟨X⟩ τ ∶ ⟨X⟩S→ ⟨X⟩T ∶XA

.

166

5. Yoneda Lemma

the left general Yoneda morphism

[X,A]XA

⟨X,⟨A⟩⟩ // [X,A]XA

[X ∶A]∼⟨X∶A⟩∼

//

X⟨⟨A⟩A⟩

[X ∶A]∼

for ⟨X, ⟨A⟩⟩ is the same thing as the hom

[X,A]XA

⟨X,A⟩ // [X,A]XA

[X ∶A]∼⟨X∶A⟩∼

//

⟨XA⟩

[X ∶A]∼

of the left general Yoneda functor for [X,A]; that is, for any natural transformation τ ∶ S→T ∶X→A, the module morphism

τ A ∶ T ⟨A⟩→ S ⟨A⟩ ∶XA

coincides withτ ⟨A⟩ ∶ T ⟨A⟩→ S ⟨A⟩ ∶XA

.

Proof. Both Xτ and ⟨X⟩ τ map each ⟨X⟩S-arrow h ∶ x a to the ⟨X⟩T-arrow h τa ∶ x a(see Remark 2.3.6(1)).

Theorem 5.3.9. Given a category E and a module M ∶XA, the triangle

⟨E,M⟩⟨XM⟩E

yy

⟨E,⟨XM⟩⟩&&

⟨X ∶ E⟩⟨⟩

// ⟨E, ⟨X ∶⟩⟩

commutes; that is, the composition

[E,X] ⟨E,M⟩ //

XE

[E,A]ME

[X ∶ E] ⟨X∶E⟩ //

⟨XM⟩E

[X ∶ E]

[E, [X ∶]]⟨E,⟨X∶⟩⟩

//

⟨⟩

[E, [X ∶]]

of the right general Yoneda morphism for ⟨E,M⟩ and the right exponential transpositionyields the cell

[E,X][E,X]

⟨E,M⟩ // [E,A][E,M]

[E, [X ∶]]⟨E,⟨X∶⟩⟩

//

⟨E,⟨XM⟩⟩

[E, [X ∶]]

, postcomposition with the right Yoneda morphism for M.

167

5. Yoneda Lemma

the triangle⟨E,M⟩

E⟨MA⟩xx

⟨E,⟨MA⟩⟩''

⟨E ∶A⟩∼⟨⟩

// ⟨E, ⟨∶A⟩∼⟩


[E,X]EM

⟨E,M⟩ // [E,A]EA

[E ∶A]∼

⟨E∶A⟩∼ //

E⟨MA⟩

[E ∶A]∼

[E, [∶A]∼]⟨E,⟨∶A⟩∼⟩

//

⟨⟩

[E, [∶A]∼]

of the left general Yoneda morphism for ⟨E,M⟩ and the left exponential transposition yieldsthe cell

[E,X][E,M]

⟨E,M⟩ // [E,A][E,A]

[E, [∶A]∼]⟨E,⟨∶A⟩∼⟩

//

⟨E,⟨MA⟩⟩

[E, [∶A]∼]

, postcomposition with the left Yoneda morphism for M.

Proof. By Proposition 2.2.3, the diagram

[E, ⌊M⌋]⟨XM⟩E

xx

[E,⟨XM⟩]''

[X ∶ E] // [E, [X ∶]]

commutes. The hom of this diagram composed with ⟨E, IM⟩ as shown in

⟨E,M⟩⟨E,IM⟩

⟨E, ⌊M⌋⟩⟨⟨XM⟩E⟩

xx

⟨E,⟨⟨XM⟩⟩⟩''

⟨X ∶ E⟩⟨⟩

// ⟨E, ⟨X ∶⟩⟩

yields the desired commutative triangle by Remark 5.3.4(1) and Remark 5.2.6(1).

Remark 5.3.10. Theorem5.3.9 says that, given a cylinder α ∶ S T ∶ EM, the identity

⟨⟨XM⟩α⟩= ⟨⟨XM⟩⟩ αholds; that is, the right exponential transpose of the module morphism

⟨XM⟩α ∶ ⟨X⟩S→ ⟨M⟩T ∶X E

168

5. Yoneda Lemma

is the natural transformation given by the composition

ES

||T

""X

X

M //

α

A

M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

of α and the right Yoneda morphism forM. the identity

⟨α ⟨MA⟩⟩ = α ⟨⟨MA⟩⟩holds; that is, the left exponential transpose of the module morphism

α ⟨MA⟩ ∶ T ⟨A⟩→ S ⟨M⟩ ∶ EA

is the natural transformation given by the composition

ES

T

##X

M

M //

α

A

A

[∶A]∼⟨∶A⟩∼

//

⟨MA⟩

[∶A]∼

of α and the left Yoneda morphism forM.

Note. The following is a pointwise description of Remark 5.3.10.

Corollary 5.3.11. For any cylinder

ES

T

##X M

//

α

A

and any object e ∈ ∥E∥, the identity

⟨Xα⟩e =Xαe

holds; that is, the right slice

⟨Xα⟩e ∶ ⟨⟨X⟩S⟩e→ ⟨⟨M⟩T⟩e ∶X ∗

at e of the module morphism generated by X direct along α is given by the right module

Xαe ∶ ⟨X⟩ (S ⋅ e)→ ⟨M⟩ (T ⋅ e) ∶X ∗

generated by X direct along the component of α at e.

169

5. Yoneda Lemma

the identitye ⟨αA⟩ = αeA

holds; that is, the left slice

e ⟨αA⟩ ∶ e ⟨T ⟨A⟩⟩→ e ⟨S ⟨M⟩⟩ ∶ ∗A

at e of the module morphism generated by A inverse along α is given by the left module

αeA ∶ (e ⋅T) ⟨A⟩→ (e ⋅ S) ⟨M⟩ ∶ ∗A

generated by A inverse along the component of α at e.

Proof. By Remark 5.3.10, ⟨Xα⟩e = ⟨⟨XM⟩α⟩e is the same thing as the component of thecylinder [⟨⟨XM⟩⟩ α] at e, which is given by the image of αe under the cell ⟨XM⟩, i.e.given by Xαe.

Theorem 5.3.12. Let E be a category and M ∶XA be a module. The right general Yoneda morphism

[E,X]XE

⟨E,M⟩ // [E,A]ME

[X ∶ E]⟨X∶E⟩

//

⟨XM⟩E

[X ∶ E]

for ⟨E,M⟩ is fully faithful. Specically, for each pair of functors S ∶ E→X and T ∶ E→A,the assignment α ↦Xα yields a bijection

(S) ⟨E,M⟩ (T) ≅ (⟨X⟩S) ⟨X ∶ E⟩ (⟨M⟩T)

from the set of cylinders S T ∶ E M to the set of module morphisms ⟨X⟩S → ⟨M⟩T ∶X E, whose inverse sends each module morphism Φ ∶ ⟨X⟩S → ⟨M⟩T to the cylinderXΦ ∶ S T ∶ EM dened by

[XΦ]e = 1(e ⋅ S) ⋅Φ

for e ∈ ∥E∥, where 1(e ⋅ S) ⋅Φ is the image of the identity X-arrow e ⋅S→ S ⋅e (i.e. ⟨X⟩S-arrowe ⋅ S e) under the function

(e ⋅ S) ⟨X⟩ (S ⋅ e) = (e ⋅ S) ⟨⟨X⟩S⟩e (e ⋅ S)⟨Φ⟩e // (e ⋅ S) ⟨⟨M⟩T⟩e = (e ⋅ S) ⟨M⟩ (T ⋅ e)

. The left general Yoneda morphism

[E,X]EM

⟨E,M⟩ // [E,A]EA

[E ∶A]∼⟨E∶A⟩∼

//

E⟨MA⟩

[E ∶A]∼

for ⟨E,M⟩ is fully faithful. Specically, for each pair of functors S ∶ E→X and T ∶ E→A,the assignment α ↦ αA yields a bijection

(S) ⟨E,M⟩ (T) ≅ (T ⟨A⟩) ⟨E ∶A⟩ (S ⟨M⟩)

170

5. Yoneda Lemma

from the set of cylinders S T ∶ E M to the set of module morphisms T ⟨A⟩ → S ⟨M⟩ ∶E A, whose inverse sends each module morphism Φ ∶ T ⟨A⟩ → S ⟨M⟩ to the cylinderΦA ∶ S T ∶ EM dened by

[ΦA]e = Φ ⋅ 1(T ⋅ e)

for e ∈ ∥E∥, where Φ ⋅1(T ⋅ e) is the image of the identity A-arrow e⋅T→ T ⋅e (i.e. T ⟨A⟩-arrowe T ⋅ e) under the function

(e ⋅T) ⟨A⟩ (T ⋅ e) = e ⟨T ⟨A⟩⟩ (T ⋅ e) e⟨Φ⟩(T ⋅ e) // e ⟨S ⟨M⟩⟩ (T ⋅ e) = (e ⋅ S) ⟨M⟩ (T ⋅ e)

.

Proof. By Theorem5.3.9, the cell ⟨XM⟩E is fully faithful i so is the cell ⟨E, ⟨XM⟩⟩.But since the cell ⟨XM⟩ is fully faithful (see Theorem5.2.11), so is ⟨E, ⟨XM⟩⟩ byProposition 4.3.20. For the second assertion, it suces to show that a natural transformationα ∶ S T ∶ EM is recovered from the module morphism ⟨Xα⟩ by αe = 1(e ⋅ S)⋅ (e ⋅ S) ⟨Xα⟩e.But by Theorem5.2.11 and Corollary 5.3.11,

αe = 1(e ⋅ S) ⋅ (e ⋅ S) ⟨Xαe⟩ = 1(e ⋅ S) ⋅ (e ⋅ S) ⟨Xα⟩e

.

Remark 5.3.13. Theorem5.2.11 is regarded as a special case of Theorem5.3.12 where E is theterminal category.

Corollary 5.3.14. (General Yoneda embedding). Let X and A be categories. The right general Yoneda functor [XA] ∶ [A,X] → [X ∶A] is fully faithful. Specically,

for each pair of functors S,T ∶A→X, the assignment τ ↦ ⟨X⟩ τ yields a bijection

(S) ⟨A,X⟩ (T) ≅ (⟨X⟩S) ⟨X ∶A⟩ (⟨X⟩T)

from the set of natural transformations S → T ∶ A → X to the set of module morphisms⟨X⟩S → ⟨X⟩T ∶ X A, whose inverse sends each module morphism Φ ∶ ⟨X⟩S → ⟨X⟩T tothe natural transformation [Φ] ∶ S→ T dened by

[Φ]a = 1(a ⋅ S) ⋅Φ

for a ∈ ∥A∥, where 1(a ⋅ S) ⋅Φ is the image of the identity X-arrow a ⋅S→ S ⋅a (i.e. ⟨X⟩S-arrowa ⋅ S a) under the function

(a ⋅ S) ⟨X⟩ (S ⋅ a) = (a ⋅ S) ⟨⟨X⟩S⟩a (a ⋅ S)⟨Φ⟩a // (a ⋅ S) ⟨⟨X⟩T⟩a = (a ⋅ S) ⟨X⟩ (T ⋅ a)

. The left general Yoneda functor [XA] ∶ [X,A] → [X ∶A]∼ is fully faithful. Specically,

for each pair of functors S,T ∶X→A, the assignment τ ↦ τ ⟨A⟩ yields a bijection

(S) ⟨X,A⟩ (T) ≅ (T ⟨A⟩) ⟨X ∶A⟩ (S ⟨A⟩)

from the set of natural transformations S → T ∶ X → A to the set of module morphismsT ⟨A⟩ → S ⟨A⟩ ∶ X A, whose inverse sends each module morphism Φ ∶ T ⟨A⟩ → S ⟨A⟩ tothe natural transformation [Φ] ∶ S→ T dened by

[Φ]x = Φ ⋅ 1(T ⋅ x)

171

5. Yoneda Lemma

for x ∈ ∥X∥, where Φ⋅1(T ⋅ x) is the image of the identity A-arrow x⋅T→ T⋅x (i.e. T ⟨A⟩-arrowx T ⋅ x) under the function

(x ⋅T) ⟨A⟩ (T ⋅ x) = x ⟨T ⟨A⟩⟩ (T ⋅ x) x⟨Φ⟩(T ⋅ x) // x ⟨S ⟨A⟩⟩ (T ⋅ x) = (x ⋅ S) ⟨A⟩ (T ⋅ x)

.

Proof. Since the hom of the right general Yoneda functor XA is the same thing as theright general Yoneda morphism for ⟨A, ⟨X⟩⟩ (see Proposition 5.3.8), this is a special case ofTheorem5.3.12 whereM is given by the hom of a category.

Remark 5.3.15. Corollary 5.2.13 (Yoneda embedding) is regarded as a special case of Corol-lary 5.3.14 where A (resp. X) is the terminal category.

Theorem 5.3.16. Let X and A be categories. The right general Yoneda morphism for ⟨A, ⟨Xd⟩⟩ composed with the iso cell in Corol-

lary 5.1.10 as shown in

[A,X]1

⟨XdA⟩ //

1

[X ∶A]

[A,X]XA

⟨A,⟨Xd⟩⟩ // [A, [X ∶]]⟨Xd⟩A =

[X ∶A]⟨X∶A⟩

//

⟨X⟨Xd⟩⟩A

[X ∶A]

yields a representation

⟨XdA⟩ ≅ [XA] ⟨X ∶A⟩ ∶ [A,X] [X ∶A]

of the right general Yoneda module XdA by the right general Yoneda functor XA. Fora functor G ∶A→X and a moduleM ∶XA, the representation sends each right cylinderα ∶ GM to the module morphism Xα ∶ ⟨X⟩G→M in Example 5.3.5(1).

The left general Yoneda morphism for ⟨X, ⟨eA⟩⟩ composed with the iso cell in Corol-lary 5.1.10 as shown in

[X ∶A]∼

⟨XeA⟩ //

1

[X,A]1

[X, [∶A]∼]X⟨eA⟩ =

⟨X,⟨eA⟩⟩ // [X,A]XA

[X ∶A]∼⟨X∶A⟩∼

//

X⟨⟨eA⟩A⟩

[X ∶A]∼

yields a corepresentation

⟨XeA⟩ ≅ ⟨X ∶A⟩∼ [XA] ∶ [X ∶A]∼ [X,A]

of the left general Yoneda module XeA by the left general Yoneda functor XA. For afunctor F ∶ X → A and a module M ∶ X A, the representation sends each left cylinderα ∶M F to the module morphism αA ∶ F ⟨A⟩→M in Example 5.3.5(1).

172

5. Yoneda Lemma

Proof. See Proposition 5.1.4 for the identity ⟨Xd⟩A =. The rst assertion now follows fromthe fully faithfulness of the general Yoneda morphism (Theorem5.3.12). The second assertionis immediate from Remark 5.1.11 and Proposition 5.3.7.

Note. The following is a componentwise description of Theorem5.3.16.

Corollary 5.3.17. (General Yoneda Lemma). Given a module M ∶ X A and a functor G ∶ A → X, the assignment α ↦ Xα (see

Example 5.3.5(1)) yields a bijection

(G) ⟨XdA⟩ (M) ≅ (⟨X⟩G) ⟨X ∶A⟩ (M)

from the set of right cylinders G M to the set of module morphisms ⟨X⟩G →M, whoseinverse sends each module morphism Φ ∶ ⟨X⟩G →M to the right cylinder XΦ ∶ G Mdened by

[XΦ]a = 1(a ⋅G) ⋅Φfor a ∈ ∥A∥, where 1(a ⋅G) ⋅Φ is the image of the identity X-arrow a⋅G→ G⋅a (i.e. ⟨X⟩G-arrowa ⋅G a) under the function

(a ⋅G) ⟨X⟩ (G ⋅ a) = (a ⋅G) ⟨⟨X⟩G⟩a (a ⋅G)⟨Φ⟩a // (a ⋅G) ⟨M⟩a

. Moreover, the bijection is natural in G and M. Given a module M ∶ X A and a functor F ∶ X → A, the assignment α ↦ α A (see

Example 5.3.5(1)) yields a bijection

(M) ⟨XeA⟩ (F) ≅ (F ⟨A⟩) ⟨X ∶A⟩ (M)

from the set of left cylinders M F to the set of module morphisms F ⟨A⟩ →M, whoseinverse sends each module morphism Φ ∶ F ⟨A⟩ → M to the left cylinder ΦA ∶ M Fdened by

[ΦA]x = Φ ⋅ 1(F ⋅ x)

for x ∈ ∥X∥, where Φ ⋅1(F ⋅ x) is the image of the identity A-arrow x ⋅F→ F⋅x (i.e. F ⟨A⟩-arrowx F ⋅ x) under the function

(x ⋅ F) ⟨A⟩ (F ⋅ x) = x ⟨F ⟨A⟩⟩ (F ⋅ x) x⟨Φ⟩(F ⋅ x) // x ⟨M⟩ (F ⋅ x)

. Moreover, the bijection is natural in F and M.

Proof. The bijective correspondence follows from Theorem5.3.12 by identifying right cylindersG M with two-sided cylinders G 1A ∶ A M. The naturality of the bijection in G andM is just a restatement of Theorem5.3.16.

Remark 5.3.18.(1) The representation (resp. corepresentation) in Theorem5.3.16 is called the general Yoneda

representation (resp. corepresentation) and denoted by XA (resp. XA) as in

[A,X]XA

⟨XdA⟩ // [X ∶A]1

[X ∶A]∼

1

⟨XeA⟩ // [X,A]XA

[X ∶A]⟨X∶A⟩

//

XA

[X ∶A] [X ∶A]∼⟨X∶A⟩∼

//

XA

[X ∶A]∼

173

5. Yoneda Lemma

; with this notation, the bijections of the general Yoneda lemma are written as

(G) ⟨XA⟩ (M) ∶ (G) ⟨XdA⟩ (M) ≅ (⟨X⟩G) ⟨X ∶A⟩ (M)

and(M) ⟨XA⟩ (F) ∶ (M) ⟨XeA⟩ (F) ≅ (F ⟨A⟩) ⟨X ∶A⟩ (M)

.(2) The Yoneda lemma (Theorem5.2.10 and Corollary 5.2.16) and the Yoneda representation

(Remark 5.2.17) are special cases of the general Yoneda lemma and the general Yonedarepresentation where A (resp. X) is the terminal category. Namely, the bijections inRemark 5.2.17 are identied with

(r) ⟨X∗⟩ (M) ∶ (r) ⟨Xd∗⟩ (M) ≅ (⟨X⟩ r) ⟨X ∶ ∗⟩ (M)

and(M) ⟨∗A⟩ (r) ∶ (M) ⟨∗eA⟩ (r) ≅ (r ⟨A⟩) ⟨∗ ∶A⟩ (M)

(see Remark 5.1.7(3)) respectively.

Corollary 5.3.19. Let Θ ∶M→ N ∶XA be a module morphism and consider the bijectionin Corollary 5.3.17. For any right cylinder α ∶ GM,

X[α Θ] = ⟨Xα⟩ Θ

, and for any module morphism Φ ∶ ⟨X⟩G→M ∶XA,

X⟨Φ Θ⟩ = [XΦ] Θ

. For any left cylinder α ∶M F,

[α Θ]A = ⟨αA⟩ Θ

, and for any module morphism Φ ∶ F ⟨A⟩→M ∶XA,

⟨Φ Θ⟩A = [ΦA] Θ

.

Proof. Immediate by the naturality of the bijection.

Theorem 5.3.20.

Given a pair of natural transformations β ∶ P → F Q ∶X →D and α ∶ F S → T ∶ E →A asin

XP

Q

##β

D

S ##

EFoo

TA

α

174

5. Yoneda Lemma

, their pasting composite [β S] [Q α] ∶ P S→ T Q is given by the composition

XP

Q

##D

S

⟨D⟩F //

β

E

T

A⟨A⟩

//

Dα

A

of the cylinder β (see Example 4.3.5(1)) and the cell Dα (see Example 5.3.5(3)). Given a pair of natural transformations β ∶ P F → Q ∶X →D and α ∶ S → T F ∶ E →A as

inX

P

Q

##β

E F //

S ##

D

TA

α

, their pasting composite [P α] [β T] ∶ P S→ T Q is given by the composition

XP

Q

##E

S

F⟨D⟩ //

β

D

T

A⟨A⟩

//

αD

A

of the cylinder β (see Example 4.3.5(1)) and the cell αD (see Example 5.3.5(3)).

Proof. For any x ∈ ∥X∥,

[[β S] [Q α]]x = [β S]x [Q α]x = (βx ⋅ S) α(x ⋅Q)

and[β ⟨Dα⟩]x = βx ⋅ ⟨Dα⟩

. We thus need to show that βx ⋅ ⟨Dα⟩ = (βx ⋅ S) α(x ⋅Q). But by replacing h ∶ d→ F ⋅ e withβx ∶ x ⋅ P→ F ⋅Q ⋅ x in the commutative diagram of Example 5.3.5(3), we have

x ⋅ Pβx

x ⋅ P ⋅ Sβx ⋅ S

βx ⋅ ⟨Dα⟩))

x ⋅Q ⋅ F x ⋅Q ⋅ F ⋅ S α(x ⋅Q)// T ⋅Q ⋅ x

as required.

Corollary 5.3.21.

Given a pair of natural transformations η ∶ 1X → G F ∶ X → X and ε ∶ G H → 1A ∶ A → Aas in

X1

F

##η

X

H ##

AGoo

1A

ε

175

5. Yoneda Lemma

, the triangle

F ⟨A⟩

ηA &&

[[η H] [F ε]]⟨A⟩ // H ⟨A⟩

⟨X⟩GXε

88

commutes, where [η H] [F ε] ∶ H→ F is the pasting composite of η and ε. Given a pair of natural transformations ε ∶ G F → 1A ∶ A → A and η ∶ 1X → H F ∶ X → X

as inA

G

1

##ε

X F //

1 ##

A

HX

η

, the triangle

⟨X⟩G

Xε &&

⟨X⟩[[G η] [ε H]] // ⟨X⟩H

F ⟨A⟩ηA

88

commutes, where [G η] [ε H] ∶ G→ H is the pasting composite of ε and η.

Proof. Indeed,

[[η H] [F ε]] ⟨A⟩ = [η ⟨Xε⟩] ⟨A⟩ ∗1

= [η ⟨Xε⟩]A ∗2

= ⟨ηA⟩ ⟨Xε⟩ ∗3

(∗1 by Theorem5.3.20; ∗2 by Proposition 5.3.8; ∗3 by Corollary 5.3.19).

5.4. Yoneda morphisms for cones

Denition 5.4.1. Let E be a category andM ∶XA be a module. The right general Yoneda morphism for ⟨∗E,M⟩ (see Denition 4.6.5) is the cell

X⟨∗E,M⟩ //

X

[E,A]ME

[X ∶]⟨X∶∗E⟩

//

⟨XM⟩∗E

[X ∶ E]

sending each cone α ∶ r K ∶ ∗EM to the wedge

Xα = ⟨XM⟩α ∶ ⟨X⟩ r ⟨M⟩K ∶X ∗E

(see Example 5.3.5(4)). The left general Yoneda morphism for ⟨E∗,M⟩ (see Denition 4.6.5) is the cell

[E,X]EM

⟨E∗,M⟩ //A

A

[E ∶A]∼⟨E∗∶A⟩∼

//

E∗⟨MA⟩

[∶A]∼

176

5. Yoneda Lemma

sending each cone α ∶ K r ∶ E∗M to the wedge

αA = α ⟨MA⟩ ∶ r ⟨A⟩ K ⟨M⟩ ∶ E∗A

(see Example 5.3.5(4)).

Remark 5.4.2. By virtue of Theorem4.6.19 and Remark 4.10.2(2), the cell ⟨XM⟩∗E is formally given by the pasting composition

X[∆E,X] //

X

[E,X]XE

⟨E,M⟩ // [E,A]ME

[X ∶][X∶∆E]

// [X ∶ E]⟨X∶E⟩

//

⟨XM⟩E

[X ∶ E]

of the right general Yoneda morphism for ⟨E,M⟩ and the commutative diagram in Propo-sition 2.3.7.

the cell E∗⟨MA⟩ is formally given by the pasting composition

[E,X]EM

⟨E,M⟩ // [E,A]EA

A

A

[∆E,A]oo

[E ∶A]∼⟨E∶A⟩∼

//

E⟨MA⟩

[E ∶A]∼ [∶A]∼[∆E∶A]oo

of the left general Yoneda morphism for ⟨E,M⟩ and the commutative diagram in Proposi-tion 2.3.7.

Proposition 5.4.3. The right (resp. left) general Yoneda morphism for ⟨∗E,M⟩ (resp.⟨E∗,M⟩) is fully faithful.

Proof. Since the cell ⟨XM⟩∗E is obtained from the cell ⟨XM⟩ E (the right generalYoneda morphism for ⟨E,M⟩) by the pasting composition in Remark 5.4.2, the fully faith-fulness of ⟨XM⟩∗E follows from that of ⟨XM⟩ E (see Theorem5.3.12) by applyingProposition 1.2.33.

Proposition 5.4.4. Given a category E and a module M ∶XA, the triangle

⟨∗E,M⟩⟨XM⟩∗E

xx

⟨∗E,⟨XM⟩⟩''

⟨X ∶ ∗E⟩ // ⟨∗E, ⟨X ∶⟩⟩


X⟨∗E,M⟩ //

X

[E,A]ME

[X ∶]1

⟨X∶∗E⟩ //

⟨XM⟩∗E

[X ∶ E]

[X ∶]⟨∗E,⟨X∶⟩⟩

//

[E, [X ∶]]

177

5. Yoneda Lemma

of the right general Yoneda morphism for ⟨∗E,M⟩ and the right exponential transpositionof wedges X ∗E (see Notation 4.10.3) yields the cell

X

X

⟨∗E,M⟩ // [E,A][E,M]

[X ∶]⟨∗E,⟨X∶⟩⟩

//

⟨∗E,⟨XM⟩⟩

[E, [X ∶]]

, postcomposition with the right Yoneda morphism for M. the triangle

⟨E∗,M⟩E∗⟨MA⟩

ww

⟨E∗,⟨MA⟩⟩''

⟨E∗ ∶A⟩∼ // ⟨E∗, ⟨∶A⟩∼⟩


[E,X]EM

⟨E∗,M⟩ //A

A

[E ∶A]∼

⟨E∗∶A⟩∼ //

E∗⟨MA⟩

[∶A]∼

1

[E, [∶A]∼]⟨E∗,⟨∶A⟩∼⟩

//

[∶A]∼

of the left general Yoneda morphism for ⟨E∗,M⟩ and the left exponential transposition ofwedges E∗A (see Notation 4.10.3) yields the cell

[E,X][E,M]

⟨E∗,M⟩ //A

A

[E, [∶A]∼]⟨E∗,⟨∶A⟩∼⟩

//

⟨E∗,⟨MA⟩⟩

[∶A]∼

, postcomposition with the left Yoneda morphism for M.

Proof. Consider the cells and commutative diagrams as in

X[∆E,X] //

X

[E,X]XE

⟨E,M⟩ // [E,A]ME

[X ∶] [X∶∆E] //

1

[X ∶ E]

⟨X∶E⟩ //

⟨XM⟩E

[X ∶ E]

[X ∶][∆E,[X∶]]

// [E, [X ∶]]⟨E,[X∶]⟩

//

⟨⟩

[E, [X ∶]]

. The horizontal composition yields

X⟨∗E,M⟩ //

X

[E,A]ME

[X ∶]1

⟨X∶∗E⟩ //

⟨XM⟩∗E

[X ∶ E]

[X ∶]⟨∗E,⟨X∶⟩⟩

//

[E, [X ∶]]

178

5. Yoneda Lemma

by Remark 5.4.2 and Remark 4.10.4, and the vertical composition yields

X[∆E,X] //

X

[E,X] ⟨E,M⟩ //

[E,X]

[E,A][E,M]

[X ∶][∆E,[X∶]]

// [E, [X ∶]]⟨E,⟨X∶⟩⟩

//

⟨E,⟨XM⟩⟩

[E, [X ∶]]

by Theorem5.3.9, which then produces

X

X

⟨∗E,M⟩ // [E,A][E,M]

[X ∶]⟨∗E,⟨X∶⟩⟩

//

⟨∗E,⟨XM⟩⟩

[E, [X ∶]]

by Corollary 4.6.20. The assertion now follows from Proposition 1.2.34.

Remark 5.4.5. Proposition 5.4.4 says that for any cone α ∶ r K ∶ ∗EM the identity

⟨⟨XM⟩α⟩= ⟨⟨XM⟩⟩ α

holds; that is, the right exponential transpose of the wedge

⟨XM⟩α ∶ ⟨X⟩ r ⟨M⟩K ∶X ∗E

is the cone given by the composition

∗r

E∆oo

K

X

X

M //

α

A

M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

of α and the right Yoneda morphism forM. for any cone α ∶ K r ∶ E∗M the identity

⟨α ⟨MA⟩⟩ = α ⟨⟨MA⟩⟩

holds; that is, the left exponential transpose of the wedge

α ⟨MA⟩ ∶ r ⟨A⟩ K ⟨M⟩ ∶ E∗A


E

K

∆ // ∗r

X

M

M //

α

A

A

[∶A]∼⟨∶A⟩∼

//

⟨MA⟩

[∶A]∼

of α and the left Yoneda morphism forM.

179

5. Yoneda Lemma

Note. The following denition is a special case of Denition 5.4.1 whereM is given by the homof a category.

Denition 5.4.6. Let E and C be categories. The right general Yoneda morphism for ⟨∗E,C⟩ (see Denition 4.9.3) is the cell

C⟨∗E,C⟩ //

C

[E,C]CE

[C ∶]⟨C∶∗E⟩

//

⟨C∗E⟩

[C ∶ E]

sending each cone α ∶ r K ∶ ∗E→C to the wedge ⟨C⟩α ∶ ⟨C⟩ r ⟨C⟩K ∶C ∗E. The left general Yoneda morphism for ⟨E∗,C⟩ (see Denition 4.9.3) is the cell

[E,C]EC

⟨E∗,C⟩ // C

C

[E ∶C]∼⟨E∗∶C⟩∼

//

⟨E∗C⟩

[∶C]∼

sending each cone α ∶ K r ∶ E∗→C to the wedge α ⟨C⟩ ∶ r ⟨C⟩ K ⟨C⟩ ∶ E∗C.

Remark 5.4.7.(1) By virtue of Remark 4.9.4(2) and Remark 4.10.2(2),

the cell ⟨C∗E⟩ is formally given by the pasting composition

C[∆E,C] //

C

[E,C]CE

⟨E,C⟩ // [E,C]CE

[C ∶][C∶∆E]

// [C ∶ E]⟨C∶E⟩

//

⟨CE⟩

[C ∶ E]

of the hom of the right general Yoneda functor for [E,C] and the commutative diagramin Proposition 2.3.7.

the cell ⟨E∗C⟩ is formally given by the pasting composition

[E,C]EC

⟨E,C⟩ // [E,C]EC

C

C

[∆E,C]oo

[E ∶C]∼⟨E∶C⟩∼

//

⟨EC⟩

[E ∶C]∼ [∶C]∼[∆E∶C]oo

of the hom of the left general Yoneda functor for [E,C] and the commutative diagramin Proposition 2.3.7.

(2) Comparing the compositions above and those in Remark 5.4.2, and noting Proposition 5.3.8,we see that the right general Yoneda morphism for ⟨∗E,C⟩ is just a special instance of the right

general Yoneda morphism for ⟨∗E,M⟩ whereM is given by the hom of C; that is,

⟨C∗E⟩ = ⟨C⟨C⟩⟩∗E

.

180

5. Yoneda Lemma

the left general Yoneda morphism for ⟨E∗,C⟩ is just a special instance of the left generalYoneda morphism for ⟨E∗,M⟩ whereM is given by the hom of C; that is,

⟨E∗C⟩ = E∗⟨⟨C⟩C⟩

.

Note. The following is a special case of Proposition 5.4.3 where M is given by the hom of acategory.

Proposition 5.4.8. The right (resp. left) general Yoneda morphism for ⟨∗E,C⟩ (resp. ⟨E∗,C⟩)is fully faithful.

Proof. Since the cell ⟨C∗E⟩ is obtained from the cell ⟨CE⟩ (the hom of the right generalYoneda functor for [E,C]) by the pasting composition in Remark 5.4.7(1), the fully faithfulnessof ⟨C∗E⟩ follows from that of ⟨CE⟩ (see Corollary 5.3.14) by applying Proposition 1.2.33.


Proposition 5.4.9. Given categories E and C, the triangle

⟨∗E,C⟩C∗Exx

⟨∗E,C⟩''

⟨C ∶ ∗E⟩ // ⟨∗E, ⟨C ∶⟩⟩


C⟨∗E,C⟩ //

C

[E,C]CE

[C ∶]1

⟨C∶∗E⟩ //

C∗E

[C ∶ E]

[C ∶]⟨∗E,⟨C∶⟩⟩

//

[E, [C ∶]]

of the right general Yoneda morphism for ⟨∗E,C⟩ and the right exponential transposition ofwedges C ∗E (see Notation 4.10.3) yields the cell

C

C

⟨∗E,C⟩ // [E,C][E,C]

[C ∶]⟨∗E,⟨C∶⟩⟩

//

⟨∗E,C⟩

[E, [C ∶]]

, postcomposition with the right Yoneda functor for C. the triangle

⟨E∗,C⟩E∗C

xx

⟨E∗,C⟩''

⟨E∗ ∶C⟩∼ // ⟨E∗, ⟨∶C⟩∼⟩

181

5. Yoneda Lemma


[E,C]EC

⟨E∗,C⟩ // C

C

[E ∶C]∼

⟨E∗∶C⟩∼ //

E∗C

[∶C]∼

1

[E, [∶C]∼]⟨E∗,⟨∶C⟩∼⟩

//

[∶C]∼

of the left general Yoneda morphism for ⟨E∗,C⟩ and the left exponential transposition ofwedges E∗C (see Notation 4.10.3) yields the cell

[E,C][E,C]

⟨E∗,C⟩ // C

C

[E, [∶C]∼]⟨E∗,⟨∶C⟩∼⟩

//

⟨E∗,C⟩

[∶C]∼

, postcomposition with the left Yoneda functor for C.

Proof. Since ⟨C∗E⟩ = ⟨C⟨C⟩⟩∗E (see Remark 5.4.7(2)) and ⟨C⟩ = ⟨C⟨C⟩⟩ (seeProposition 5.2.8), the desired commutative diagram is obtained from that in Proposition 5.4.4by replacingM with the hom of C.

Remark 5.4.10. Proposition 5.4.9 says that for any cone α ∶ r K ∶ ∗E→C the identity

⟨⟨C⟩α⟩= [C] α

holds; that is, the right exponential transpose of the wedge

⟨C⟩α ∶ ⟨C⟩ r ⟨C⟩K ∶C ∗E


∗r

E∆oo

K

C

C

⟨C⟩ //

α

C

C

[C ∶]⟨C∶⟩

//

⟨C⟩

[C ∶]

of α and the right Yoneda functor for C. for any cone α ∶ K r ∶ E∗→C the identity

⟨α ⟨C⟩⟩ = α [C]

holds; that is, the left exponential transpose of the wedge

α ⟨C⟩ ∶ r ⟨C⟩ K ⟨C⟩ ∶ E∗C

182

5. Yoneda Lemma


E

K

∆ // ∗r

C

C

⟨C⟩ //

α

C

C

[∶C]∼⟨∶C⟩∼

//

⟨C⟩

[∶C]∼

of α and the left Yoneda functor for C.

5.5. Correspondences between frames and cells

Note. The following is a special case of the general Yoneda lemma (Corollary 5.3.17) whereMis a representable module.

Theorem 5.5.1. Given a pair of functors

XF

//AGoo

the assignment ε↦Xε (see Example 5.3.5(2)) yields a bijection

(G F) ⟨A,A⟩ (1A) ≅ (⟨X⟩G) ⟨X ∶A⟩ (F ⟨A⟩)

from the set of natural transformations G F→ 1A ∶A→A to the set of module morphisms⟨X⟩G → F ⟨A⟩ ∶ X A, whose inverse sends each module morphism Φ ∶ ⟨X⟩G → F ⟨A⟩ tothe natural transformation XΦ ∶ G F→ 1A dened by

[XΦ]a = 1(a ⋅G) ⋅Φ

for a ∈ ∥A∥, where 1(a ⋅G) ⋅Φ is the image of the identity X-arrow a⋅G→ G⋅a (i.e. ⟨X⟩G-arrowa ⋅G a) under the function

(a ⋅G) ⟨X⟩ (G ⋅ a) = (a ⋅G) ⟨⟨X⟩G⟩a (a ⋅G)⟨Φ⟩a // (a ⋅G) ⟨F ⟨A⟩⟩a = (a ⋅G ⋅ F) ⟨A⟩a

. Moreover, the bijection is natural in G and F. the assignment η ↦ ηA (see Example 5.3.5(2)) yields a bijection

(1X) ⟨X,X⟩ (G F) ≅ (F ⟨A⟩) ⟨X ∶A⟩ (⟨X⟩G)

from the set of natural transformations 1X → G F ∶X→X to the set of module morphismsF ⟨A⟩ → ⟨X⟩G ∶ X A, whose inverse sends each module morphism Φ ∶ F ⟨A⟩ → ⟨X⟩G tothe natural transformation ΦA ∶ 1X → G F dened by

[ΦA]x = Φ ⋅ 1(F ⋅ x)

for x ∈ ∥X∥, where Φ ⋅1(F ⋅ x) is the image of the identity A-arrow x ⋅F→ F⋅x (i.e. F ⟨A⟩-arrowx F ⋅ x) under the function

(x ⋅ F) ⟨A⟩ (F ⋅ x) = x ⟨F ⟨A⟩⟩ (F ⋅ x) x⟨Φ⟩(F ⋅ x) // x ⟨⟨X⟩G⟩ (F ⋅ x) = x ⟨X⟩ (G ⋅ F ⋅ x)

. Moreover, the bijection is natural in G and F.

183

5. Yoneda Lemma

Proof. ReplacingM with F ⟨A⟩ in Corollary 5.3.17, we have a bijection

(G) ⟨XdA⟩ (F ⟨A⟩) ≅ (⟨X⟩G) ⟨X ∶A⟩ (F ⟨A⟩)

, natural in G and F. But by denition,

(G) ⟨XdA⟩ (F ⟨A⟩) =∏A

G ⟨F ⟨A⟩⟩ =∏A

[G F] ⟨A⟩ [1A] = (G F) ⟨A,A⟩ (1A)

.

Note. The following is a special case of the general Yoneda lemma (Corollary 5.3.17) whereMis given by the composite module S ⟨M⟩T.

Theorem 5.5.2. Given a functor F ∶D→ E and a module M ∶XA, there is a canonical module isomorphism

ΨFM ∶ ⟨F M⟩ ≅ ⟨⟨E⟩F,M⟩ ∶ [E,X] [D,A]

, natural in F andM, giving for each pair of functors S ∶ E→X and T ∶D→A a bijection

(S) ⟨ΨFM⟩ (T) ∶ (S) ⟨F M⟩ (T) ≅ (S) ⟨⟨E⟩F,M⟩ (T)

from the set of cylinders F S T ∶ D M to the set of cells S T ∶ ⟨E⟩F M.Specically, the bijection sends each cylinder

E

S

DFoo

T

X M//

α

A

to the cell

E⟨E⟩F //

S

D

T

X M//

Eα

A

in Example 5.3.5(3); the inverse sends each cell

E⟨E⟩F //

S

D

T

X M//

Θ

A

to the cylinder

E

S

D

T

Foo

X M//

EΘ

A

dened by[EΘ]d = 1(d ⋅ F) ⋅Θ

184

5. Yoneda Lemma

for d ∈ ∥D∥, where 1(d ⋅ F) ⋅Θ is the image of the identity E-arrow d ⋅F→ F ⋅d (i.e. ⟨E⟩F-arrowd ⋅F d) under the cell Θ. Moreover, ΨF

M is natural inM withM varying in MOD; thatis, the square

⟨F M⟩ΨFM //

⟨FΦ⟩

⟨⟨E⟩F,M⟩⟨⟨E⟩F,Φ⟩

⟨F N ⟩ΨFN

// ⟨⟨E⟩F,N ⟩

commutes for every cell Φ ∶M N . there is a canonical module isomorphism

ΨFM ∶ ⟨F M⟩ ≅ ⟨F ⟨E⟩ ,M⟩ ∶ [D,X] [E,A]

, natural in F andM, giving for each pair of functors S ∶D→X and T ∶ E→A a bijection

(S) ⟨ΨFM⟩ (T) ∶ (S) ⟨F M⟩ (T) ≅ (S) ⟨F ⟨E⟩ ,M⟩ (T)

from the set of cylinders S T F ∶ D M to the set of cells S T ∶ F ⟨E⟩ M.Specically, the bijection sends each cylinder

DF //

S

E

T

X M//

α

A

to the cell

DF⟨E⟩ //

S

E

T

X M//

αE

A

in Example 5.3.5(3); the inverse sends each cell

DF⟨E⟩ //

S

E

T

X M//

Θ

A

to the cylinder

D

S

F // E

T

X M//

ΘE

A

dened by[ΘE]d = Θ ⋅ 1(F ⋅ d)

for d ∈ ∥D∥, where Θ ⋅1(F ⋅ d) is the image of the identity E-arrow d ⋅F→ F ⋅d (i.e. F ⟨E⟩-arrowd F ⋅d) under the cell Θ. Moreover, ΨF

M is natural inM withM varying in MOD; thatis, the square

⟨F M⟩ΨFM //

⟨FΦ⟩

⟨F ⟨E⟩ ,M⟩⟨F⟨E⟩,Φ⟩

⟨F N ⟩ΨFN

// ⟨F ⟨E⟩ ,N ⟩

commutes for every cell Φ ∶M N .

185

5. Yoneda Lemma

Proof. In Corollary 5.3.17, replace M ∶ X A with S ⟨M⟩T ∶ E D and replace G ∶ A → Xwith F ∶D→ E. Then we have a bijection

(F) ⟨ED⟩ (S ⟨M⟩T) ∶ (F) ⟨EdD⟩ (S ⟨M⟩T) ≅ (⟨E⟩F) ⟨E ∶D⟩ (S ⟨M⟩T)

(see Remark 5.3.18(1) for ED), natural in F, S, T, andM. But

(F) ⟨EdD⟩ (S ⟨M⟩T) =∏D

F ⟨S ⟨M⟩T⟩ = (F S) ⟨D,M⟩ (T) = (S) ⟨F M⟩ (T)

and(⟨E⟩F) ⟨E ∶D⟩ (S ⟨M⟩T) = (S) ⟨⟨E⟩F,M⟩ (T)

by the denitions. Hence an isomorphism ΨFM ∶ ⟨F M⟩ ≅ ⟨⟨E⟩F,M⟩ is given by

(S) ⟨ΨFM⟩ (T) = (F) ⟨ED⟩ (S ⟨M⟩T)

. The last assertion holds since for any cell

X

P

M //A

Q

Y N//

Φ

B

, the square

(S) ⟨F M⟩ (T)(S)⟨ΨF

M⟩(T)//

(S)⟨FΦ⟩(T)

(S) ⟨⟨E⟩F,M⟩ (T)(S)⟨⟨E⟩F,Φ⟩(T)

(S P) ⟨F N ⟩ (Q T)(S P)⟨ΨF

N ⟩(Q T)// (S P) ⟨⟨E⟩F,N ⟩ (Q T)

, i.e.

(F) ⟨EdD⟩ (S ⟨M⟩T) (F)⟨ED⟩(S⟨M⟩T) //

(F)⟨EdD⟩(S⟨Φ⟩T)

(⟨E⟩F) ⟨E ∶D⟩ (S ⟨M⟩T)(⟨E⟩F)⟨E∶D⟩(S⟨Φ⟩T)

(F) ⟨EdD⟩ (S ⟨P ⟨N ⟩Q⟩T)(F)⟨ED⟩(S⟨P⟨N ⟩Q⟩T)

// (⟨E⟩F) ⟨E ∶D⟩ (S ⟨P ⟨N ⟩Q⟩T)

commutes by the naturality of ED.

Remark 5.5.3.(1) Although derived as a special case of the general Yoneda lemma, Theorem5.5.2 is more

versatile; Corollary 5.3.17 turns to be a special case of the bijection in Theorem5.5.2 whereS and T are the identities.

(2) Theorem5.3.12 is viewed as a special case of Theorem5.5.2 by depicting a cylinder

ES

T

##X M

//

α

A

as

X

1

ESoo

T

ET //

S

A

1

X M//

α

A X M//

α

A

186

5. Yoneda Lemma

The category X acts on α and generates a cell

X⟨X⟩S //

1

E

T

X M//

Xα

A

direct along α, and the assignment α ↦Xα yields a bijection from the set of cylindersS T ∶ EM to the set of cells 1X T ∶ ⟨X⟩SM, i.e. the set of module morphisms⟨X⟩S→ ⟨M⟩T ∶X E.

The category A acts on α and generates a cell

ET⟨A⟩ //

S

A

1

X M//

αA

A

inverse along α, and the assignment α ↦ αA yields a bijection from the set of cylindersS T ∶ EM to the set of cells S 1A ∶ T ⟨A⟩M, i.e. the set of module morphismsT ⟨A⟩→ S ⟨M⟩ ∶ EA.

(3) Theorem5.5.1 is also viewed as a special case of Theorem5.5.2 by depicting natural trans-formations ε ∶ G F→ 1A ∶A→A and η ∶ 1X → G F ∶X→X as

X

F

AGoo

1

XF //

1

A

G

A⟨A⟩

//

ε

A X⟨X⟩

//

η

X

The category X acts on ε and generates a cell

X⟨X⟩G //

F

A

1

A⟨A⟩

//

Xε

A

direct along ε, and the assignment ε ↦ Xε yields a bijection from the set of naturaltransformations G F→ 1A ∶A→A to the set of cells F 1A ∶ ⟨X⟩G ⟨A⟩, i.e. the setof module morphisms ⟨X⟩G→ F ⟨A⟩ ∶XA.

The category A acts on η and generates a cell

XF⟨A⟩ //

1

A

G

X⟨X⟩

//

ηA

X

inverse along η, and the assignment η ↦ ηA yields a bijection from the set of naturaltransformations 1X → G F ∶X→X to the set of cells 1X G ∶ F ⟨A⟩ ⟨X⟩, i.e. the setof module morphisms F ⟨A⟩→ ⟨X⟩G ∶XA.

Corollary 5.5.4. Given a category E and a module M ∶ X A, there is a canonical moduleisomorphism

ΨEM ∶ ⟨E,M⟩→ ⟨⟨E⟩ ,M⟩ ∶ [E,X] [E,A]

187

5. Yoneda Lemma

, giving for each pair of functors S ∶ E→X and T ∶ E→A a bijection

(S) ⟨ΨEM⟩ (T) ∶ (S) ⟨E,M⟩ (T) ≅ (S) ⟨⟨E⟩ ,M⟩ (T)

from the set of cylinders S T ∶ E M to the set of cells S T ∶ ⟨E⟩ M. Specically, thebijection sends each cylinder

ES

T

##X M

//

α

A

to the cell

E⟨E⟩ //

S

E

T

X M//

⟨α⟩

A

which maps each E-arrow h ∶ e→ e′ to the M-arrow

(h ⋅ S) αe′ = αe (T ⋅ h) ∶ e ⋅ S T ⋅ e′

as indicated ine

h

e ⋅ Sh ⋅ S

αe //

h ⋅ ⟨α⟩''

T ⋅ eT ⋅ h

e

h

e′ e′ ⋅ S αe′// T ⋅ e′ e′

; the inverse sends each cell

E⟨E⟩ //

S

E

T

X M//

Θ

A

to the cylinderE

S

T

##X M

//

[Θ]

A

dened by[Θ]e = 1e ⋅Θ

for e ∈ ∥E∥, where 1e ⋅Θ is the image of the identity e→ e under the function

e ⟨E⟩e e⟨Θ⟩e // e ⟨S ⟨M⟩T⟩e = (e ⋅ S) ⟨M⟩ (T ⋅ e)

. Moreover, ΨEM is natural in M with M varying in MOD; that is, the square

⟨E,M⟩ΨEM //

⟨E,Φ⟩

⟨⟨E⟩ ,M⟩⟨⟨E⟩,Φ⟩

⟨E,N ⟩ΨEN// ⟨⟨E⟩ ,N ⟩


188

5. Yoneda Lemma

Proof. Since the hom ⟨E⟩ is represented and corepresented by the identity functor 1E ∶ E →E, and since ⟨E,M⟩ = ⟨1E M⟩ = ⟨1E M⟩ (see Remark 4.5.4(2)) and ⟨E,Φ⟩ = ⟨1E Φ⟩ =⟨1E Φ⟩ (see Remark 4.5.8(2)), the assertion follows as a special case of Theorem5.5.2 whereF is given by 1E, with ΨE

M dened by

ΨEM = Ψ1E

M = Ψ1EM

; the cell ⟨α⟩ is given by Eα = αE and the cylinder [Θ] is given by EΘ = ΘE.

Remark 5.5.5. If the moduleM in Corollary 5.5.4 is replaced by the hom of a category C, wehave a bijection between the set of natural transformations α ∶ S → T ∶ E → C and the set ofcells

E⟨E⟩ //

S

E

T

C⟨C⟩

//

⟨α⟩

C

. A morphism between two paralleling functors S,T ∶ E → C is thus dened either by a cellshown above or by a natural transformation α ∶ S → T ∶ E → C, which is a frame of the endmodules S ⟨C⟩T ∶ E E. The relation between cells and frames can be compared with thatbetween linear maps and matrices.

Corollary 5.5.6. Given a category E and a module M ∶XA, there is a canonical module isomorphism

Ψ∗EM ∶ ⟨∗E,M⟩ ≅ ⟨∆∗E,M⟩ ∶X [E,A]

(see Example 1.1.23(9) for ∆∗E), giving for each pair of an object x ∈ ∥X∥ and a functorK ∶ E→A a bijection

(x) ⟨Ψ∗EM ⟩ (K) ∶ (x) ⟨∗E,M⟩ (K) ≅ (x) ⟨∆∗E,M⟩ (K)

from the set of cones x K ∶ ∗E M to the set of conical cells x K ∶ ∆∗E M.Specically, the bijection sends each cone

∗x

E∆Eoo

K

X M//

α

A

to the conical cell

∗ ∆∗E //

x

E

K

X M//

⟨α⟩

A

which maps each ⟨∆∗E⟩-arrow ∗ ∶ ∗ e to the component of α at e; the inverse sends eachconical cell

∗ ∆∗E //

x

E

K

X M//

Θ

A

189

5. Yoneda Lemma

to the cone

∗x

E∆Eoo

K

X M//

[Θ]

A

dened by[Θ]e = ∗ ⋅Θ

for e ∈ ∥E∥, where ∗⋅Θ is the image of the ⟨∆∗E⟩-arrow ∗ ∶ ∗ e under the cell Θ. Moreover,Ψ∗EM is natural in M with M varying in MOD; that is, the square

⟨∗E,M⟩Ψ∗EM //

⟨∗E,Φ⟩

⟨∆∗E,M⟩⟨∆∗E,Φ⟩

⟨∗E,N ⟩Ψ∗EN

// ⟨∆∗E,N ⟩

commutes for every cell Φ ∶M N . there is a canonical module isomorphism

ΨE∗M ∶ ⟨E∗,M⟩ ≅ ⟨∆E∗,M⟩ ∶ [E,X]A

(see Example 1.1.23(9) for ∆E∗), giving for each pair of an object a ∈ ∥A∥ and a functorK ∶ E→X a bijection

(K) ⟨ΨE∗M ⟩ (a) ∶ (K) ⟨E∗,M⟩ (a) ≅ (K) ⟨∆E∗,M⟩ (a)

from the set of cones K a ∶ E∗ M to the set of conical cells K a ∶ ∆E∗ M.Specically, the bijection sends each cone

E∆E //

K

∗a

X M//

α

A

to the conical cell

E∆E∗ //

K

∗a

X M//

⟨α⟩

A

which maps each ⟨∆E∗⟩-arrow ∗ ∶ e ∗ to the component of α at e; the inverse sends eachconical cell

E∆E∗ //

K

∗a

X M//

Θ

A

to the cone

E

K

∆E // ∗a

X M//

[Θ]

A

190

5. Yoneda Lemma

dened by[Θ]e = Θ ⋅ ∗

for e ∈ ∥E∥, where Θ ⋅∗ is the image of the ⟨∆E∗⟩-arrow ∗ ∶ e ∗ under the cell Θ. Moreover,ΨE∗M is natural in M with M varying in MOD; that is, the square

⟨E∗,M⟩ΨE∗M //

⟨E∗,Φ⟩

⟨∆E∗,M⟩⟨∆E∗,Φ⟩

⟨E∗,N ⟩ΨE∗N

// ⟨∆E∗,N ⟩


Proof. Since ∆∗E is the corepresentable module of the functor ∆E ∶ E→ ∗, and since ⟨∗E,M⟩ =⟨∆E M⟩ (see Theorem4.6.19) and ⟨∗E,Φ⟩ = ⟨∆E Φ⟩ (see Corollary 4.6.20), the assertionfollows as a special case of Theorem5.5.2 where F is given by ∆E, with Ψ∗E

M dened by

Ψ∗EM = Ψ∆E

M

; the conical cell ⟨α⟩ is given by ∗α and the cone [Θ] is given by ∗Θ.

Theorem 5.5.7. Given an endomodule M ∶ E E, there is a canonical bijection

ΦEM ∶∏

E

M ≅ (⟨E⟩) ⟨E ∶ E⟩ (M)

from the set of frames of M to the set of module morphisms ⟨E⟩ →M ∶ E E. The bijectionsends each frame α of M to the module morphism ⟨α⟩ ∶ ⟨E⟩ →M which maps each E-arrowh ∶ e→ e′ to the M-arrow h αe′ = αe h ∶ e→ e′ as indicated in

e

h

αe //

h ⋅ ⟨α⟩&&

e

he′ αe′

// e′

; the inverse sends each module morphism Φ ∶ ⟨E⟩→M to the frame [Φ] of M dened by

[Φ]e = 1e ⋅Φ

for e ∈ ∥E∥, where 1e ⋅Φ is the image of the identity e→ e under the function

e ⟨Φ⟩e ∶ e ⟨E⟩e→ e ⟨M⟩e

. Moreover, the bijection is natural in M.

Proof. This follows from Corollary 5.5.4 by setting X =A = E and S = T = 1E and noting that(1E) ⟨E,M⟩ (1E) =∏EM and (1E) ⟨⟨E⟩ ,M⟩ (1E) = (⟨E⟩) ⟨E ∶ E⟩ (M) by the denitions.

Note. In Theorem5.5.8 and Theorem5.5.9, the hom of a category E is regarded as a left module⟨E⟩ ∶ ∗ E∼ ×E (resp. a right module ⟨E⟩ ∶ E∼ ×E ∗).

Theorem 5.5.8.

191

5. Yoneda Lemma

Given a left module M ∶ ∗ E∼ ×E, there is a canonical bijection

ΦEM ∶∏

E

M ≅ (⟨E⟩) ⟨∶ E∼ ×E⟩ (M)

from the set of cylindrical frames of M to the set of module morphisms ⟨E⟩ → M ∶ ∗ E∼ ×E. The bijection sends each cylindrical frame α of M to the module morphism ⟨α⟩ ∶⟨E⟩→M which maps each E-arrow h ∶ e→ e′ to theM-arrow αe′ (h,e′) = αe (e,h) ∶ ∗(e,e′) as indicated in

∗αe′

αe //

h ⋅ ⟨α⟩''

(e,e)(e,h)

(e′,e′)(h,e′)

// (e,e′)

. The inverse sends each module morphism Φ ∶ ⟨E⟩→M to the cylindrical frame [Φ] ofMdened by

[Φ]e = Φ ⋅ 1e

for e ∈ ∥E∥, where Φ ⋅ 1e is the image of the identity e→ e under the function

⟨Φ⟩ (e,e) ∶ ⟨E⟩ (e,e)→ ⟨M⟩ (e,e)

. Moreover, the bijection is natural in M. Given a right module M ∶ E∼ ×E ∗, there is a canonical bijection

ΦEM ∶∏

E

M ≅ (⟨E⟩) ⟨E∼ ×E ∶⟩ (M)

from the set of cylindrical frames ofM to the set of module morphisms ⟨E⟩→M ∶ E∼ ×E∗. The bijection sends each cylindrical frame α ofM to the module morphism ⟨α⟩ ∶ ⟨E⟩→Mwhich maps each E-arrow h ∶ e→ e′ to theM-arrow (e′,h) αe′ = (h,e) αe ∶ (e′,e) ∗ asindicated in

(e′,e)(h,e)

(e′,h) //

h ⋅ ⟨α⟩''

(e′,e′)αe′(e,e) αe// ∗

. The inverse sends each module morphism Φ ∶ ⟨E⟩→M to the cylindrical frame [Φ] ofMdened by

[Φ]e = 1e ⋅Φfor e ∈ ∥E∥, where 1e ⋅Φ is the image of the identity e→ e under the function

(e,e) ⟨Φ⟩ ∶ (e,e) ⟨E⟩→ (e,e) ⟨M⟩

. Moreover, the bijection is natural in M.

Proof. This is a restatement of Theorem5.5.7 with M ∶ E E regarded as a right moduleM ∶ E∼ ×E ∗.

Theorem 5.5.9. Given a category E and a module M ∶XA,

192

5. Yoneda Lemma

there is a canonical module isomorphism

ΨEM ∶ ⟨E,M⟩ ≅ ⟨⟨E⟩ ,M⟩ ∶X [E∼ ×E,A]

, giving for each pair of an object x ∈ ∥X∥ and a bifunctor K ∶ E∼ ×E→X a bijection

(x) ⟨ΨEM⟩ (K) ∶ (x) ⟨E,M⟩ (K) ≅ (x) ⟨⟨E⟩ ,M⟩ (K)

from the set of cylinders x K ∶ E M to the set of cells x K ∶ ⟨E⟩ M. Specically,the bijection sends each cylinder

∗x

E∼ ×E

K

X M//

α

A

to the cell

∗ ⟨E⟩ //

x

E∼ ×E

K

X M//

⟨α⟩

A


αe′ K (h,e′) = αe K (e,h) ∶ x K (e,e′)

as indicated inx

αe′

αe //

⟨α⟩ ⋅ h((

K (e,e)K(e,h)

K (e′,e′)K(h,e′)

// K (e,e′)


∗ ⟨E⟩ //

x

E∼ ×E

K

X M//

Θ

A

to the cylinder [Θ] ∶ x K ∶ EM dened by

[Θ]e = Θ ⋅ 1e

for e ∈ ∥E∥, where Θ ⋅ 1e is the image of the identity e→ e under the function

⟨E⟩ (e,e) ⟨Θ⟩(e,e) // ⟨x ⟨M⟩K⟩ (e,e) = x ⟨M⟩ (K (e,e))


⟨E,M⟩ΨEM //

⟨E,Φ⟩

⟨⟨E⟩ ,M⟩⟨⟨E⟩,Φ⟩

⟨E,N ⟩ΨEN// ⟨⟨E⟩ ,N ⟩


193

5. Yoneda Lemma

there is a canonical module isomorphism

ΨEM ∶ ⟨E,M⟩ ≅ ⟨⟨E⟩ ,M⟩ ∶ [E∼ ×E,X]A

, giving for each pair of an object a ∈ ∥A∥ and a bifunctor K ∶ E∼ ×E→X a bijection

(K) ⟨ΨEM⟩ (a) ∶ (K) ⟨E,M⟩ (a) ≅ (K) ⟨⟨E⟩ ,M⟩ (a)

from the set of cylinders K a ∶ E M to the set of cells K a ∶ ⟨E⟩ M. Specically,the bijection sends each cylinder

E∼ ×E

K

∗a

X M//

α

A

to the cell

E∼ ×E

K

⟨E⟩ // ∗a

X M//

⟨α⟩

A


K (e′,h) αe′ = K (h,e) αe ∶ K (e′,e) a

as indicated in

K (e′,e)K(h,e)

K(e′,h) //

h ⋅ ⟨α⟩((

K (e′,e′)αe′

K (e,e) αe// a


E∼ ×E

K

⟨E⟩ // ∗a

X M//

Θ

A

to the cylinder [Θ] ∶ K a ∶ EM dened by

[Θ]e = 1e ⋅Θ

for e ∈ ∥E∥, where 1e ⋅Θ is the image of the identity e→ e under the function

(e,e) ⟨E⟩ (e,e)⟨Θ⟩ // (e,e) ⟨K ⟨M⟩a⟩ = (K (e,e)) ⟨M⟩a


⟨E,M⟩ΨEM //

⟨E,Φ⟩

⟨⟨E⟩ ,M⟩⟨⟨E⟩,Φ⟩

⟨E,N ⟩ΨEN// ⟨⟨E⟩ ,N ⟩


194

5. Yoneda Lemma

Proof. ReplacingM with x ⟨M⟩K in Theorem5.5.8, we have a bijection

ΦEx⟨M⟩K ∶∏

E

x ⟨M⟩K ≅ (⟨E⟩) ⟨∶ E∼ ×E⟩ (x ⟨M⟩K)

, natural in x and K. But

∏E

x ⟨M⟩K = (x) ⟨E,M⟩ (K)

and(⟨E⟩) ⟨∶ E∼ ×E⟩ (x ⟨M⟩K) = (x) ⟨⟨E⟩ ,M⟩ (K)

by the denitions. Hence an isomorphism ΨEM ∶ ⟨E,M⟩ ≅ ⟨⟨E⟩ ,M⟩ is given by

(x) ⟨ΨEM⟩ (K) = ΦE

x⟨M⟩K

. The last assertion holds since for any cell

X

P

M //A

Q

Y N//

Φ

B

, the square

x ⟨E,M⟩Kx⟨ΨE

M⟩K//

x⟨E,Φ⟩K

x ⟨⟨E⟩ ,M⟩K

x⟨⟨E⟩,Φ⟩K

(x ⋅ P) ⟨E,N ⟩ (Q K)(x ⋅ P)⟨ΨE

N ⟩(Q K)// (x ⋅ P) ⟨⟨E⟩ ,N ⟩ (Q K)

, i.e.

∏E x ⟨M⟩KΦE

x⟨M⟩K //

∏E x⟨Φ⟩K

(⟨E⟩) ⟨∶ E∼ ×E⟩ (x ⟨M⟩K)(⟨E⟩)⟨∶E∼×E⟩(x⟨Φ⟩K)

∏E x ⟨P ⟨N ⟩Q⟩KΦE

x⟨P⟨N ⟩Q⟩K

// (⟨E⟩) ⟨∶ E∼ ×E⟩ (x ⟨P ⟨N ⟩Q⟩K)

commutes by the naturality of ΦEM.

195

6. Universals

6.1. Units of one-sided modules

Denition 6.1.1.

An arrow u ∶ r ∗ of a right module M ∶ X ∗ is called a unit1 if the right modulemorphism Xu ∶ ⟨X⟩ r→M ∶X ∗ is iso.

An arrow u ∶ ∗ r of a left moduleM ∶ ∗A is called a unit if the left module morphismuA ∶ r ⟨A⟩→M ∶ ∗A is iso.

Remark 6.1.2.(1) By the Yoneda lemma (Theorem5.2.10), representations and units correspond one-to-one;

if u ∶ r ∗ is a unit of a right module M ∶ X ∗, then the object r represents Mby the isomorphism ⟨Xu⟩−1 ∶M → ⟨X⟩ r; conversely, if an object r ∈ ∥X∥ and a rightmodule isomorphism Υ ∶ M → ⟨X⟩ r form a representation of M, then the M-arrow1r ⋅Υ−1 ∶ r ∗ is a unit ofM.

if u ∶ ∗ r is a unit of a left module M ∶ ∗ A, then the object r represents Mby the isomorphism ⟨uA⟩−1 ∶M → r ⟨A⟩; conversely, if an object r ∈ ∥A∥ and a leftmodule isomorphism Υ ∶ M → r ⟨A⟩ form a representation of M, then the M-arrowΥ−1 ⋅ 1r ∶ ∗ r is a unit ofM.

(2) Let u ∶ r ∗ be a unit of a right moduleM ∶ X ∗. Given anM-arrow m ∶ x ∗, its

inverse image under Xu is called the adjunct of m along u and written m/u; that is,

m/u ∶= m ⋅ ⟨Xu⟩−1

; this is the unique X-arrow x→ r making the triangle

xm/u

m

%%r u

// ∗

commute. An M-arrow u ∶ r ∗ is a unit if and only if to every M-arrow m ∶ x ∗there is a unique X-arrow m/u ∶ x→ r as above.

Let u ∶ ∗ r be a unit of a left module M ∶ ∗ A. Given an M-arrow m ∶ ∗ a, itsinverse image under uA is called the adjunct of m along u and written u/m; that is,

u/m ∶= ⟨uA⟩−1 ⋅m

; this is the unique A-arrow r→ a making the triangle

∗ u //

m %%

ru/ma

1A unit is called a universal element in the literature.

196

6. Universals

commute. An M-arrow u ∶ ∗ r is a unit if and only if to every M-arrow m ∶ ∗ athere is a unique A-arrow u/m ∶ r→ a as above.

Proposition 6.1.3.

an arrow of a right module M is a unit if and only if it is a terminal object of the commacategory ⌈M↓⌉.

an arrow of a left module M is a unit if and only if it is an initial object of the commacategory ⌈M↓⌉.

Proof. Immediate by the last sentence of Remark 6.1.2(2).

Proposition 6.1.4.

A right module M is representable if and only if the comma category ⌈M↓⌉ has a terminalobject.

A left module M is representable if and only if the comma category ⌈M↓⌉ has an initialobject.

Proof. SinceM is representable i it has a unit (see Remark 6.1.2(1)), this is immediate fromProposition 6.1.3.

Theorem 6.1.5.

Suppose that a right module M ∶ X ∗ has a unit u ∶ r ∗. Then an X-arrow f ∶ s → r isiso if and only if the composite f u ∶ s ∗ is a unit ofM; to put it the other way round, anM-arrow v ∶ s ∗ is a unit if and only if its adjunct v/u ∶ s→ r along u is an iso X-arrow.

Suppose that a left moduleM ∶ ∗A has a unit u ∶ ∗ r. Then an A-arrow f ∶ r→ s is isoif and only if the composite u f ∶ ∗ s is a unit of M; to put it the other way round, anM-arrow v ∶ ∗ s is a unit if and only if its adjunct u/v ∶ r→ s along u is an iso A-arrow.

Proof. By the naturality of the Yoneda morphism

X(f u) = ⟨X⟩ f Xu

. Since u is a unit, Xu is iso. Hence X(f u) is iso i ⟨X⟩ f is iso. But X(f u) is iso if u is a unit, and, since the Yoneda functor is fully faithful, ⟨X⟩ f is iso i f is iso.

Theorem 6.1.6.

Let M ∶ X ∗ be a right module. If u ∶ r ∗ and v ∶ s ∗ are two units, then the adjunctv/u ∶ s→ r of v along u and the adjunct u/v ∶ r→ s of u along v, as indicated in

s

v/u

v

(( ∗r

u/v

OO

u

66

, are the inverse of each other. Let M ∶ ∗ A be a left module. If u ∶ ∗ r and v ∶ ∗ s are two units, then the adjunct

u/v ∶ r→ s of v along u and the adjunct v/u ∶ s→ r of u along v, as indicated in

r

u/v

∗

u66

v (( s

v/u

OO

, are the inverse of each other.

197

6. Universals

Proof. Sinceu/v v/u u = u/v v = u

and u is a unit, u/v v/u = 1r by the uniqueness of the factorization. Symmetrically, v/u u/v =1s.

Corollary 6.1.7. If a right (resp. left) module is representable, then a representing object isunique up to isomorphism.

Proof. Since a representing object has a unit associated with it (see Remark 6.1.2(1)), this isimmediate from Theorem6.1.6 (or Theorem6.1.5).

Denition 6.1.8.

A right module cell

XM //

P

∗1

Y N//

Φ

∗

is said to(1) preserve units if Φ sends each unit u ∶ r ∗ ofM to a unit u ⋅Φ ∶ r ⋅ P ∗ of N ;(2) reect units if anM-arrow u ∶ r ∗ is a unit whenever the N -arrow u ⋅Φ ∶ r ⋅P ∗ is a

unit;(3) create units if for every unit v ∶ s ∗ of N there is exactly oneM-arrow u ∶ r ∗ with

u ⋅Φ = v, and if this u is a unit. A left module cell

∗ M //

1

A

Q

∗ N//

Φ

B

is said to(1) preserve units if Φ sends each unit u ∶ ∗ r ofM to a unit u ⋅Φ ∶ ∗ Q ⋅ r of N ;(2) reect units if anM-arrow u ∶ ∗ r is a unit whenever the N -arrow u ⋅Φ ∶ ∗ Q ⋅ r is a

unit;(3) create units if for every unit v ∶ ∗ s of N there is exactly oneM-arrow u ∶ ∗ r with

u ⋅Φ = v, and if this u is a unit.

Proposition 6.1.9. If a right (resp. left) module cell creates units, then it reects units.

Proof. Obvious by the denitions.

Proposition 6.1.10. If a right (resp. left) module cell preserves a unit, then it preserves everyunit; that is, if a right module cell Φ ∶ P ∗ ∶M N sends a unit u ∶ r ∗ ofM to a unit u ⋅Φ ∶ r ⋅P ∗

of N , then Φ sends any other unit v ∶ s ∗ of M to a unit v ⋅Φ ∶ s ⋅ P ∗ of N as well. if a left module cell Φ ∶ ∗ Q ∶M N sends a unit u ∶ ∗ r ofM to a unit u ⋅Φ ∶ ∗ Q ⋅ r

of N , then Φ sends any other unit v ∶ ∗ s of M to a unit v ⋅Φ ∶ ∗ Q ⋅ s of N as well.

Proof. If u is a unit ofM, another unit v ofM is written as v = v/uu, with v/u an isomorphismby Theorem6.1.5. Φ thus sends v to theN -arrow v⋅Φ = (v/u ⋅ P) (u ⋅Φ) (see Proposition 1.2.3),with (v/u ⋅ P) an isomorphism (because any functor preserves isomorphisms). Hence if u ⋅Φ isa unit of N , so is v ⋅Φ again by Theorem6.1.5.

198

6. Universals

Proposition 6.1.11. Consider a cell Φ as in Denition 6.1.8. If N has a unit and Φ createsunits, then M has a unit as well and Φ preserves units.

Proof. ClearlyM has a unit. Since Φ preserves the units it has created, it preserves every unitby Proposition 6.1.10.

Proposition 6.1.12. If a right (resp. left) module cell is iso, then it preserves, reects, andcreates units.

Proof. Evident.

6.2. Universal arrows

Denition 6.2.1. LetM ∶XA be a module. An M-arrow u ∶ r a is inverse universal if the right module morphism Xu ∶ ⟨X⟩ r →

⟨M⟩a ∶ X ∗ is iso. Given an object a ∈ ∥A∥, an inverse universal M-arrow u ∶ r a orthe pair (r,u), or the object r itself, is called a universal of a inverse alongM.

AnM-arrow u ∶ x r is direct universal if the left module morphism uA ∶ r ⟨A⟩→ x ⟨M⟩ ∶∗ A is iso. Given an object x ∈ ∥X∥, a direct universal M-arrow u ∶ x r or the pair(r,u), or the object r itself, is called a universal of x direct alongM.

Remark 6.2.2.(1) An M-arrow u ∶ x r is direct universal if and only if the M∼-arrow u ∶ r x is inverse

universal.(2) By Remark 5.2.6(2),

anM-arrow u ∶ r a is inverse universal if and only if it is a unit of the right module⟨M⟩a ∶X ∗, and a unit u ∶ r ∗ of a right moduleM ∶X ∗ is the same thing as aninverse universal arrow of M regarded as a two-sided module from X to the terminalcategory.

an M-arrow u ∶ x r is direct universal if and only if it is a unit of the left modulex ⟨M⟩ ∶ ∗ A, and a unit u ∶ ∗ r of a left moduleM ∶ ∗ A is the same thing as adirect universal arrow ofM regarded as a two-sided module from the terminal categoryto A.

(3) Remark 6.1.2(2) is repeated below in terms of universal arrows. Let u ∶ r a be an inverse universalM-arrow. Given anM-arrow m ∶ x a, its inverse

image under Xu is called the adjunct of m along u and written m/u; that is,

m/u ∶= m ⋅ ⟨Xu⟩−1

; this is the unique X-arrow x→ r making the triangle

xm/u

m

%%r u

// a

commute. An M-arrow u ∶ r a is inverse universal if and only if to every M-arrowm ∶ x a there is a unique X-arrow m/u ∶ x→ r as above.

Let u ∶ x r be a direct universal M-arrow. Given an M-arrow m ∶ x a, its inverseimage under uA is called the adjunct of m along u and written u/m; that is,

u/m ∶= m ⋅ ⟨uA⟩−1

199

6. Universals

; this is the unique A-arrow r→ a making the triangle

xu //

m %%

ru/ma

commute. An M-arrow u ∶ x r is direct universal if and only if to every M-arrowm ∶ x a there is a unique A-arrow u/m ∶ r→ a as above.

(4) AnM-arrow u ∶ r s is called two-way universal if it is both inverse and direct universal.

Example 6.2.3. Let F ∶D→ E be a functor. An inverse universal of an object e ∈ ∥E∥ along the representable module F ⟨E⟩ ∶ D E is

an F ⟨E⟩-arrow u ∶ r e (i.e. E-arrow u ∶ r ⋅ F → e) such that to every F ⟨E⟩-arrow f ∶ d e(i.e. E-arrow f ∶ d ⋅ F→ e), there is a unique D-arrow f/u ∶ d→ r such that the triangle

d

f/u

d ⋅ F(f/u) ⋅ F

f

&&r r ⋅ F u

// e

commutes. An E-arrow u ∶ r ⋅ F → e is said to be universal from F to e if the F ⟨E⟩-arrowu ∶ r e is inverse universal.

A direct universal of an object e ∈ ∥E∥ along the corepresentable module ⟨E⟩F ∶ E D isan ⟨E⟩F-arrow u ∶ e r (i.e. E-arrow u ∶ e → F ⋅ r) such that to every ⟨E⟩F-arrow f ∶ e d(i.e. E-arrow f ∶ e→ F ⋅ d), there is a unique D-arrow u/f ∶ r→ d such that the triangle

eu //

f &&

F ⋅ rF ⋅ (u/f)

r

u/f

F ⋅ d d

commutes. An E-arrow u ∶ e → F ⋅ r is said to be universal from e to F if the ⟨E⟩F-arrowu ∶ e r is direct universal.

Remark 6.2.4. The module F ⟨E⟩ ∶ D E and ⟨E⟩F ∶ E D abstract the commutativediagrams above into

d

f/u

f

%%

eu //

f %%

r

u/f

r u// e d

and present a simpler and more conceptual view of universals.

Proposition 6.2.5. For any arrow f ∶ r → s of a category C, the following conditions areequivalent:(1) C-arrow f ∶ r→ s is iso;(2) ⟨C⟩-arrow f ∶ r s is inverse universal;(3) ⟨C⟩-arrow f ∶ r s is direct universal;(4) ⟨C⟩-arrow f ∶ r s is two-way universal,where ⟨C⟩ is the hom of C.

Proof. By denition, f ∶ r s is universal inverse along ⟨C⟩ i the right module morphismC f ∶ ⟨C⟩ r → ⟨C⟩ s is iso. By Proposition 5.2.8 C f = ⟨C⟩ f, and by the fully faithfulness ofthe Yoneda functor, ⟨C⟩ f is iso i f is iso. The conditions (1) and (2) are thus equivalent. Theequivalence of the conditions (1) and (3) is proved dually, and the equivalence of the conditions(1) and (4) follows.

200

6. Universals

Note. Theorem6.2.6 and Theorem6.2.7 are restatements of Theorem6.1.5 and Theorem6.1.6in terms of universal arrows.

Theorem 6.2.6. Let M ∶XA be a module. Suppose that an object a ∈ ∥A∥ has a universal u ∶ r a inverse alongM. Then an X-arrow

f ∶ s → r is iso if and only if the composite f u ∶ s a is an inverse universal M-arrow;to put it the other way round, an M-arrow v ∶ s a is inverse universal if and only if itsadjunct v/u along u is an iso X-arrow.

Suppose that an object x ∈ ∥X∥ has a universal u ∶ x r direct alongM. Then an A-arrowf ∶ r→ s is iso if and only if the composite u f ∶ x s is a direct universalM-arrow; to putit the other way round, an M-arrow v ∶ x s is direct universal if and only if its adjunctu/v along u is an iso A-arrow.

Proof. Since an inverse universal M-arrow u ∶ r a (resp. v ∶ s a) is the same thing asa unit of the right module ⟨M⟩a ∶ X ∗ (see Remark 6.2.2(2)), the assertion follows fromTheorem6.1.5.

Theorem 6.2.7. Let M ∶XA be a module. If u ∶ r a and v ∶ s a are two inverse universal M-arrows, then the adjunct v/u ∶ s → r

of v along u and the adjunct u/v ∶ r→ s of u along v, as indicated in

s

v/u

v

(( a

r

u/v

OO

u

66

, are the inverse of each other. If u ∶ x r and v ∶ x s are two direct universal M-arrows, then the adjunct u/v ∶ r → s of

v along u and the adjunct v/u ∶ r→ s of u along v, as indicated in

r

u/v

x

u66

v (( s

v/u

OO

, are the inverse of each other.

Proof. Since an inverse universal M-arrow u ∶ r a (resp. v ∶ s a) is the same thing asa unit of the right module ⟨M⟩a ∶ X ∗ (see Remark 6.2.2(2)), the assertion follows fromTheorem6.1.6.

Corollary 6.2.8. Let M ∶XA be a module. A universal of an object a ∈ ∥A∥ inverse along M, if exists, is unique up to isomorphism. A universal of an object x ∈ ∥X∥ direct along M, if exists, is unique up to isomorphism.

Proof. Immediate from Theorem6.2.7 (or Theorem6.2.6).

Theorem 6.2.9. Let F ∶D→ E be a functor and e be an object of E. If there is an iso E-arrow u ∶ r ⋅F→ e universal from F to e, then every E-arrow v ∶ s ⋅F→ e

universal from F to e is iso. If there is an iso E-arrow u ∶ e→ F ⋅ r universal from e to F, then every E-arrow v ∶ e→ F ⋅ s

universal from e to F is iso.

201

6. Universals

Proof. Consider the commutative diagram

s

v/u

s ⋅ F(v/u) ⋅ F

v

&&r r ⋅ F u

// e

. We need to show that if u is iso, so is v. For this, it suces to show that (v/u) ⋅F is iso. Butthis holds because v/u is iso by Theorem6.2.6 and any functor preserves isomorphisms.

Theorem 6.2.10. Let M ∶XA be a module. Suppose that an object a ∈ ∥A∥ has a universal u ∶ r a inverse along M and let f ∶ a → b

be an iso A-arrow. Then the composite u f ∶ r b is also an inverse universal M-arrow. Suppose that an object x ∈ ∥X∥ has a universal u ∶ x r direct alongM and let f ∶ y → x be

an iso X-arrow. Then the composite f u ∶ y r is also a direct universal M-arrow.

Proof. By the naturality of the Yoneda morphism (see Denition 5.2.5),

X(u f) =Xu ⟨M⟩ f

. On the right side of the equation, Xu is iso because u is inverse universal, and ⟨M⟩ f is isobecause f is iso. Hence X(u f) is iso, i.e. u f is inverse universal.

Theorem 6.2.11. Let M ∶XA be a module and consider a commutative square

xµ //

g

a

f

y ν// b

consisting ofM-arrows µ and ν, an iso X-arrow g, and an iso A-arrow f. In this square, if µis inverse (resp. direct) universal, so is ν.

Proof. Suppose that µ is inverse universal. By Theorem6.2.10, u f is inverse inverse universal.Hence, by Theorem6.2.6, ν = g−1 u f is inverse universal.

Theorem 6.2.12.

Consider a composable pair of a module and a functor

XM //A E

Koo

. AnM-arrow u ∶ r K ⋅e is inverse universal if and only if so is the ⟨M⟩K-arrow u ∶ r e. Consider a composable pair of a module and a functor

EK //X

M //A

. AnM-arrow u ∶ e ⋅K r is direct universal if and only if so is the K ⟨M⟩-arrow u ∶ e r.

Proof. u ∶ r e is an inverse universal ⟨M⟩K-arrow i u ∶ r ∗ is a unit of ⟨⟨M⟩K⟩e,and u ∶ r K ⋅ e is an inverse universal M-arrow i u ∶ r ∗ is a unit of ⟨M⟩ (K ⋅ e). But⟨⟨M⟩K⟩e = ⟨M⟩ (K ⋅ e).

Corollary 6.2.13. Let F ∶D→ E be a functor. An E-arrow u ∶ r→ F ⋅ d is iso if and only if the ⟨E⟩F-arrow u ∶ r d is inverse universal.

202

6. Universals

An E-arrow u ∶ d ⋅ F→ r is iso if and only if the F ⟨E⟩-arrow u ∶ d r is direct universal.

Proof. By Proposition 6.2.5, u ∶ r → F ⋅ d is iso i it is an inverse universal ⟨E⟩-arrow. Theassertion thus follows from Theorem6.2.12.

Theorem 6.2.14.

Consider a composable pair of a functor and a module

EK //X

M //A

with K fully faithful. If anM-arrow u ∶ r ⋅K a is inverse universal, so is the K ⟨M⟩-arrowu ∶ r a.

Consider a composable pair of a functor and a module

XM //A E

Koo

with K fully faithful. If an M-arrow u ∶ x K ⋅ r is direct universal, so is the ⟨M⟩K-arrowu ∶ x r.

Proof. By Example 5.2.9(1), Eu is given by the composition ⟨K⟩ r K ⟨Xu⟩. Since K is fullyfaithful, ⟨K⟩ r is iso; since u ∶ r ⋅K a is an inverse universalM-arrow, Xu is iso, and henceso is K ⟨Xu⟩. Eu is thus iso.

Corollary 6.2.15. Let F ∶D→ E be a fully faithful functor. If an E-arrow u ∶ r ⋅ F → e is iso, then it is universal from F to e, i.e. the F ⟨E⟩-arrow

u ∶ r e is inverse universal. If an E-arrow u ∶ e → F ⋅ r is iso, then it is universal from e to F, i.e. the ⟨E⟩F-arrow

u ∶ e r is direct universal.

Proof. By Proposition 6.2.5, u ∶ r ⋅ F e is an inverse universal ⟨E⟩-arrow. Hence, by Theo-rem6.2.14, u ∶ r e is an inverse universal F ⟨E⟩-arrow.

Theorem 6.2.16. Consider a pair of functors

EF // C

⟨C⟩ // C DGoo

with both F and G fully faithful. If a C-arrow u ∶ r ⋅ F → G ⋅ s is iso, then the F ⟨C⟩G-arrowu ∶ r s is two-way universal.

Proof. By Corollary 6.2.15, the F ⟨C⟩-arrow u ∶ r G ⋅ s (resp. ⟨C⟩G-arrow u ∶ r ⋅ F s) isinverse (resp. direct) universal; hence, by Theorem6.2.12, the F ⟨C⟩G-arrow u ∶ r s is inverse(resp. direct) universal.

Denition 6.2.17. A cell

XM //

P

A

Q

Y N//

Φ

B

is said to(1) preserve

inverse universals if Φ sends each inverse universal M-arrow u ∶ r a to an inverseuniversal N -arrow u ⋅Φ ∶ r ⋅ P Q ⋅ a.

203

6. Universals

direct universals if Φ sends each direct universalM-arrow u ∶ x r to a direct universalN -arrow u ⋅Φ ∶ x ⋅ P Q ⋅ r.

(2) reect inverse universals if an M-arrow u ∶ r a is inverse universal whenever the N -arrow

u ⋅Φ ∶ r ⋅ P Q ⋅ a is inverse universal. direct universals if an M-arrow u ∶ x r is direct universal whenever the N -arrow

u ⋅Φ ∶ x ⋅ P Q ⋅ r is direct universal.(3) create

inverse universals if for every object a ∈ ∥A∥ and for every inverse universal N -arrowv ∶ s Q ⋅ a there is exactly oneM-arrow u ∶ r a with u ⋅Φ = v, and if this u is inverseuniversal.

direct universals if for every object x ∈ ∥X∥ and for every direct universal N -arrowv ∶ x ⋅ P s there is exactly oneM-arrow u ∶ x r with u ⋅Φ = v, and if this u is directuniversal.

Remark 6.2.18. By Remark 6.2.2(2), Φ preserves (reects, creates) inverse universals if and only if each right slice (see Deni-

tion 2.1.8) preserves (reects, creates) units in the sense of Denition 6.1.8. Φ preserves (reects, creates) direct universals if and only if each left slice (see Deni-

tion 2.1.8) preserves (reects, creates) units in the sense of Denition 6.1.8.

Proposition 6.2.19. If a cell creates inverse (resp. direct) universals, then it reects inverse(resp. direct) universals.

Proof. Obvious by the denitions.

Remark 6.2.20. See also Theorem6.3.12.

Proposition 6.2.21. Consider a cell Φ as in Denition 6.2.17. If Φ is fully faithful and P is iso, then Φ preserves, reects, and creates inverse universals. If Φ is fully faithful and Q is iso, then Φ preserves, reects, and creates direct universals.

Proof. If Φ is fully faithful and P is iso, then each right slice of Φ is iso by Proposition 2.1.9.The assertion thus follows from Proposition 6.1.12 by noting Remark 6.2.18.

Proposition 6.2.22.

Given a cell and a commutative square of functors as in

XM //

P

A

Q

DKoo

F

Y N//

Φ

B EL

oo

, if the cell Φ preserves (reects, creates) inverse universals, so does the composite

X

P

⟨M⟩K //D

F

Y⟨N ⟩L

//

⟨Φ⟩K

E

.

204

6. Universals

Given a cell and a commutative square of functors as in

DK //

F

XM //

P

A

Q

EL

//Y N//

Φ

B

, if the cell Φ preserves (reects, creates) direct universals, so does the composite

D

F

K⟨M⟩ //A

Q

EL⟨N ⟩

//

K⟨Φ⟩

B

.

Proof. Suppose that Φ preserves (resp. reects) inverse universals. By Theorem6.2.12, wesee that the cell ⟨Φ⟩K also preserves (resp. reects) inverse universals. Now suppose thatΦ creates inverse universals. To see that ⟨Φ⟩K also creates inverse universals, let d ∈ ∥D∥and suppose that an ⟨N ⟩L-arrow v ∶ s F ⋅ d is inverse universal. Then, by Theorem6.2.12,N -arrow v ∶ s L ⋅ F ⋅ d = Q ⋅ K ⋅ d is inverse universal. Hence there is exactly one M-arrowu ∶ r K ⋅ d whose image under Φ is the N -arrow v ∶ s Q ⋅ K ⋅ d; that is, there is exactlyone ⟨M⟩K-arrow u ∶ r d whose image under ⟨Φ⟩K is the ⟨N ⟩L-arrow v ∶ s F ⋅ d. SincetheM-arrow u ∶ r K ⋅d is inverse universal, again by Theorem6.2.12, so is the ⟨M⟩K-arrowu ∶ r d.

Theorem 6.2.23. A right (resp. left) Yoneda morphism preserves and reects inverse (resp.direct) universal.

Proof. The right Yoneda morphism for a moduleM ∶X A (see Denition 5.2.5) sends eachM-arrow u ∶ r a to the right module Xu ∶ ⟨X⟩ r → ⟨M⟩a ∶ X ∗. But by denition,u ∶ r a is inverse universal i Xu is iso, and by Proposition 6.2.5, Xu is iso i Xu isinverse universal.

Denition 6.2.24. LetM ∶XA be a module. Given a pair of inverse universal M-arrows u ∶ r a and v ∶ s b, the conjugate of an

A-arrow f ∶ a → b inverse along (u, v) is the X-arrow given by (u f) /v, the adjunct of thecomposite u f along v, i.e. the unique X-arrow g ∶ r→ s making the square

rg

u // a

fs v

// b

commute. This commutative square, or the assignment f ↦ g, is called an inverse conjuga-tion by (u, v).

Given a pair of direct universal M-arrows u ∶ x r and v ∶ y s, the conjugate of anX-arrow g ∶ x → y direct along (u, v) is the A-arrow given by u/ (g v), the adjunct of thecomposite g v along u, i.e. the unique A-arrow f ∶ r→ s making the square

xu //

g

r

f

y v// s

commute. This commutative square, or the assignment g ↦ f, is called a direct conjugationby (u, v).

205

6. Universals

Proposition 6.2.25. Let M ∶XA be a module.(1) Conjugation is functorial in the following sense:

a) for any inverse (resp. direct) universal M-arrow u, the identities form an inverse(resp. direct) conjugation

r1

u // s1

r u// s

.b) if in the diagram

rg

u // s

fr′

g′

u′// s′

f′

r′′u′′

// s′′

, each of the two inner squares is an inverse (resp. direct) conjugation, so is the outerrectangle.

(2) Conjugation preserve isomorphisms; that is, given an inverse (resp. direct) conjugation

rg

u // a

fs v

// b

, if f (resp. g) is an isomorphism, so is g (resp. f).(3) Conjugation is universal in the following sense.

Given an inverse conjugation

rg

u // a

fs v

// b

and any commutative square

x

h

m // a

f

y n// b

, the adjunct of m along u and the adjunct of n along v yield a unique pair of X-arrowsmaking the diagram

x

m/u

h

m

''y

n/v

n

''

rg

u // a

f

s v// b

commute. Given a direct conjugation

xg

u // r

f

y v// s

206

6. Universals

and any commutative square

xg

m // a

h

y n// b

, the adjunct of m along u and the adjunct of n along v yield a unique pair of A-arrowsmaking the diagram

xg

u //

m

''

rf

u/m

y

v //

n

''

s

v/n

a

h

b

commute.

Proof.(1) Evident.(2) Any functorial operation preserves isomorphisms.(3) The uniqueness of such a pair follows from the uniqueness of the adjunct. The proof is

thus complete if we show that the square

x

h

m/u // rg

yn/v

// s

commutes. But since

h n/v v = h n = m f = m/u u f = m/u g v

and v is inverse universal, we have

h n/v = m/u g

by the uniqueness of the factorization.

6.3. Units of two-sided modules

Denition 6.3.1. LetM ∶XA be a module. A right cylinder

XM

//µ ARoo

or the pair (R, µ) is called a counit ofM if the module morphism Xµ ∶ ⟨X⟩R→M ∶XAis iso.

A left cylinder

XM //

R//µ A

or the pair (R, µ) is called a unit ofM if the module morphism µA ∶ R ⟨A⟩ →M ∶X Ais iso.

207

6. Universals

Remark 6.3.2. By the general Yoneda lemma (Corollary 5.3.17), representations and unitscorrespond one-to-one; if a right cylinder µ ∶ R M is a counit of a module M ∶ X A, then the functor R

corepresentsM by the isomorphism ⟨Xµ⟩−1 ∶M→ ⟨X⟩R; conversely, if a functor R ∶A→Xand a module isomorphism Υ ∶M → ⟨X⟩R form a corepresentation of M, then the rightcylinder XΥ−1 ∶ RM is a counit ofM.

if a left cylinder µ ∶M R is a unit of a moduleM ∶XA, then the functor R representsM by the isomorphism ⟨µA⟩−1 ∶ M → R ⟨A⟩; conversely, if a functor R ∶ X → A and amodule isomorphism Υ ∶ M → R ⟨A⟩ form a representation of M, then the left cylinderΥ−1A ∶M R is a unit ofM.

Proposition 6.3.3. Let M ∶XA be a module. A right cylinder µ ∶ RM is a counit of M if and only if each component µa ∶ a ⋅ R a is

an inverse universal M-arrow. A left cylinder µ ∶M R is a unit of M if and only if each component µx ∶ x R ⋅ x is a

direct universal M-arrow.

Proof. By Proposition 2.1.3, Xµ is iso i its each slice ⟨Xµ⟩a is iso. Since ⟨Xµ⟩a = Xµa

(see Corollary 5.3.11), ⟨Xµ⟩a is iso i µa is inverse universal.

Remark 6.3.4.(1) Proposition 6.3.3 gives an alternative denition of units of a module.(2) By Proposition 6.3.3 and Remark 6.2.2(2), under the identication in Remark 4.3.2(3),

a unit of a right moduleM ∶X ∗ is the same thing as a counit ofM regarded as thetwo-sided module from X to the terminal category.

a unit of a left module M ∶ ∗ A is the same thing as a unit of M regarded as thetwo-sided module from the terminal category to A.

Proposition 6.3.5. Let M ∶XA be a module. A counit of M is a universal of M inverse along the right general Yoneda module XdA,

i.e. a unit (in the sense of Denition 6.1.1) of the right module ⟨XdA⟩ (M). A unit of M is a universal of M direct along the left general Yoneda module XeA, i.e. a

unit (in the sense of Denition 6.1.1) of the left module (M) ⟨XeA⟩.

Proof. Let µ ∶ R M be a counit and α ∶ G M be a right cylinder. We need to show thatthere is a unique natural transformation α/µ ∶ G→ R ∶A→X making the triangle

G

α/µ

α

%%R µ

//M

commute. For an A-arrow f ∶ a→ b, consider the naturality squares

a ⋅ Rf ⋅ R

µa // a

f

a ⋅Gf ⋅G

αa // a

f

b ⋅ R µb// b b ⋅G αb

// b

given by µ and α. Since µa and µb are inverse universal by Proposition 6.3.3, the left squareforms an inverse conjugation. Hence, by Proposition 6.2.25(3), the adjunct of αa along µa and

208

6. Universals

the adjunct of αb along µb yield a unique pair of X-arrows making the diagram

a ⋅G

αa/µa

f ⋅G

αa

''b ⋅G

αb/µb

αb

((

a ⋅ Rf ⋅ R

µa // a

f

b ⋅ R µb// b

commute. The family of X-arrows αa/µa, one for each a ∈ ∥A∥, thus forms a unique naturaltransformation α/µ ∶ G→ R such that α/µ µ = α.

Remark 6.3.6. The converse does not hold. See Example 6.3.7.

Example 6.3.7. Let 2 denote the discrete category consisting of objects 0,1, 2 denote theinterval category, and letM ∶ 2 2 be a module which looks like

0 //

%%

0

1 // 1

. ThenM admits only one right cylinder

2M

//µ 2∆0oo

along it. µ is a universal ofM inverse along the module 2d2; however, µ is not a unit ofMsince µ1 ∶ 0 1 is not an inverse universalM-arrow.

Theorem 6.3.8. Let M ∶XA be a module. If µ ∶ RM and ν ∶ SM are two counits of M, then R and S are isomorphic. If µ ∶M R and ν ∶M S are two units of M, then R and S are isomorphic.

Proof. Since, as we saw in Proposition 6.3.5, R and S are universals of M inverse along theright general Yoneda module XdA, they are isomorphic by Corollary 6.2.8.

Corollary 6.3.9. A representing functor of a module, if exists, is unique up to isomorphism.

Proof. By Remark 6.3.2, this is just a restatement of Theorem6.3.8.

Theorem 6.3.10. Let M ∶XA be a module. If there is a family of inverse universal M-arrows µa ∶ ra a, one for each object a ∈ ∥A∥,

then there is a unique functor R ∶ A → X with R (a) = ra such that µ ∶= (µa)a∈∥A∥ forms aright cylinder RM; moreover, µ is a counit of M.

If there is a family of direct universal M-arrows µx ∶ x rx, one for each object x ∈ ∥X∥,then there is a unique functor R ∶ X → A with R (x) = rx such that µ ∶= (µx)x∈∥X∥ forms aleft cylinder µ ∶M R; moreover, µ is a unit of M.

209

6. Universals

Proof. The arrow function of R is given by the inverse conjugation

raR ⋅ f

µa // a

f

rb µb// b

(see Denition 6.2.24) for each A-arrow f ∶ a → b. R is functorial by Proposition 6.2.25(1) andthe uniqueness R follows from the uniqueness of a conjugate. Since each µa is inverse universal,µ forms a counit ofM by Proposition 6.3.3.

Note. The axiom of choice is used in the proof of the following.

Corollary 6.3.11. Given a module M ∶XA, the following conditions are equivalent:

(1) M is corepresentable;(2) M has a counit;(3) for every object a ∈ ∥A∥, the right module ⟨M⟩a ∶X ∗ is representable;(4) for every object a ∈ ∥A∥, the right module ⟨M⟩a ∶ X ∗ has a unit; that is, every

a ∈ ∥A∥ has a universal inverse along M. the following conditions are equivalent:

(1) M is representable;(2) M has a unit;(3) for every object x ∈ ∥X∥, the left module x ⟨M⟩ ∶ ∗A is representable;(4) for every object x ∈ ∥X∥, the left module x ⟨M⟩ ∶ ∗A has a unit; that is, every x ∈ ∥X∥

has a universal direct along M.

Proof.(1)⇔ (2) See Remark 6.3.2.(3)⇔ (4) See Remark 6.1.2(1).(2)⇒ (4) Immediate from Proposition 6.3.3.(4)⇒ (2) A family of inverse universal M-arrows µa ∶ ra a, one chosen for each a ∈ ∥A∥,

yields a counit ofM by Theorem6.3.10.

Theorem 6.3.12. Consider a cell Φ as in Denition 6.2.17. If N has a counit and Φ creates inverse universal, then M has a counit as well and Φ

preserves inverse universals. If N has a unit and Φ creates direct universal, then M has a unit as well and Φ preserves

direct universals.

Proof. By the equivalence of (2) and (4) in Corollary 6.3.11, and by Remark 6.2.18, this isreduced to Proposition 6.1.11.

Theorem 6.3.13. Let M ∶XA be a module. For a counit

XM

//µ ARoo

of M, the following conditions are equivalent:(1) the functor R is fully faithful;(2) each component of µ is direct universal (hence two-way universal).

210

6. Universals

For a unit

XM //

R//µ A

of M, the following conditions are equivalent:(1) the functor R is fully faithful;(2) each component of µ is inverse universal (hence two-way universal).

Proof. Let a and b be objects of A. The commutativity of the naturality square

a ⋅ Rf ⋅ R

µa // a

f

b ⋅ R µb// b

for every A-arrow f ∶ a→ b, translates into the commutativity of the triangle

a ⟨A⟩bf↦f ⋅ Rww

f↦µa f

&&(a ⋅ R) ⟨X⟩ (R ⋅ b)

h↦h µb// (a ⋅ R) ⟨M⟩b

. Since µb is inverse universal, the assignment h ↦ h µb is bijective. Hence the assignmentf ↦ f ⋅ R is bijective (i.e. R is fully faithful) i the assignment f ↦ µa f is bijective (i.e. µa isdirect universal).

6.4. Lifts

Denition 6.4.1.

A cylinderE

R

K

##X M

//

µ

A

is called(1) inverse universal if it is an inverse universal ⟨E,M⟩-arrow (see Denition 4.3.6);(2) pointwise inverse universal if each component µe ∶ e ⋅ R K ⋅ e is an inverse universalM-arrow.

Given a functor K ∶ E→A, an inverse universal (resp. pointwise inverse universal) cylinderµ ∶ R K ∶ EM or the pair (R, µ), or the functor R itself, is called a lift (resp. pointwiselift) of K inverse alongM.

A cylinderE

K

R

##X M

//

µ

A

is called(1) direct universal if it is a direct universal ⟨E,M⟩-arrow (see Denition 4.3.6);(2) pointwise direct universal if each component µe ∶ e ⋅ K R ⋅ e is a direct universalM-arrow.

211

6. Universals

Given a functor K ∶ E → X, a direct universal (resp. pointwise direct universal) cylinderµ ∶ K R ∶ EM or the pair (R, µ), or the functor R itself, is called a lift (resp. pointwiselift) of K direct alongM.

Remark 6.4.2.(1) A cylinder µ ∶ R K ∶ E M is inverse universal if and only if to every cylinder α ∶ L

K ∶ E M there is a unique natural transformation α/µ ∶ L → R such that α = α/µ µ.Dually, a cylinder µ ∶ K R ∶ E M is direct universal if and only if to every cylinderα ∶ K L ∶ E M, there is a unique natural transformation µ/α ∶ R → L such thatα = µ µ/α.

(2) A cylinder µ ∶ R S ∶ E M is called two-way universal (resp. pointwise two-wayuniversal) if it is both inverse and direct universal (resp. pointwise inverse and directuniversal).

Proposition 6.4.3.

For a cylinder µ ∶ R K ∶ EM the following conditions are equivalent:(1) µ is pointwise inverse universal;(2) the module morphism ⟨XM⟩µ ∶ ⟨X⟩R→ ⟨M⟩K ∶X E is iso;(3) the composite

ER

||K

""X

X

M //

µ

A

M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

of µ and the right Yoneda morphism for M forms a natural isomorphism E→ [X ∶]. For a cylinder µ ∶ K R ∶ EM the following conditions are equivalent:

(1) µ is pointwise direct universal;(2) the module morphism µ ⟨MA⟩ ∶ R ⟨A⟩→ K ⟨M⟩ ∶ EA is iso;(3) the composite

EK

R

##X

M

M //

µ

A

A

[∶A]∼⟨∶A⟩∼

//

⟨MA⟩

[∶A]∼

of µ and the left Yoneda morphism for M forms a natural isomorphism E→ [∶A]∼.

Proof.(1)⇔ (3) The component of the natural transformation ⟨⟨XM⟩⟩ µ at e ∈ ∥E∥ is given

by the right module morphism Xµe = ⟨XM⟩µe. Since a natural transformation is iso ieach component is an isomorphism, ⟨⟨XM⟩⟩ µ is iso i Xµe is iso, i.e. µe is inverseuniversal, for each e ∈ ∥E∥.

(2)⇔ (3) By Proposition 2.1.3, ⟨XM⟩µ is iso i ⟨⟨XM⟩µ⟩ is iso. But ⟨⟨XM⟩µ⟩=⟨⟨XM⟩⟩ µ (see Remark 5.3.10).

Remark 6.4.4. Proposition 6.4.3 gives alternative denitions of the pointwise universality of acylinder.

212

6. Universals

Proposition 6.4.5.

A counit

XM

//µ ARoo

of a module M is the same thing as a pointwise lift

AR

1

##X M

//

µ

A

of the identity A→A inverse along M. A unit

XM //

R//µ A

of a module M is the same thing as a pointwise lift

X1

R

##X M

//

µ

A

of the identity X→X direct along M.

Proof. By denition, (R, µ) is a unit ofM i the module morphism ⟨XM⟩µ ∶ ⟨X⟩R →M isiso. But by Proposition 6.4.3, this is the case i (R, µ) is a pointwise lift of the identity A→Ainverse alongM.

Remark 6.4.6. Proposition 6.3.3 now follows from Proposition 6.4.5.

Proposition 6.4.7.

A pointwise liftE

R

K

##X M

//

µ

A

of K inverse along M is the same thing as a counit

X⟨M⟩K

//µ ERoo

of the composite module ⟨M⟩K. A pointwise lift

EK

R

##X M

//

µ

A

of K direct along M is the same thing as a unit

XK⟨M⟩ //

R//µ A

of the composite module K ⟨M⟩.

213

6. Universals

Proof. Immediate from Proposition 6.4.3 and Proposition 5.3.7: by denition, a right cylinderµ ∶ R ⟨M⟩K is a unit i the module morphism ⟨X⟨M⟩K⟩µ ∶ ⟨X⟩R → ⟨M⟩K is iso, byProposition 6.4.3, a two-sided cylinder µ ∶ R K ∶ E M is a pointwise lift i the mod-ule morphism ⟨XM⟩µ ∶ ⟨X⟩R → ⟨M⟩K is iso, and these module isomorphisms coincide byProposition 5.3.7.

Proposition 6.4.8. A pointwise inverse (resp. direct) universal cylinder is inverse (resp.direct) universal.

Proof. Let µ ∶ R K ∶ E M be a pointwise inverse universal cylinder. By Proposition 6.4.7,µ is a unit of ⟨M⟩K. Hence, by Proposition 6.3.5, µ is a universal of ⟨M⟩K inverse alongXdA. Hence, by the identity in Theorem5.1.8, µ is a universal of K inverse along ⟨E,M⟩.

Proposition 6.4.9. For a natural transformation τ ∶ R→ S ∶ E→C, i.e. a cylinder τ ∶ R S ∶E ⟨C⟩, the following conditions are equivalent:(1) τ is a natural isomorphism in C;(2) τ is an inverse universal cylinder along the hom ⟨C⟩;(3) τ is a direct universal cylinder along the hom ⟨C⟩;(4) τ is a two-way universal cylinder along the hom ⟨C⟩;(5) τ is a pointwise inverse universal cylinder along the hom ⟨C⟩;(6) τ is a pointwise direct universal cylinder along the hom ⟨C⟩;(7) τ is a pointwise two-way universal cylinder along the hom ⟨C⟩.

Proof. Since a natural transformation τ ∶ R → S ∶ E → C is the same thing as an arrowτ ∶ R → S in the category [E,C], and since a cylinder τ ∶ R S ∶ E ⟨C⟩ is the same thing asan arrow τ ∶ R S in the module ⟨E, ⟨C⟩⟩ = ⟨E,C⟩ (see Remark 4.3.7(3)), the equivalence ofconditions (1), (2), (3), and (4) follows by applying Proposition 6.2.5 to the category [E,C].Since a natural transformation is iso i each component is an isomorphism, the equivalence ofconditions (1), (5), (6), and (7) follows again from Proposition 6.2.5.

Note. By Proposition 6.4.7, the following is a special case of Theorem6.3.10 (and vice versa byProposition 6.4.5).

Theorem 6.4.10. Let M be a module and K be a functor as in Denition 6.4.1. If there is a family of inverse universalM-arrows µe ∶ re K ⋅e, one for each object e ∈ ∥E∥,

then there is a unique functor R ∶ E → X with e ⋅ R = re such that µ ∶= (µe)e∈∥E∥ forms acylinder R K ∶ EM, and µ is pointwise inverse universal.

If there is a family of direct universalM-arrows µe ∶ e ⋅K re, one for each object e ∈ ∥E∥,then there is a unique functor R ∶ E → A with re = R ⋅ e such that µ ∶= (µe)e∈∥E∥ forms acylinder K R ∶ EM, and µ is pointwise direct universal.

Proof. Since an M-arrow µe ∶ re K ⋅ e is inverse universal i so is the ⟨M⟩K-arrow µe ∶re e (see Theorem6.2.12), by Proposition 6.4.7, the assertion is reduced to an instance ofTheorem6.3.10 whereM is given by the composite module ⟨M⟩K.

Theorem 6.4.11. Let E be a category and M ∶XA be a module. If a cylinder µ ∶ R K ∶ E M is inverse universal (resp. pointwise inverse universal),

then a natural transformation τ ∶ S→ R ∶ E→X is iso if and only if the cylinder τ µ ∶ SK ∶ EM is inverse universal (resp. pointwise inverse universal).

If a cylinder µ ∶ K R ∶ E M is direct universal (resp. pointwise direct universal), thena natural transformation τ ∶ R → S ∶ E → A is iso if and only if the cylinder µ τ ∶ K S ∶EM is direct universal (resp. pointwise direct universal).

214

6. Universals

Proof. Suppose that µ is inverse universal, i.e. an inverse universal ⟨E,M⟩-arrow. Then theassertion is just an instance of Theorem6.2.6 whereM is given by ⟨E,M⟩. Now suppose thatµ is pointwise inverse universal. Since a natural transformation is iso i each component is anisomorphism, it suces to show that, given e ∈ ∥E∥, [τ µ]e is an inverse universal M-arrowi τe is an iso X-arrow. But since [τ µ]e = τe µe, this follows again from Theorem6.2.6.

Corollary 6.4.12. Let E be a category and M ∶XA be a module. If a functor K ∶ E → A has a pointwise lift inverse along M, then every lift of K inverse

along M is pointwise. If a functor K ∶ E→X has a pointwise lift direct along M, then every lift of K direct alongM is pointwise.

Proof. Let µ ∶ R K ∶ EM and ν ∶ S K ∶ EM be two lifts of K inverse alongM. Then,by Theorem6.2.7, there are natural isomorphisms ν/µ ∶ S → R and µ/ν ∶ R → S inverse to eachother making the diagram

S

ν/µ

ν

((K

R

µ/ν

OO

µ

66

commute. Hence, by Theorem6.4.11, if one of µ and ν is pointwise inverse universal, so is theother.

Theorem 6.4.13. Let H be a functor and µ be a cylinder as in

D

HE

S

T

##X M

//

µ

A

. If µ is pointwise inverse (resp. direct) universal, so is the composite H µ (see Deni-tion 4.3.24).

Proof. Obvious since [H µ]d = µ(H ⋅ d) for each d ∈ ∥D∥.

Corollary 6.4.14. If a cylinder µ is pointwise inverse (resp. direct) universal, so is thecylinder [D, µ] (see Remark 4.3.35(2)) for any category D; moreover, each component of [D, µ]is a pointwise inverse (resp. direct) universal cylinder.

Proof. Since the component of [D, µ] at a functor H is given by the cylinder H µ, the asser-tion follows from Theorem6.4.13 by noting that (see Proposition 6.4.8) a pointwise universalcylinder is universal.

Corollary 6.4.15. Given a functor H ∶D → E and a module M ∶ X A, the precompositioncell ⟨H,M⟩ ∶ ⟨E,M⟩ ⟨D,M⟩ (see Denition 4.3.28) preserves pointwise universality; that is,⟨H,M⟩ sends each pointwise inverse (resp. direct) universal cylinder µ ∶ EM to a pointwiseinverse (resp. direct) universal cylinder H µ ∶DM.


215

6. Universals

Note. By Proposition 6.4.5, Theorem6.4.16 and Corollary 6.4.17 below are special cases ofTheorem6.4.13 and Corollary 6.4.14 (and vice versa by Proposition 6.4.7).

Theorem 6.4.16. Let µ be a right (resp. left) cylinder and K be a functor as in

XM

//µ ARoo

EKoo E

K //XM //

R//µ A

. If µ is a counit (resp. unit) of M, then the composite K µ (see Denition 4.3.26) is apointwise inverse (resp. direct) universal cylinder.

Proof. Obvious since [K µ]e = µ(K ⋅ e) for each e ∈ ∥E∥.

Corollary 6.4.17. If a right (resp. left) cylinder µ is a counit (resp. unit), so is the right (resp.left) cylinder [E, µ] (see Remark 4.3.35(3)) for any category E; moreover, each component of[E, µ] is a pointwise inverse (resp. direct) universal cylinder.

Proof. Since the component of [E, µ] at a functor K is given by the cylinder K µ, the asser-tion follows from Theorem6.4.16 by noting that (see Proposition 6.4.8) a pointwise universalcylinder is universal.

Theorem 6.4.18. Let µ be a cylinder as in Theorem6.4.13 and let D be an essentially widesubcategory of E. Then µ is pointwise inverse (resp. direct) universal if and only if so is itsrestriction to D (see Remark 4.3.25(2)); that is, if and only if the component of µ at eachd ∈ ∥D∥ is inverse (resp. direct) universal.

Proof. The forward implication is immediate from Theorem6.4.13. Assume now that τ ispointwise universal on D. We need to show that the component τe at each e ∈ ∥E∥ is universal.Since D is essentially wide in E, there is an object d ∈ ∥D∥ and an iso E-arrow h ∶ d→ e, givinga naturality square

d ⋅ S µd //

h ⋅ S

T ⋅ dT ⋅ h

e ⋅ S µe// T ⋅ e

with h ⋅S and T ⋅h iso (because any functor preserves isomorphisms). Now, since µd is universalby assumption, so is µe by Theorem6.2.11.

Remark 6.4.19. Since a natural isomorphism is the same thing as a pointwise universal cylinderalong the hom of a category (see Proposition 6.4.9), the lemma in Preliminary 19 is a specialcase of Theorem6.4.18.

6.5. Kan lifts

Note. Example 4.3.5(1) allows the following denition.

Denition 6.5.1. Given a pair of functors

DF // E C

Koo

216

6. Universals

a natural transformationC

R

K

##D

F//

µ

E

from R F to K or the pair (R, µ), or the functor R itself, is called a right Kan lift (resp.pointwise right Kan lift) of K along F if the cylinder

CR

K

##D

F⟨E⟩//

µ

E

is inverse universal (resp. pointwise inverse universal). a natural transformation

CK

R

##E

µ

DF

oo

from K to F R or the pair (R, µ), or the functor R itself, is called a left Kan lift (resp.pointwise left Kan lift) of K along F if the cylinder

CK

R

##E

⟨E⟩F//

µ

D

is direct universal (resp. pointwise direct universal).

Remark 6.5.2.(1) A Kan lift (resp. pointwise Kan lift) is thus a special instance of a lift (resp. pointwise

lift) dened in Denition 6.4.1 whereM is representable.(2) A natural transformation µ ∶ R F→ K forms a right Kan lift if and only if to every natural

transformation α ∶ L F→ K there is a unique natural transformation α/µ ∶ L→ R such thatα = [α/µ F] µ (where [α/µ F] µ is the pasting composite of α/µ and µ). Dually, anatural transformation µ ∶ K → F R forms a left Kan lift if and only if to every naturaltransformation α ∶ K→ F L there is a unique natural transformation µ/α ∶ R→ L such thatα = µ [µ/α F].

(3) By Proposition 6.4.7, a pointwise right Kan lift of K along F is the same thing as a counit of the composite

module F ⟨E⟩K ∶DC. a pointwise left Kan lift of K along F is the same thing as a unit of the composite module

K ⟨E⟩F ∶CD.(4) Using the terms introduced in Example 6.2.3, a pointwise Kan lift is described as follows:

a natural transformation µ ∶ R F → K forms a pointwise right Kan lift of K along F ifand only if each component µc ∶ c ⋅ R ⋅ F→ K ⋅ c is universal from F to K ⋅ c.

a natural transformation µ ∶ K→ F R forms a pointwise left Kan lift of K along F if andonly if each component µc ∶ c ⋅K→ F ⋅ R ⋅ c is universal from c ⋅K to F.

(5) See Example 6.5.5(1) for an example of a non-pointwise Kan lift.

Theorem 6.5.3.

217

6. Universals

Consider functors as inC

R

K

##D

F//

µ

E

and assume that µ ∶ R F → K is a natural isomorphism. Under this condition, if F is fullyfaithful, then µ forms a pointwise right Kan lift of K along F. The converse holds if weassume in addition that R is essentially surjective.

Consider functors as inC

K

R

##E

µ

DF

oo

and assume that µ ∶ K → F R is a natural isomorphism. Under this condition, if F is fullyfaithful, then µ forms a pointwise left Kan lift of K along F. The converse holds if we assumein addition that R is essentially surjective.

Proof. By Example 5.2.9(2), the diagram

⟨D⟩ (R ⋅ c) Dµc //

⟨F⟩(R ⋅ c)

F ⟨E⟩ (K ⋅ c)

⟨F ⟨E⟩F⟩ (R ⋅ c) = F ⟨E⟩ (F ⋅ R ⋅ c)F⟨E⟩µc

33

commutes for every c ∈ ∥C∥. Since µc is an iso E-arrow by the assumption, F ⟨E⟩µc is iso.Hence Dµc is iso i ⟨F⟩ (R ⋅ c) is iso. If F is fully faithful, then ⟨F⟩ (R ⋅ c) and hence Dµc isiso for every c ∈ ∥C∥. µ is thus pointwise inverse universal. Conversely, if µ is pointwise inverseuniversal, then Dµc and hence ⟨F⟩ (R ⋅ c) is iso for every c ∈ ∥C∥. F is thus fully faithful byProposition 2.1.5 under the condition that R is essentially surjective.

Remark 6.5.4. If F is not fully faithful in Theorem6.5.3, then a natural isomorphism µ, evenan identity, need not form a Kan lift. See Example 6.5.5(2) for an example.

Example 6.5.5.

(1) Example 6.3.7 is repeated below in terms of Kan lifts. Let 2 and 2 be as in Example 6.3.7and let F ∶ 2 → 2 be the inclusion functor. Then the representable module of F looks likethis:

0 //

%%

0

1 // 1

. The identity 2→ 2 has only one natural transformation

2∆0

||

1

""2

F//

µ

2

along F. The constant functor ∆0 and µ form a right Kan lift of the identity 2 → 2 alongF; however, the lift is not pointwise since µ1 ∶ 0 ⋅ F→ 1 is not universal from F to 1.

218

6. Universals

(2) Consider functors as in∗

1

E∆

// ∗

, where ∗ is the terminal category and E is any category. Given e ∈ ∥E∥, the functore ∶ ∗→ E and the identity natural transformation

∗e

1

##E

∆//

µ

∗

form a right Kan lift of the identity ∗→ ∗ along ∆ only when e is a terminal object of E.

219

7. Limits

7.1. Limits

Denition 7.1.1. Let E be a category andM ∶XA be a module. A cone

∗r

E∆oo

K

X M//

π

A

is called universal if it is an inverse universal ⟨∗E,M⟩-arrow (see Denition 4.6.5). Given afunctor K ∶ E → A, a universal cone π ∶ r K ∶ ∗E M or the pair (r, π), or the object ritself, is called a limit of K alongM, with the object r denoted by ∏E K (or just by ∏K).

A cone

E

K

∆ // ∗r

X M//

π

A

is called universal if it is a direct universal ⟨E∗,M⟩-arrow (see Denition 4.6.5). Given afunctor K ∶ E → X, a universal cone π ∶ K r ∶ E∗ M or the pair (r, π), or the object ritself, is called a colimit of K alongM, with the object r denoted by ∐E K (or just by ∐K).

Remark 7.1.2.(1) A cone π ∶ r K ∶ ∗EM is universal if and only if to every cone α ∶ x K ∶ ∗EM there

is a unique X-arrow α/π ∶ x → r such that α = α/π π. Dually, a cone π ∶ K r ∶ E∗ Mis universal if and only if to every cone α ∶ K a ∶ E∗ M there is a unique A-arrowπ/α ∶ r→ a such that α = π π/α.

(2) A limit∏E K (resp. colimit∐E K ) is called a product (resp. coproduct) when E is discrete.

Theorem 7.1.3.

Given a module and functors as in

XM //A E

Koo DFoo

, a cone

∗r

D∆oo

F

X⟨M⟩K

//

π

E

forms a limit of F along the composite module ⟨M⟩K if and only if π depicted as in

∗r

D∆oo

K F

X M//

π

A

220

7. Limits

forms a limit of the composite functor K F along M. Given a module and functors as in

DF // E

K //XM //A

, a cone

D

F

∆ // ∗r

EK⟨M⟩

//

π

A

forms a colimit of F along the composite module K ⟨M⟩ if and only if π depicted as in

D

F K

∆ // ∗r

X M//

π

A

forms a colimit of the composite functor F K along M.

Proof. By Proposition 6.2.21, the cell

X

1

⟨∗D,⟨M⟩K⟩ // [D,E][D,K]

X⟨∗D,M⟩

//

1

[D,A]

(see Proposition 4.6.7) preserves and reects inverse universals; that is, an ⟨∗D, ⟨M⟩K⟩-arrowπ ∶ r F is inverse universal i so is the ⟨∗D,M⟩-arrow π ∶ r [D,K] ⋅ F; that is, a coneπ ∶ r F ∶ ∗D ⟨M⟩K is universal i so is the cone π ∶ r K F ∶ ∗DM.

Denition 7.1.4. A moduleM ∶XA is said to have limits over a category E if every functor E→A has a limit alongM. have colimits over a category E if every functor E→X has a colimit alongM.

Proposition 7.1.5.

A moduleM ∶XA has limits over a category E if and only if the module ⟨∗E,M⟩ has acounit.

A moduleM ∶XA has colimits over a category E if and only if the module ⟨E∗,M⟩ hasa unit.

Proof. Since a limit of a functor K ∶ E→A is dened as a universal of K inverse along ⟨∗E,M⟩,the assertion follows from the equivalence of (2) and (4) in Corollary 6.3.11.

Denition 7.1.6. A moduleM ∶XA is called complete (resp. cocomplete) if it has limits(resp. colimits) over any small category E.

Proposition 7.1.7.

A moduleM is complete if and only if for any small category E the module ⟨∗E,M⟩ has acounit.

A moduleM is cocomplete if and only if for any small category E the module ⟨E∗,M⟩ hasa unit.

221

7. Limits



XM //

P

A

Q

Y N//

Φ

B

is said to preserve (reect, create) limits over a category E if the postcomposition cell ⟨∗E,Φ⟩ (see

Denition 4.6.15) preserves (reects, creates) inverse universals. preserve (reect, create) colimits over a category E if the postcomposition cell ⟨E∗,Φ⟩ (see

Denition 4.6.15) preserves (reects, creates) direct universals.

Remark 7.1.9.(1) Namely, Φ is said to

a) preserve limits over E if each universal cone π ∶ r K ∶ ∗E M yields by composition with

Φ a universal cone π Φ ∶ r ⋅ P Q K ∶ ∗E N . colimits over E if each universal cone π ∶ K r ∶ E∗ M yields by composition

with Φ a universal cone π Φ ∶ K P Q ⋅ r ∶ E∗ N .b) reect

limits over E if a cone π ∶ r K ∶ ∗E M is universal whenever the cone π Φ ∶r ⋅ P Q K ∶ ∗E N is universal.

colimits over E if a cone π ∶ K r ∶ E∗ M is universal whenever the coneπ Φ ∶ K P Q ⋅ r ∶ E∗ N is universal.

c) create limits over E if for every functor K ∶ E → A and for every universal cone κ ∶ s

Q K ∶ ∗E N there is exactly one cone π ∶ r K ∶ ∗E M with π Φ = κ, and ifthis π is universal.

colimits over E if for every functor K ∶ E→X and for every universal cone κ ∶ K Ps ∶ E∗ N there is exactly one cone π ∶ K r ∶ E∗ M with π Φ = κ, and if thisπ is universal.

(2) A cell Φ is said to preserve (reect, create) limits (resp. small limits) if it preserves (reects, creates) limits

over any category (resp. small category). preserve (reect, create) colimits (resp. small colimits) if it preserves (reects, creates)

colimits over any category (resp. small category).(3) We also say that a cell Φ ∶M N is

continuous ifM is compete and Φ preserves small limits. cocontinuous ifM is cocomplete and Φ preserves small colimits.

Proposition 7.1.10. Consider a cell Φ as in Denition 7.1.8. If Φ is fully faithful and P is iso, then Φ preserves, reects, and creates limits. If Φ is fully faithful and Q is iso, then Φ preserves, reects, and creates colimits.

Proof. This is equivalent to saying that if Φ is fully faithful and P is iso, then for any categoryE the cell

[E,X] ⟨E∗,M⟩ //

[E,P]

A

Q

[E,Y]⟨E∗,N ⟩

//

⟨E∗,Φ⟩

B

222

7. Limits

preserves, reects, and creates inverse universals. Since the functorial operations [E,−] and⟨E∗,−⟩ preserve isomorphisms, the functor [E,P] is an isomorphism and the cell ⟨E∗,Φ⟩ isfully faithful. The assertion thus follows from Proposition 6.2.21.

Proposition 7.1.11. Let E be a category and let Φ be a cell as in Denition 7.1.8. If N has limits over E and Φ creates limits from them, then M has limits over E as well

and Φ preserves them. If N has colimits over E and Φ creates colimits from them, thenM has colimits over E as

well and Φ preserves them.

Proof. Noting Proposition 7.1.5, we see that this is an instance of Theorem6.3.12 where Φ isgiven by the postcomposition cell ⟨∗E,Φ⟩.

Proposition 7.1.12. Consider a cell Φ as in Denition 7.1.8. If N is complete and Φ creates small limits, thenM is complete as well and Φ is continuous. If N is cocomplete and Φ creates small colimits, then M is cocomplete as well and Φ is

cocontinuous.


7.2. Limits with parameters

Denition 7.2.1. Let E and D be categories andM ∶XA be a module. A cone

∗R

D

K

∆Doo

[E,X]⟨E,M⟩

//

ω

[E,A]

along the module ⟨E,M⟩ (see Denition 4.3.6) is called(1) universal if it is an inverse universal ⟨∗D, ⟨E,M⟩⟩-arrow (see Denition 4.6.5);(2) pointwise universal if each slice

∗R

D

K

∆Doo

[E,X][e,X]

⟨E,M⟩ //

ω

[E,A][e,A]

X M//

⟨e,M⟩

A

(see Remark 4.8.8(2)) is a universal cone, i.e. an inverse universal ⟨∗D,M⟩-arrow.Given a functor K ∶D→ [E,A], a pointwise universal cone ω ∶ R K ∶ ∗D ⟨E,M⟩ or thepair (R, ω), or the functor R itself, is called a pointwise limit of K along ⟨E,M⟩, with thefunctor R denoted by ∏E

D K (or just by ∏E K). A cone

D

K

∆D // ∗R

[E,X]⟨E,M⟩

//

ω

[E,A]

along the module ⟨E,M⟩ (see Denition 4.3.6) is called

223

7. Limits

(1) universal if it is a direct universal ⟨D∗, ⟨E,M⟩⟩-arrow (see Denition 4.6.5);(2) pointwise universal if each slice

D

K

∆D // ∗R

[E,X][e,X]

⟨E,M⟩ //

ω

[E,A][e,A]

X M//

⟨e,M⟩

A

(see Remark 4.8.8(2)) is a universal cone, i.e. a direct universal ⟨D∗,M⟩-arrow.Given a functor K ∶D→ [E,X], a pointwise universal cone ω ∶ K R ∶D∗ ⟨E,M⟩ or thepair (R, ω), or the functor R itself, is called a pointwise colimit of K along ⟨E,M⟩, with thefunctor R denoted by ∐E

D K (or just by ∐E K).

Proposition 7.2.2.

The following conditions are equivalent:(1) a cone

∗R

D

K

∆Doo

[E,X]⟨E,M⟩

//

ω

[E,A]

is universal (resp. pointwise universal);(2) its transpose

E

R

E

K⊺

Eoo

X⟨∗D,M⟩

//

ω⊺

[D,A]

(see Denition 4.8.7) is an inverse universal (resp. pointwise inverse universal) cylinder. The following conditions are equivalent:

(1) a cone

D

K

∆D // ∗R

[E,X]⟨E,M⟩

//

ω

[E,A]


E

K⊺

E // E

R

[D,X]⟨D∗,M⟩

//

ω⊺

A

(see Denition 4.8.7) is a direct universal (resp. pointwise direct universal) cylinder.

Proof. By the isomorphism in Remark 4.8.8(1), ω is universal i ω⊺ is inverse universal. Sincethe slice of ω at each e ∈ ∥E∥ is given by the component of ω⊺ at e, ω is pointwise universal iω⊺ is pointwise inverse universal.

224

7. Limits

Remark 7.2.3. By Proposition 7.2.2 and by the bijectiveness of transposition, we see that givena moduleM ∶XA, a limit (resp. pointwise limit) of a functor K ∶ D → [E,A] along the module ⟨E,M⟩ is

the same thing as a lift (resp. pointwise lift) of K⊺ ∶ E → [D,A] inverse along the module⟨∗D,M⟩.

a colimit (resp. pointwise colimit) of a functor K ∶D → [E,X] along the module ⟨E,M⟩ isthe same thing as a lift (resp. pointwise lift) of K⊺ ∶ E → [D,X] direct along the module⟨D∗,M⟩.

Proposition 7.2.4. A pointwise universal cone is universal.

Proof. This is reduced to Proposition 6.4.8 by the equivalence of (1) and (2) in Proposition 7.2.2.

Theorem 7.2.5. Let E and D be categories and M ∶XA be a module. If a functor K ∶D→ [E,A] has a pointwise limit along ⟨E,M⟩, then every limit of K along

⟨E,M⟩ is pointwise. If a functor K ∶ D → [E,X] has a pointwise colimit along ⟨E,M⟩, then every colimit of K

along ⟨E,M⟩ is pointwise.

Proof. By the equivalence of (1) and (2) in Proposition 7.2.2, this is reduced to an instance ofCorollary 6.4.12 whereM is given by the module ⟨∗D,M⟩.

Theorem 7.2.6. Given a module M and a functor K as in Denition 7.2.1, the family ofevaluations ⟨e,M⟩, one for each object e ∈ ∥E∥, collectively creates a pointwise limit (resp.colimit) for K in the following sense: if there is a family of universal cones ωe ∶ re [e,A] K ∶ ∗D M, one for each e ∈ ∥E∥,

then there is a unique cone ω ∶ R K ∶ ∗D ⟨E,M⟩ such that ω ⟨e,M⟩ = ωe, and ω ispointwise universal.

if there is a family of universal cones ωe ∶ K [e,X] re ∶D∗ M, one for each e ∈ ∥E∥,then there is a unique cone ω ∶ K R ∶ D∗ ⟨E,M⟩ such that ω ⟨e,M⟩ = ωe, and ω ispointwise universal.

Proof. By the equivalence of (1) and (2) in Proposition 7.2.2, this is reduced to an instance ofTheorem6.4.10 whereM is given by the module ⟨∗D,M⟩.

Denition 7.2.7. Let E and D be categories andM ∶XA be a module. A wedge

E

R

E ×D

K

E×∆Doo

X M//

ω

A

is called(1) universal if it is an inverse universal ⟨E × ∗D,M⟩-arrow (see Denition 4.8.3);(2) pointwise universal if each left slice

∗R(e)

D

K⊢(e)

∆Doo

X M//

[ω⊢]e

A

(see Denition 4.8.5) is a universal cone, i.e. an inverse universal ⟨∗D,M⟩-arrow.

225

7. Limits

Given a functor K ∶ E×D→A, a pointwise universal wadge ω ∶ R K ∶ E×∗DM or thepair (R, ω), or the functor R itself, is called an E-parameterized limit of K along M, withthe functor R denoted by ∏E

D K (or just by ∏E K). A wedge

E ×D

K

E×∆D // E

R

X M//

ω

A

is called(1) universal if it is a direct universal ⟨E ×D∗,M⟩-arrow (see Denition 4.8.3);(2) pointwise universal if each left slice

D

K⊢(e)

∆D // ∗R(e)

X M//

[ω⊢]e

A

(see Denition 4.8.5) is a universal cone, i.e. a direct universal ⟨D∗,M⟩-arrow.Given a functor K ∶ E×D→X, a pointwise universal wadge ω ∶ K R ∶ E×D∗M or thepair (R, ω), or the functor R itself, is called an E-parameterized colimit of K alongM, withthe functor R denoted by ∐E


Proposition 7.2.8.

The following conditions are equivalent:(1) a wedge

E

R

E ×D

K

E×∆Doo

X M//

ω

A

is universal (resp. pointwise universal);(2) its right exponential transpose

∗R

D

K⊣

∆Doo

[E,X]⟨E,M⟩

//

ω⊣

[E,A]

(see Denition 4.8.5) is a universal (resp. pointwise universal) cone;(3) its left exponential transpose

E

R

E

K⊢

Eoo

X⟨∗D,M⟩

//

ω⊢

[D,A]

(see Denition 4.8.5) is an inverse universal (resp. pointwise inverse universal) cylinder. The following conditions are equivalent:

226

7. Limits

(1) a wedge

E ×D

K

E×∆D // E

R

X M//

ω

A


D

K⊣

∆D // ∗R

[E,X]⟨E,M⟩

//

ω⊣

[E,A]

(see Denition 4.8.5) is a universal (resp. pointwise universal) cone;(3) its left exponential transpose

E

K⊢

E // E

R

[D,X]⟨D∗,M⟩

//

ω⊢

A

(see Denition 4.8.5) is a direct universal (resp. pointwise direct universal) cylinder.

Proof. By the isomorphisms in Remark 4.8.6, ω is universal i ω⊣ is universal i ω⊢ is inverseuniversal. Since the left slice of ω at each e ∈ ∥E∥ is given by the component of ω⊢ at e, ω ispointwise universal i ω⊢ is pointwise inverse universal. By Proposition 7.2.2, ω⊣ is pointwiseuniversal i [ω⊣]⊺ is pointwise inverse universal. But [ω⊣]⊺ = ω⊢ by the commutative diagramin Remark 4.8.8(1).

Remark 7.2.9. By Proposition 7.2.8 and by the bijectiveness of exponential transposition, wesee that given a moduleM ∶XA, the following are the same thing:

(1) an E-parameterized limit of a functor K ∶ E ×D→A alongM;(2) a pointwise limit of K⊣ ∶D→ [E,A] along the module ⟨E,M⟩;(3) a pointwise lift of K⊢ ∶ E→ [D,A] inverse along the module ⟨∗D,M⟩.

the following are the same thing:(1) an E-parameterized colimit of a functor K ∶ E ×D→X alongM;(2) a pointwise colimit of K⊣ ∶D→ [E,X] along the module ⟨E,M⟩;(3) a pointwise lift of K⊢ ∶ E→ [D,X] direct along the module ⟨D∗,M⟩.

Proposition 7.2.10. A pointwise universal wedge is universal.

Proof. This is reduced to Proposition 6.4.8 by the equivalence of (1) and (3) in Proposition 7.2.8.

Theorem 7.2.11. Let M ∶XA be a module. If a wedge ω ∶ R K ∶ E × ∗D M is universal (resp. pointwise inverse universal), then a

natural transformation τ ∶ S → R is iso if and only if the wedge τ ω ∶ S K ∶ E × ∗D Mis universal (resp. pointwise inverse universal).

227

7. Limits

If a wedge ω ∶ K R ∶ E ×D∗ M is universal (resp. pointwise direct universal), then anatural transformation τ ∶ R → S is iso if and only if the wedge ω τ ∶ K S ∶ E ×D∗ Mis universal (resp. pointwise direct universal).

Proof. This is reduced to Theorem6.4.11 by the equivalence of (1) and (3) in Proposition 7.2.8:the statement faithfully translates, along the iso cell ⊢ in Remark 4.8.6, into (hence reects)an instance of Theorem6.4.11 whereM is given by the module ⟨∗D,M⟩.

Theorem 7.2.12. Let E and D be categories and let M ∶XA be a module. Suppose that M has limits over D; that is (see Proposition 7.1.5), suppose that the module

⟨∗D,M⟩ has a counit

X⟨∗D,M⟩

//ϕ [D,A]∏oo

. Then this counit yields(1) a counit

[E,X]⟨E×∗D,M⟩

//ϕE [E ×D,A]∏E

oo

of the module of wedges E×∗DM, giving, for each functor K ∶ E×D→A, a universalwadge

[ϕE]K∶

E

∏K K ∶ E × ∗DM, in fact a pointwise universal wadge, with each left slice

[E

∏K] (e) =∏ [K⊢ (e)] [[[ϕE]K]⊢]

e= ϕK⊢(e)

giving a limit of the functor K⊢ (e) ∶D→A.(2) a counit

[E,X]⟨∗D,⟨E,M⟩⟩

//ϕE [D, [E,A]]∏E

oo

of the module of cones ∗D ⟨E,M⟩, giving, for each functor K ∶ D → [E,A], auniversal cone

[ϕE]K∶

E

∏K K ∶ ∗D ⟨E,M⟩, in fact a pointwise universal cone, with each slice

[E

∏K] (e) =∏ [K⊺ (e)] [[[ϕE]K]⊺]

e= ϕK⊺(e)

giving a limit of the functor K⊺ (e) ∶D→A. Suppose thatM has colimits over D; that is (see Proposition 7.1.5), suppose that the module

⟨D∗,M⟩ has a unit

[D,X]⟨D∗,M⟩ //

∐//ϕ A

. Then this unit yields

228

7. Limits

(1) a unit

[E ×D,X]⟨E×D∗,M⟩ //

∐E//ϕE [E,A]

of the module of wedges E×D∗M, giving, for each functor K ∶ E×D→X, a universalwadge

[ϕE]K∶

E

∐K K ∶ E ×D∗M, in fact pointwise universal wadge, with each left slice

[E

∐K] (e) =∐ [K⊢ (e)] [[[ϕE]K]⊢]

e= ϕK⊢(e)

giving a colimit of the functor K⊢ (e) ∶D→X.(2) a unit

[D, [E,X]]⟨D∗,⟨E,M⟩⟩ //

∐E//ϕE [E,A]

of the module of cones D∗ ⟨E,M⟩, giving, for each functor K ∶ D → [E,X], auniversal cone

[ϕE]K∶

E

∐K K ∶D∗ ⟨E,M⟩, in fact a pointwise universal cone, with each slice

[E

∐K] (e) =∐ [K⊺ (e)] [[[ϕE]K]⊺]

e= ϕK⊺(e)

giving a colimit of the functor K⊺ (e) ∶D→X.

Proof. By Corollary 6.4.17, the counit ϕ of the module ⟨∗D,M⟩ yields a counit

[E,X]⟨E,⟨∗D,M⟩⟩

//[E,ϕ] [E, [D,A]][E,∏]oo

of the module ⟨E, ⟨∗D,M⟩⟩, and, from this counit, the iso cells in Remark 4.8.6 and Re-mark 4.8.8(1) create a counit ϕE of the module ⟨E × ∗D,M⟩ and a counit ϕE of the module⟨∗D, ⟨E,M⟩⟩.

Corollary 7.2.13. If a module M ∶ X A has limits (resp. colimits) over a category D,so is the module ⟨E,M⟩ for any category E, and every limit (resp. colimit) over D is givenpointwise.

Proof. By Theorem7.2.12, every functor K ∶ D → [E,A] has a pointwise limit ∏E K along⟨E,M⟩, and every limit of K is pointwise by Theorem7.2.5.

Theorem 7.2.14. If a module M ∶ X A is complete (resp. cocomplete), so is the module⟨E,M⟩ for any category E, and all small limits (resp. colimits) are given pointwise.

Proof. Immediate from 7.2.13.

229

7. Limits

Theorem 7.2.15. If a moduleM ∶XA is complete (resp. cocomplete), then for any functorH ∶ E′ → E the precomposition cell

[E,X] ⟨E,M⟩ //

[H,X]

[E,A][H,A]

[E′,X]⟨E′,M⟩

//

⟨H,M⟩

[E′,A]

(see Denition 4.3.28) is continuous (resp. cocontinuous).

Proof. Assume that M is complete. We need to show that for any small category D, thepostcomposition cell ⟨∗D, ⟨H,M⟩⟩ preserves inverse universals. Since every universal coneω ∶ ∗D ⟨E,M⟩ is pointwise universal by Theorem7.2.14, and ω is pointwise universal iso is its transpose ω⊺ ∶ E ⟨∗D,M⟩ (see Proposition 7.2.2), the problem translates along thenaturality square

⟨∗D, ⟨E,M⟩⟩⟨∗D,⟨H,M⟩⟩

oo ⊺ // ⟨E, ⟨∗D,M⟩⟩⟨H,⟨∗D,M⟩⟩

⟨∗D, ⟨E′,M⟩⟩ oo ⊺// ⟨E′, ⟨∗D,M⟩⟩

into showing that the precomposition cell ⟨H, ⟨∗D,M⟩⟩ preserves pointwise universality. Butwe have already seen this in Corollary 6.4.15.

7.3. Limits in a category

Note. A limit of a functor in a category is dened as below as a special case of Denition 7.1.1whereM is given by the hom of a category and coincides with the usual denition of a limitin the literature.

Denition 7.3.1.

A cone π ∶ r K ∶ ∗E→C is called universal if it is an inverse universal ⟨∗E,C⟩-arrow (seeDenition 4.9.3). Given a functor K ∶ E → C, a universal cone π ∶ r K ∶ ∗E → C or thepair (r, π), or the object r itself, is called a limit of K in C.

A cone π ∶ K r ∶ E∗ → C is called universal if it is a direct universal ⟨E∗,C⟩-arrow (seeDenition 4.9.3). Given a functor K ∶ E → C, a universal cone π ∶ K r ∶ E∗ → C or thepair (r, π), or the object r itself, is called a colimit of K in C.

Remark 7.3.2. A cone π ∶ r K ∶ ∗E → C is universal if and only if to every cone α ∶ s K ∶ ∗E → C there is a unique C-arrow α/π ∶ s → r such that α = α/π π. Dually, a coneπ ∶ K r ∶ E∗→C is universal if and only if to every cone α ∶ K t ∶ E∗→C there is a uniqueC-arrow π/α ∶ r→ t such that α = π π/α.

Denition 7.3.3. A category C is said to have limits (resp. colimits) over a category E ifevery functor E→C has a limit (resp. colimits) in C.


Proposition 7.3.4.

A category C has limits over a category E if and only if the module ⟨∗E,C⟩ has a counit. A category C has colimits over a category E if and only if the module ⟨E∗,C⟩ has a unit.

230

7. Limits

Proof. Since a limit of a functor K ∶ E→C is dened as a universal of K inverse along ⟨∗E,C⟩,the assertion follows from the equivalence of (2) and (4) in Corollary 6.3.11.

Remark 7.3.5. Since the module ⟨∗E,C⟩ (resp. ⟨E∗,C⟩) is represented (resp. corepresented)by the diagonal functor [∆E,C] (see Remark 4.9.4(2)), Proposition 7.3.4 is alternatively statedas below in terms of adjunctions (see Section 8.2). A category C has limits over a category E if and only if the diagonal functor [∆E,C] has

a right adjoint. A category C has colimits over a category E if and only if the diagonal functor [∆E,C] has

a left adjoint.

Denition 7.3.6. A category C is called complete (resp. cocomplete) if it has limits (resp.colimits) over any small category E.

Remark 7.3.7. A category C is complete (resp. cocomplete) if and only if the hom ⟨C⟩ iscomplete (resp. cocomplete) in the sense of Denition 7.1.6.


Proposition 7.3.8.

A category C is complete if and only if for any small category E the module ⟨∗E,C⟩ has acounit.

A category C is cocomplete if and only if for any small category E the module ⟨E∗,C⟩ hasa unit.


Note. The preservation (reection, creation) of limits by a functor is dened as below as aspecial case of Denition 7.1.8 where Φ is given by the hom of a functor and coincides with theusual notion of preservation (reection, creation) of limits dened in the literature.

Denition 7.3.9. A functor H ∶C→ B is said to preserve (reect, create) limits over a category E if the postcomposition cell ⟨∗E,H⟩ (see

Denition 4.9.5) preserves (reects, creates) inverse universals. preserve (reect, create) colimits over a category E if the postcomposition cell ⟨E∗,H⟩ (see


Remark 7.3.10.(1) A functor H preserves (reects, creates) limits (resp. colimits) over a category E if and

only if the hom cell ⟨H⟩ does the same in the sense of Denition 7.1.8.(2) A functor H is said to

preserve (reect, create) limits (resp. small limits) if it preserves (reects, creates) limitsover any category (resp. small category).

preserve (reect, create) colimits (resp. small colimits) if it preserves (reects, creates)colimits over any category (resp. small category).

(3) We also say that functor H ∶C→ B is continuous if C is compete and H preserves small limits. cocontinuous if C is cocomplete and H preserves small colimits.

Note. Proposition 7.3.11 and Proposition 7.3.12 below are special cases of Proposition 7.1.11and Proposition 7.1.12 where Φ is given by the hom of a functor.

231

7. Limits

Proposition 7.3.11. Let Let H ∶C→ B be a functor. If B has limits over E and H creates limits from them, then C has limits over E as well

and H preserves them. If B has colimits over E and H creates colimits from them, then C has colimits over E as

well and H preserves them.

Proof. Noting Proposition 7.3.8, we see that this is an instance of Theorem6.3.12 where Φ isgiven by the postcomposition cell ⟨∗E,H⟩.

Proposition 7.3.12. Let H ∶C→ B be a functor. If B is complete and H creates small limits, then C is complete as well and H is continuous. If B is cocomplete and H creates small colimits, then C is cocomplete as well and H is

cocontinuous.

Proof. Immediate from 7.3.11.

Theorem 7.3.13. Let M ∶XA be a module. Any corepresentation

X

1

M //A

R

X⟨X⟩

//

Υ

X

of M preserves, reects, and creates limits. Any representation

X

R

M //A

1

A⟨A⟩

//

Υ

A

of M preserves, reects, and creates colimits.


Corollary 7.3.14. Let M ∶XA be a module. If M is corepresentable and X is complete, then M is complete as well. If M is representable and A is cocomplete, then M is cocomplete as well.

Proof. Since a corepresentation of M creates limits by Theorem7.3.13, the assertion followsfrom Proposition 7.1.12.

Note. The following is a special case of Theorem7.2.6 where M is given by the hom of acategory.

Theorem 7.3.15. Given a functor K ∶D→ [E,C], the family of evaluations [e,C] ∶ [E,C]→C, one for each object e ∈ ∥E∥, collectively creates a pointwise limit (resp. colimit) for K inthe following sense: if there is a family of universal cones ωe ∶ re [e,C] K ∶ ∗D → C, one for each e ∈ ∥E∥,

then there is a unique cone ω ∶ R K ∶ ∗D → [E,C] such that ω [e,C] = ωe, and ω ispointwise universal.

if there is a family of universal cones ωe ∶ K [e,C] re ∶ D∗ → C, one for each e ∈ ∥E∥,then there is a unique cone ω ∶ K R ∶ D∗ → [E,C] such that ω [E,C] = ωe, and ω ispointwise universal.

232

7. Limits

Proof. Since a cone in C (resp. [E,C]) is the same thing as a cone along the hom ⟨C⟩(resp. ⟨E,C⟩), the assertion follows from Theorem7.2.6 by replacing M with the hom of Cand recalling that ⟨E, ⟨C⟩⟩ = ⟨E,C⟩ (see Remark 4.3.7(3)) and ω ⟨e, ⟨C⟩⟩ = ω [e,C] (seeRemark 4.3.15(2) and Remark 4.3.32).

Note. Theorem7.3.16 and Theorem7.3.17 below are special cases of Theorem7.2.14 and The-orem7.2.15 whereM is given by the hom of a category.

Theorem 7.3.16. If a category C is complete (resp. cocomplete), so is the functor category[E,C] for any category E, and all small limits (resp. colimits) are given pointwise.

Proof. Since C (resp. [E,C]) is complete i its hom ⟨C⟩ (resp. ⟨E,C⟩) is complete in thesense of Denition 7.1.6, the assertion follows from Theorem7.2.14 by replacing M with thehom of C and recalling that ⟨E, ⟨C⟩⟩ = ⟨E,C⟩ (see Remark 4.3.7(3)).

Theorem 7.3.17. If a category C is complete (resp. cocomplete), then for any functor H ∶ E′ →E the precomposition functor [H,C] ∶ [E,C]→ [E′,C] is continuous (resp. cocontinuous).

Proof. SinceC is complete i its hom ⟨C⟩ is complete in the sense of Denition 7.1.6, and [H,C]is continuous i its hom ⟨H,C⟩ is continuous in the sense of Denition 7.1.8, the assertion followsfrom Theorem7.2.15 by replacing M with the hom of C and recalling that ⟨H, ⟨C⟩⟩ = ⟨H,C⟩(see Remark 4.3.29(3)).

7.4. Limits of modules

Theorem 7.4.1. Given a small left module M ∶ ∗ E, i.e. a functor M ∶ E → Set with Esmall, a limit of M is given by an equalizer π as in

∏EMπ

π //∏e∈∥E∥ ⟨M⟩e∏e∈∥E∥(χe,e′)e′∈∥E∥

∏e′∈∥E∥ ⟨M⟩e′∏e′∈∥E∥(∆e,e′)e∈∥E∥

//∏e∈∥E∥∏e′∈∥E∥ [e ⟨E⟩e′, ⟨M⟩e′]

, whereχe,e′ ∶ ⟨M⟩e→ [e ⟨E⟩e′, ⟨M⟩e′] ; m↦ [h↦ m h]

is the function given by the exponential transpose of the composition betweenM-arrows andE-arrows, and

∆e,e′ ∶ ⟨M⟩e′ → [e ⟨E⟩e′, ⟨M⟩e′] ; m↦ [h↦ m]is the diagonal function.

a colimit of M is given by a coequalizer π as in

∐e∈∥E∥∐e′∈∥E∥ ⟨M⟩e × e ⟨E⟩e′∐e∈∥E∥(∆e,e′)e′∈∥E∥

∐e′∈∥E∥(χe,e′)e∈∥E∥ //∐e′∈∥E∥ ⟨M⟩e′π

∐e∈∥E∥ ⟨M⟩e π//∐EM

, whereχe,e′ ∶ ⟨M⟩e × e ⟨E⟩e′ → ⟨M⟩e′; (m,h)↦ m h

233

7. Limits

is the function given by the composition between M-arrows and E-arrows, and

∆e,e′ ∶ ⟨M⟩e × e ⟨E⟩e′ → ⟨M⟩e; (m,h)↦ m

is the projection.

Proof. By the claim below, a limit ofM is given by a universal fork on ∏e∈∥E∥ (χe,e′)e′∈∥E∥ and

∏e′∈∥E∥ (∆e,e′)e∈∥E∥, i.e. by an equalizer of them.

Claim. Given a small set S, a family of functions αe ∶ S → ⟨M⟩e, one for each e ∈ ∥E∥, formsa cone S M if and only if the diagram

S(αe′)e′∈E

(αe)e∈E //∏e∈∥E∥ ⟨M⟩e∏e∈∥E∥(χe,e′)e′∈∥E∥

∏e′∈∥E∥ ⟨M⟩e′∏e′∈∥E∥(∆e,e′)e∈∥E∥

//∏e∈∥E∥∏e′∈∥E∥ [e ⟨E⟩e′, ⟨M⟩e′]

commutes, i.e. if and only if the function (αe)e∈E ∶ S → ∏e∈∥E∥ ⟨M⟩e forms a fork on thefunctions ∏e∈∥E∥ (χe,e′)e′∈∥E∥ and ∏e′∈∥E∥ (∆e,e′)e∈∥E∥.

Proof. The diagram in the claim commutes i the diagram

S(αe′)e′∈E

αe // ⟨M⟩e(χe,e′)e′∈∥E∥

∏e′∈∥E∥ ⟨M⟩e′∏e′∈∥E∥∆e,e′

//∏e′∈∥E∥ [e ⟨E⟩e′, ⟨M⟩e′]

commutes for each e ∈ ∥E∥, and this diagram commutes i the diagram

Sαe′

αe // ⟨M⟩eχe,e′

⟨M⟩e′∆e,e′

// [e ⟨E⟩e′, ⟨M⟩e′]

commutes for each e′ ∈ ∥E∥, and this diagram commutes i the triangle

⟨M⟩e

⟨M⟩h

Sαe 55

αe′ )) ⟨M⟩e′

commutes for each E-arrow h ∶ e→ e′ since for any s ∈ S,

s ⋅ [αe ⟨M⟩h] = (s ⋅ αe) h = h ⋅ [(s ⋅ αe) ⋅ χe,e′] = h ⋅ [s ⋅ [αe χe,e′]]

ands ⋅ αe′ = h ⋅ [(s ⋅ αe′) ⋅∆e,e′] = h ⋅ [s ⋅ [αe′ ∆e,e′]]

by the denitions of χe,e′ and ∆e,e′ .

Corollary 7.4.2. Set is complete and cocomplete.

234

7. Limits

Proof. Since Set has equalizers and coequalizers, the assertion follows from Theorem7.4.1.

Corollary 7.4.3. For any pair of categories X and A, the module category [X ∶A] is completeand cocomplete, and all small limits and colimits are given pointwise.

Proof. Since Set is complete and cocomplete, so is the category [X ∶A] ∶= [X∼ ×A,Set] byTheorem7.3.16.

Corollary 7.4.4. For any pair of functors S ∶ E → X and T ∶ D → A, the precompositionfunctor [S ∶ T] ∶ [X ∶A]→ [E ∶D] is continuous and cocontinuous.

Proof. Since Set is complete and cocomplete, the precomposition functor [S ∶ T] ∶= [S∼ ×T,Set]is continuous and cocontinuous by Theorem7.3.17.

Theorem 7.4.5.

For a small left module M ∶ ∗ E, i.e. a functor M ∶ E → Set with E small, the set offrames of M gives a limit of M with the universal cone

ϕM ∶∏∗EMM ∶ ∗ ∗E

dened byα ⋅ ⟨ϕM⟩e = αe

such that the component of ϕM at e ∈ ∥E∥ sends each frame α ofM to its component at e. For a small right module M ∶ E ∗, i.e. a functor M ∶ E∼ → Set with E small, the set of

frames of M gives a limit of M with the universal cone

ϕM ∶∏E∗MM ∶ E∗ ∗

dened byα ⋅ e ⟨ϕM⟩ = αe

such that the component of ϕM at e ∈ ∥E∥ sends each frame α ofM to its component at e.

Proof. This follows from the observation that an element α of the product ∏e∈∥E∥ ⟨M⟩e formsa frame ofM i it lies in the equalizer in Theorem7.4.1.

Remark 7.4.6. We thus have

∏∗EM ≅∏

E

M ∏E∗M ≅∏

E∼M

.

Corollary 7.4.7. Let E be a small category. The family of universal cones ϕM ∶ ∏∗EM M ∶ ∗ ∗E, one for each left moduleM ∶ ∗ E, forms a counit

Set

⟨∶∗E⟩//ϕ [∶ E]

∏∗Eoo

of the module ⟨∶ ∗E⟩.

235

7. Limits

The family of universal cones ϕM ∶ ∏E∗M M ∶ E∗ ∗, one for each right moduleM ∶ E ∗, forms a counit

Set

⟨E∗∶⟩//ϕ [E ∶]

∏E∗oo

of the module ⟨E∗ ∶⟩.Proof. By Theorem7.4.5, each component of ϕ is universal. Hence it remains to verify that ϕsatises the naturality condition, i.e. verify that the square

∏∗EM∏∗E Φ

ϕM //MΦ

∏∗EN ϕN// N

commutes for any left module morphism Φ ∶M → N . But for any frame α ∈ ∏∗EM and anyobject e ∈ ∥E∥,

α ⋅ ⟨ϕM Φ⟩e = α ⋅ ⟨ϕM⟩e ⋅ ⟨Φ⟩e = αe ⋅ ⟨Φ⟩eand

α ⋅ ⟨∏∗E

Φ ϕN ⟩e = α ⋅∏∗E

Φ ⋅ ⟨ϕN ⟩e = [α Φ] ⋅ ⟨ϕN ⟩e = [α Φ]e = αe ⋅ ⟨Φ⟩e

.

Corollary 7.4.8. Let E be a small category. Given a category X, the module ⟨X ∶ ∗E⟩ has a counit

[X ∶]⟨X∶∗E⟩

//ϕX [X ∶ E]∏X∗Eoo

, giving, for each module M ∶X E, a universal wadge

[ϕX]M ∶X

∏∗EMM ∶X ∗E

, in fact a pointwise universal wadge, with each slice

x ⟨X

∏∗EM⟩ =∏

∗Ex ⟨M⟩ x ⟨[ϕX]M⟩ = ϕx⟨M⟩

giving a limit of the left module x ⟨M⟩ ∶ ∗ E. Given a category A, the module ⟨E∗ ∶A⟩ has a counit

[∶A]⟨E∗∶A⟩

//ϕA [E ∶A]∏A

E∗oo

, giving, for each module M ∶ EA, a universal wadge

[ϕA]M ∶A

∏E∗MM ∶ E∗A

, in fact a pointwise universal wadge, with each slice

⟨A

∏E∗M⟩a =∏

E∗⟨M⟩a ⟨[ϕA]M⟩a = ϕ⟨M⟩a

giving a limit of the right module ⟨M⟩a ∶ E ∗.

236

7. Limits

Proof. Apply Theorem7.2.12 to the counit in Corollary 7.4.7.

Remark 7.4.9.(1) By Remark 4.3.2(2), the counit

[∶A]⟨E∗∶A⟩

//ϕA [E ∶A]∏A

E∗oo

is the same thing as the unit

[E ∶A]∼⟨E∗∶A⟩∼ //

∏AE∗

//ϕA [∶A]∼

.(2) Given a small category E and a moduleM ∶XA, the triangle

[E,A]⟨∗E,M⟩

xx

ME

&&[X ∶] [X ∶ E]

∏X∗E

oo

commutes, where ⟨∗E,M⟩ is the right exponential transpose of the module ⟨∗E,M⟩ ∶X [E,A] andME is the right action ofM on [E,A]. Indeed, for any K ∈ [E,A] andany x ∈X,

(x) ⟨∗E,M⟩ (K) =∏∗Ex ⟨M⟩K = x ⟨

X

∏∗E

⟨M⟩K⟩

. Dually, the triangle[E,X]

EMww

⟨E∗,M⟩&&

[E ∶A]∼∏A

E∗// [∶A]∼

commutes.

Note. Remark 7.4.9 allows the following denition.

Denition 7.4.10. Given a small category E and a moduleM ∶XA, the cylinder

[E,A]⟨∗E,M⟩

xx

ME

&&[X ∶]

⟨X∶∗E⟩//

[M∗E]

[X ∶ E]


[X ∶]⟨X∶∗E⟩

//ϕX [X ∶ E]∏X∗Eoo

[E,A]MEoo


237

7. Limits

the cylinder[E,X]

EMww

⟨E∗,M⟩&&

[E ∶A]∼⟨E∗∶A⟩∼

//

[E∗M]

[∶A]∼


[E,X] EM // [E ∶A]∼⟨E∗∶A⟩∼ //

∏AE∗

//ϕA [∶A]∼


Remark 7.4.11. By Theorem6.4.16, the cylinder [M ∗E] (resp. [E ∗M]) is pointwise inverse(resp. direct) universal; the component of [M ∗E] at a functor K ∶ E→A is the pointwise universal wadge

[M ∗E]K ∶ ⟨∗E,M⟩ (K) ⟨M⟩K ∶X ∗E

given by [M ∗E]K = [ϕX]⟨M⟩K (the component of the counit ϕX at ⟨M⟩K), and the sliceof [M ∗E]K at x ∈ ∥X∥ is the universal cone

x ⟨[M ∗E]K⟩ ∶ (x) ⟨∗E,M⟩ (K) x ⟨M⟩K ∶ ∗ ∗E

given by x ⟨[M ∗E]K⟩ = x ⟨[ϕX]⟨M⟩K⟩ = ϕx⟨M⟩K (see Corollary 7.4.8). the component of [E ∗M] at a functor K ∶ E→X is the pointwise universal wadge

[E ∗M]K ∶ (K) ⟨E∗,M⟩ K ⟨M⟩ ∶ E∗A

given by [E ∗M]K = [ϕA]K⟨M⟩ (the component of the unit ϕA at K ⟨M⟩), and the slice of[E ∗M]K at a ∈ ∥A∥ is the universal cone

⟨[E ∗M]K⟩a ∶ (K) ⟨E∗,M⟩ (a) K ⟨M⟩a ∶ E∗ ∗

given by ⟨[E ∗M]K⟩a = ⟨[ϕA]K⟨M⟩⟩a = ϕK⟨M⟩a (see Corollary 7.4.8).

Note. Given a category E and a module M ∶ X A, we have two Yoneda morphisms for⟨∗E,M⟩,

X⟨∗E,M⟩ //

X

[E,A]⟨∗E,M⟩

X⟨∗E,M⟩ //

X

[E,A]ME

[X ∶]⟨X∶⟩

//

⟨X⟨∗E,M⟩⟩

[X ∶] [X ∶]⟨X∶∗E⟩

//

⟨XM⟩∗E

[X ∶ E]

, one dened in Denition 5.2.5 and the other in Denition 5.4.1. The following reconciles thesetwo Yoneda morphisms.

Theorem 7.4.12. Let E be a small category and M ∶XA be a module. The right general Yoneda morphism for ⟨∗E,M⟩ is obtained by pasting the cylinder

[M ∗E] dened in Denition 7.4.10 to the right Yoneda morphism for ⟨∗E,M⟩ as shownin

X

X

⟨∗E,M⟩ //

⟨X⟨∗E,M⟩⟩

[E,A]

⟨∗E,M⟩

xx

ME

[X ∶]

⟨X∶∗E⟩// [X ∶ E]

[M∗E]

238

7. Limits

; that is, given a cone α ∶ r K ∶ ∗EM, the wedge

⟨XM⟩α ∶ ⟨X⟩ r ⟨M⟩K ∶X ∗E

is obtained by the composition of the right module morphism

⟨X⟨∗E,M⟩⟩α ∶ ⟨X⟩ r→ ⟨∗E,M⟩ (K) ∶X ∗

and the wedge[M ∗E]K ∶ ⟨∗E,M⟩ (K) ⟨M⟩K ∶X ∗E

. The left general Yoneda morphism for ⟨E∗,M⟩ is obtained by pasting the cylinder [E ∗M]

dened in Denition 7.4.10 to the left Yoneda morphism for ⟨E∗,M⟩ as shown in

[E,X]

EM

⟨E∗,M⟩

''

⟨E∗,M⟩ //A

A

⟨⟨E∗,M⟩A⟩

[E ∶A]∼⟨E∗∶A⟩∼

//

[E∗M]

[∶A]∼

; that is, given a cone α ∶ K r ∶ E∗M, the wedge

α ⟨MA⟩ ∶ r ⟨A⟩ K ⟨M⟩ ∶ E∗A

is obtained by the composition of the left module morphism

α ⟨⟨E∗,M⟩A⟩ ∶ r ⟨A⟩→ (K) ⟨E∗,M⟩ ∶ ∗A

and the wedge[E ∗M]K ∶ (K) ⟨E∗,M⟩ K ⟨M⟩ ∶ E∗A

.

Proof. We need to show that

x ⟨⟨XM⟩α⟩e = x ⟨X⟨∗E,M⟩⟩α ⋅ x ⟨[M ∗E]K⟩e

for x ∈ ∥X∥ and e ∈ ∥E∥. But for an X-arrow h ∶ x→ r,

h ⋅ x ⟨⟨XM⟩α⟩e = h αe

and by Remark 7.4.11 and Theorem7.4.5,

h ⋅ x ⟨X⟨∗E,M⟩⟩α ⋅ x ⟨[M ∗E]K⟩e = (h α) ⋅ ⟨ϕx⟨M⟩K⟩e = (h α)e = h αe

.

Theorem 7.4.13. Let E be a category and M ∶XA be a module. The right general Yoneda morphism for ⟨∗E,M⟩ preserves and reects inverse universals

in the following sense: a cone π ∶ r K ∶ ∗E M is universal if and only if the wedge⟨XM⟩π ∶ ⟨X⟩ r ⟨M⟩K ∶ X ∗E is pointwise universal; that is, if and only if the conex ⟨XM⟩π ∶ x ⟨X⟩ r x ⟨M⟩K ∶ ∗ ∗E is universal for every x ∈ ∥X∥.

239

7. Limits

The left general Yoneda morphism for ⟨E∗,M⟩ preserves and reects direct universals inthe following sense: a cone π ∶ K r ∶ E∗ M is universal if and only if the wedgeπ ⟨MA⟩ ∶ r ⟨A⟩ K ⟨M⟩ ∶ E∗ A is pointwise universal; that is, if and only if the coneπ ⟨MA⟩a ∶ r ⟨A⟩a K ⟨M⟩a ∶ E∗ ∗ is universal for every a ∈ ∥A∥.

Proof. First enlarge the universe if necessary so that E become small. For a cone π ∶ r K ∶ ∗E M, by Theorem7.4.12, ⟨XM⟩π is given by the composition of ⟨X⟨∗E,M⟩⟩πand [M ∗E]K. Since [M ∗E]K is a pointwise universal wedge (see Remark 7.4.11), by Theo-rem7.2.11, ⟨XM⟩π is pointwise universal i ⟨X⟨∗E,M⟩⟩π is iso, i.e. i π is inverse univer-sal.

Corollary 7.4.14. Let E be a category and M ∶XA be a module. The right Yoneda morphism for M preserves and reects limits in the following sense: a

cone π ∶ r K ∶ ∗EM is universal if and only if its composite

∗r

E∆oo

K

X

X

M //

π

A

M

[X ∶]⟨X∶⟩

//

⟨XM⟩

[X ∶]

with the right Yoneda morphism for M is pointwise universal (see Denition 7.2.1). The left Yoneda morphism for M preserves and reects colimits in the following sense: a

cone π ∶ K r ∶ E∗M is universal if and only if its composite

E

K

∆ // ∗r

X

M

M //

π

A

A

[∶A]∼⟨∶A⟩∼

//

⟨MA⟩

[∶A]∼

with the left Yoneda morphism for M is pointwise universal (see Denition 7.2.1).

Proof. Since the right exponential transpose of the wedge ⟨XM⟩π is given by the cone⟨⟨XM⟩⟩ π (see Remark 5.4.5), by the equivalence of (1) and (2) in Proposition 7.2.8,the assertion is reduced to Theorem7.4.13 (and vice versa).

Note. Theorem7.4.15 and Corollary 7.4.16 below are special cases of Theorem7.4.13 and Corol-lary 7.4.14 whereM is given by the hom of a category.

Theorem 7.4.15. Let C and E be categories. The right general Yoneda morphism for ⟨∗E,C⟩ (see Denition 5.4.6) preserves and reects

inverse universals in the following sense: a cone π ∶ r K ∶ ∗E→C is universal if and onlyif the wedge ⟨C⟩π ∶ ⟨C⟩ r ⟨C⟩K ∶C ∗E is pointwise universal; that is, if and only if thecone c ⟨C⟩π ∶ c ⟨C⟩ r c ⟨C⟩K ∶ ∗ ∗E is universal for every c ∈ ∥C∥.

The left general Yoneda morphism for ⟨E∗,C⟩ (see Denition 5.4.6) preserves and reectsdirect universals in the following sense: a cone π ∶ K r ∶ E∗ → C is universal if and onlyif the wedge π ⟨C⟩ ∶ r ⟨C⟩ K ⟨C⟩ ∶ E∗C is pointwise universal; that is, if and only if thecone π ⟨C⟩c ∶ r ⟨C⟩c K ⟨C⟩c ∶ E∗ ∗ is universal for every c ∈ ∥C∥.

240

7. Limits

Proof. By Remark 5.4.7(2), this is a special case of Theorem7.4.13 where M is given by thehom of C.

Corollary 7.4.16. Let C and E be categories. The right Yoneda functor for C preserves and reects limits in the following sense: a cone

π ∶ r K ∶ ∗E→C is universal if and only if its composite

∗r

E∆oo

K

C

C

⟨C⟩ //

π

C

C

[C ∶]⟨C∶⟩

//

⟨C⟩

[C ∶]

with the right Yoneda functor for C is pointwise universal (see Denition 7.2.1). The left Yoneda functor for C preserves and reects colimits in the following sense: a cone

π ∶ K r ∶ E∗→C is universal if and only if its composite

E

K

∆ // ∗r

C

C

⟨C⟩ //

π

C

C

[∶C]∼⟨∶C⟩∼

//

⟨C⟩

[∶C]∼

with the left Yoneda functor for C is pointwise universal (see Denition 7.2.1).

Proof. Since the right exponential transpose of the wedge ⟨C⟩π is given by the cone [C] α(see Remark 5.4.10), by the equivalence of (1) and (2) in Proposition 7.2.8, the assertion isreduced to Theorem7.4.15 (and vice versa).

Corollary 7.4.17.

A functor H ∶C→ B preserves limits over a category E if and only if for every b ∈ ∥B∥ theleft module b ⟨B⟩H ∶ ∗C does the same.

A functor H ∶ C → B preserves colimits over a category E if and only if for every b ∈ ∥B∥the right module H ⟨B⟩b ∶C ∗ does the same.

Proof. By Theorem7.4.15, for any cone π ∶ ∗E → C, H π is universal i b ⟨B⟩ [H π] =⟨b ⟨B⟩H⟩π is universal for every b ∈ ∥B∥.

Corollary 7.4.18.

A representable left moduleM ∶ ∗A preserves limits; that is, if a cone π ∶ r K ∶ ∗E→Ais universal, so is the cone ⟨M⟩π ∶ ⟨M⟩ r ⟨M⟩K ∶ ∗ ∗E.

A representable right module M ∶ X ∗ preserves colimits; that is, if a cone π ∶ K r ∶E∗→X is universal, so is the cone π ⟨M⟩ ∶ r ⟨M⟩ K ⟨M⟩ ∶ E∗ ∗

Proof. Since M is representable, M ≅ a ⟨A⟩ for some a ∈ ∥A∥. But by Theorem7.4.15, if acone π ∶ r K ∶ ∗E→A is universal, so is the cone a ⟨A⟩π ∶ a ⟨A⟩ r a ⟨A⟩K ∶ ∗ ∗E.

Note. In the following, a module and the corresponding collage are given the same name andidentied with each other.

241

7. Limits

Corollary 7.4.19.

A left module M ∶ ∗ A preserves limits over a category E if and only if the inclusionM ∶A→ ⌊M⌋ does the same.

A right module M ∶X ∗ preserves colimits over a category E if and only if the inclusionM ∶X→ ⌊M⌋ does the same.

Proof. Since the objects of the collage category ⌊M⌋ consists of all objects of A and the object∗, by Corollary 7.4.17, the inclusion M ∶ A → ⌊M⌋ preserves limits over E i the followingconditions hold:(1) for every a ∈ ∥A∥, a ⟨⌊M⌋⟩M = a ⟨A⟩ preserves limits over E;(2) ∗ ⟨⌊M⌋⟩M =M preserves limits over E.Since the rst condition always holds by Corollary 7.4.18, the assertion follows.

Remark 7.4.20. Suppose that a left module M ∶ ∗ A preserves limits over a category E. Under this

condition, Corollary 7.4.19 says that, if a cone π ∶ r K ∶ ∗E → A is universal in A, thenit is universal in ⌊M⌋ as well; that is, to every cone α ∶ ∗ K ∶ ∗E M there is a uniqueM-arrow α/π ∶ ∗ r, the adjunct of α along π inM, such that α = α/π π.

Suppose that a right module M ∶ X ∗ preserves colimits over a category E. Under thiscondition, Corollary 7.4.19 says that, if a cone π ∶ K r ∶ E∗ → X is universal in X, thenit is universal in ⌊M⌋ as well; that is, to every cone α ∶ K ∗ ∶ E∗ M there is a uniqueM-arrow π/α ∶ r ∗, the adjunct of α along π inM, such that α = π π/α.

Theorem 7.4.21. Consider the comma and collage

⌈M↓⌉M↓0

vv

M↓1

((X

M0 ((

A

M1vv⌊M⌋

of a module M ∶XA. If the inclusion M1 preserves limits over a category E, then the pair of brations M↓

0 andM↓

1 creates limits over E in the following sense: given a functor K ∶ E → ⌈M↓⌉, if M↓0 K

and M↓1 K have limits ρ ∶ r M↓

0 K and σ ∶ s M↓1 K, then there is a unique cone

π ∶ m K in ⌈M↓⌉ such that π M↓0 = ρ and π M↓

1 = σ; moreover π is universal. If the inclusionM0 preserves colimits over a category E, then the pair of brationsM↓

0 andM↓

1 creates colimits over E in the following sense: given a functor K ∶ E→ ⌈M↓⌉, if K M↓0

and K M↓1 have colimits ρ ∶ K M↓

0 r and σ ∶ K M↓1 s, then there is a unique cone

π ∶ K m in ⌈M↓⌉ such that π M↓0 = ρ and π M↓

1 = σ; moreover π is universal.

Proof. We refer to the commutative diagram

X

M0

⌈M↓⌉M↓

0oo

M↓1 //A

M1

⌊M⌋ [2, ⌊M⌋][0,⌊M⌋]oo

[1,⌊M⌋]// ⌊M⌋

in Denition 3.2.17(1). We rst observe that, by the construction of a collage, M0 preserveslimits. Now suppose thatM↓

0 K andM↓1 K have limits ρ ∶ r M↓

0 K and σ ∶ s M↓1 K.

SinceM0 andM1 preserve limits, they yield universal cones ρ ∶ r [0, ⌊M⌋] K and σ ∶ s

242

7. Limits

[1, ⌊M⌋] K. Hence, by Theorem7.3.15, there is a unique cone π ∶ m K in [2, ⌊M⌋] suchthat π [0, ⌊M⌋] = ρ and π [1, ⌊M⌋] = σ, and π is universal. Clearly, π is in ⌈M↓⌉, and theassertion follows.

Corollary 7.4.22.

If a left moduleM ∶ ∗A preserves limits over a category E, then the left comma brationM↓ ∶ ⌈M↓⌉→A creates limits over E.

If a right module M ∶ X ∗ preserves colimits over a category E, then the right commabration M↓ ∶ ⌈M↓⌉→X creates colimits over E.

Proof. Since the inclusion M ∶ A → ⌊M⌋ preserves limits over E by Corollary 7.4.19, theassertion follows as a special case of Theorem7.4.21 where X is the terminal category.

Corollary 7.4.23.

Given a left module M ∶ ∗ A, if A is complete and M is continuous, then the commacategory ⌈M↓⌉ is complete and the left comma bration M↓ ∶ ⌈M↓⌉→A is continuous.

Given a right moduleM ∶X ∗, if X is cocomplete andM is cocontinuous, then the commacategory ⌈M↓⌉ is cocomplete and the right comma brationM↓ ∶ ⌈M↓⌉→X is cocontinuous.

Proof. SinceM preserves small limits, the left comma brationM↓ ∶ ⌈M↓⌉→A creates smalllimits by Corollary 7.4.22. The assertion thus follows from Proposition 7.3.12.

243

8. Adjunctions

8.1. Adjunctions for modules

Denition 8.1.1. Given a pair of modules M ∶ X A and N ∶ Y B, and given a pair offunctors P ∶X→Y and Q ∶ B→A, a right slant cell Φ ∶ Q P ∶M N , written diagrammatically as

X

P

M //A

Φ

Y N// B

Q

OO

, is dened by a module morphism Φ ∶ ⟨M⟩Q→ P ⟨N ⟩ ∶X B. a left slant cell Φ ∶ P Q ∶ N M, written diagrammatically as

YN //

Φ

B

Q

X M//

P

OO

A

, is dened by a module morphism Φ ∶ P ⟨N ⟩→ ⟨M⟩Q ∶X B.

Remark 8.1.2.(1) Given a pair of objects x ∈ ∥X∥ and b ∈ ∥B∥,

a right slant cell Φ ∶ Q P ∶M N sends eachM-arrow m ∶ x Q ⋅ b to the N -arrowm ⋅Φ ∶ x ⋅ P b, the image of m under the function

x ⟨M⟩ (Q ⋅ b) = x ⟨⟨M⟩Q⟩b x⟨Φ⟩b // x ⟨P ⟨N ⟩⟩b = (x ⋅ P) ⟨N ⟩b

. a left slant cell Φ ∶ P Q ∶ N M sends each N -arrow n ∶ x ⋅ P b to the M-arrow

n ⋅Φ ∶ x Q ⋅ b, the image of n under the function

(x ⋅ P) ⟨N ⟩b = x ⟨P ⟨N ⟩⟩b x⟨Φ⟩b // x ⟨⟨M⟩Q⟩b = x ⟨M⟩ (Q ⋅ b)

.(2) The identity module morphismM→M yields the identity right and left slant cells

X

1

M //A

1

XM //

1

A

1

X M//A

1

OO

X M//

1

OO

A

.

Denition 8.1.3.

244

8. Adjunctions

Given a pair of right slant cells as in

XM //

P

A

X′ M′ //

P′

Φ

A′Q

OO

X′′M′′

//

Φ′

A′′Q′OO

, their composite Φ Φ′ = Φ′ Φ is the right slant cell

X

P P′

M //A

X′′M′′

//

Φ Φ′

A′′Q Q′OO

dened by the module morphism Φ Φ′ ∶ ⟨M⟩ [Q Q′] → [P P′] ⟨M′′⟩ ∶ X A′′ given bythe composition

⟨M⟩ [Q Q′] = ⟨⟨M⟩Q⟩Q′ ⟨Φ⟩Q′// ⟨P ⟨M′⟩⟩Q′ = P ⟨⟨M′⟩Q′⟩ P⟨Φ′⟩ // P ⟨P′ ⟨M′′⟩⟩ = [P P′] ⟨M′′⟩

. Given a pair of left slant cells as in

X′′ M′′//

Φ′

A′′

Q′

X′ M′ //

P′OO

Φ

A′

Q

X M//

P

OO

A

, their composite Φ′ Φ = Φ Φ′ is the left slant cell

X′′ M′′//

Φ Φ′

A′′

Q Q′

X M//

P P′OO

A

dened by the module morphism Φ Φ′ ∶ [P P′] ⟨M′′⟩ → ⟨M⟩ [Q Q′] ∶ X A′′ given bythe composition

[P P′] ⟨M′′⟩ = P ⟨P′ ⟨M′′⟩⟩ P⟨Φ′⟩ // P ⟨⟨M′⟩Q′⟩ = ⟨P ⟨M′⟩⟩Q′ ⟨Φ⟩Q′// ⟨⟨M⟩Q⟩Q′ = ⟨M⟩ [Q Q′]

.

Proposition 8.1.4. Modules and right (resp. left) slant cells among them form a categorywith the composition given in Denition 8.1.3 and the identities given in Remark 8.1.2(2).

245

8. Adjunctions

Proof. The only non-trivial part is the verication of the associativity of the composition.Consider cells as in

XM //

P

A

X′ M′ //

P′

Φ

A′Q

OO

X′′ M′′ //

P′′

Φ′

A′′Q′OO

X′′′M′′′

//

Φ′′

A′′′Q′′OO

. The composites ⟨Φ Φ′⟩ Φ′′ and Φ ⟨Φ′ Φ′′⟩ are dened by the module morphisms⟨⟨Φ⟩Q′ P ⟨Φ′⟩⟩Q′′ [P P′] ⟨Φ′′⟩ and ⟨Φ⟩ [Q′ Q′′] P ⟨⟨Φ′⟩Q′′ P′ ⟨Φ′′⟩⟩ respectively. Butby the functoriality and associativity of the composition, we have

⟨⟨Φ⟩Q′ P ⟨Φ′⟩⟩Q′′ [P P′] ⟨Φ′′⟩ = ⟨⟨Φ⟩Q′⟩Q′′ ⟨P ⟨Φ′⟩⟩Q′′ P ⟨P′ ⟨Φ′′⟩⟩= ⟨Φ⟩ [Q′ Q′′] P ⟨⟨Φ′⟩Q′′ P′ ⟨Φ′′⟩⟩

.

Denition 8.1.5. Given a pair of modulesM ∶XA and N ∶Y B, the module of right slant cellsM N ,

⟨MUN ⟩ ∶ [B,A] [X,Y]∼


[B,A] MB // [X ∶ B] ⟨X∶B⟩ // [X ∶ B] [X,Y]∼XNoo

, where MB is the right action of M on the functor category [B,A] and XN is theleft action of N on the functor category [X,Y].

the module of left slant cells N M,

⟨N QM⟩ ∶ [X,Y]∼ [B,A]


[X,Y]∼ XN // [X ∶ B] ⟨X∶B⟩ // [X ∶ B] [B,A]MBoo

, where XN is the left action of N on the functor category [X,Y] andMB is the rightaction ofM on the functor category [B,A].

Remark 8.1.6.(1) For each pair of functors Q ∶ B→A and P ∶X→Y,

the set (Q) ⟨MUN ⟩ (P) consists of all module morphisms ⟨M⟩Q → P ⟨N ⟩ ∶ X B, i.e.all right slant cells Q P ∶M N .

the set (P) ⟨N QM⟩ (Q) consists of all module morphisms P ⟨N ⟩ → ⟨M⟩Q ∶ X B, i.e.all left slant cells P Q ∶ N M.

(2) For any modulesM and N ,⟨N QM⟩ ≅ ⟨N ∼ UM∼⟩

.

246

8. Adjunctions

Denition 8.1.7.

A right slant cell

X

P

M //A

Y N//

Φ

B

Q

OO

is called an adjunction if the module morphism Φ ∶ ⟨M⟩Q → P ⟨N ⟩ ∶ X B is iso. IfΦ ∶ Q P ∶M N is an adjunction, then the pair (Q,Φ), or the functor Q itself, is calleda right adjoint of P alongM and N .

A left slant cell

YN //

Φ

B

Q

X M//

P

OO

A

is called an adjunction if the module morphism Φ ∶ P ⟨N ⟩ → ⟨M⟩Q ∶ X B is iso. IfΦ ∶ P Q ∶ N M is an adjunction, then the pair (P,Φ), or the functor P itself, is calleda left adjoint of Q along N andM.

Remark 8.1.8. The two forms of an adjunctions, M N and N M, are referred to as theright and left forms. If Φ is the left form of an adjunction, then the inverse Φ−1 gives the rightform of the adjunction.

Proposition 8.1.9.

(1) The identity right (resp. left) slant cell is an adjunction.(2) If two composable right (resp. left) slant cells are adjunctions, so is their composite.

Proof.(1) Evident.(2) Consider a pair of right slant cells as in Denition 8.1.3. Since the cell Φ Φ′ ∶MM′′ is

dened by the module morphism ⟨Φ⟩Q′ P ⟨Φ′⟩, it suces to show that ⟨Φ⟩Q′ and P ⟨Φ′⟩are isomorphisms. But this holds by Theorem1.1.25.

Proposition 8.1.10. Modules and the right (resp. left) forms of adjunctions among themconstitute a subcategory of the category of modules and right (resp. left) slant cells.


Note. Proposition 1.2.10 allows the following denition (cf. Denition 1.2.23).

Denition 8.1.11. Let J ∶ ED be a module. Given a right slant cell

X

P

M //A

Φ

Y N// B

Q

OO

, the right slant cell

[E,X] ⟨J ,M⟩ //

[E,P]

[D,A]⟨J ,Φ⟩

[E,Y]⟨J ,N ⟩

// [D,B][D,Q]OO

247

8. Adjunctions


⟨J ,M⟩ [D,Q] = ⟨J , ⟨M⟩Q⟩ ⟨J ,Φ⟩ // ⟨J ,P ⟨N ⟩⟩ = [E,P] ⟨J ,N ⟩

, postcomposition (see Denition 1.2.14) with Φ ∶ ⟨M⟩Q→ P ⟨N ⟩. Given a left slant cell

YN //

Φ

B

Q

X M//

P

OO

A

, the left slant cell

[E,Y] ⟨J ,N ⟩ //

⟨J ,Φ⟩

[D,B][D,Q]

[E,X]⟨J ,M⟩

//

[E,P]OO

[D,A]


[E,P] ⟨J ,N ⟩ = ⟨J ,P ⟨N ⟩⟩ ⟨J ,Φ⟩ // ⟨J , ⟨M⟩Q⟩ = ⟨J ,M⟩ [D,Q]

, postcomposition (see Denition 1.2.14) with Φ ∶ P ⟨N ⟩→ ⟨M⟩Q.

Remark 8.1.12. Given a pair of functors S ∶ E→X and T ∶D→ B, the right slant cell ⟨J ,Φ⟩ sends each cell Θ ∶ S Q T ∶ J M to the cell Θ Φ ∶ S P T ∶J N dened by the module morphism Θ Φ ∶ J → [S P] ⟨N ⟩T given by the composition

J Θ // S ⟨M⟩ [Q T] = S ⟨⟨M⟩Q⟩TS⟨Φ⟩T // S ⟨P ⟨N ⟩⟩T = [S P] ⟨N ⟩T

. the left slant cell ⟨J ,Φ⟩ sends each cell Θ ∶ S P T ∶ J N to the cell Θ Φ ∶ S

Q T ∶ J M dened by the module morphism Θ Φ ∶ J → S ⟨M⟩ [Q T] given by thecomposition

J Θ // [S P] ⟨N ⟩T = S ⟨P ⟨N ⟩⟩TS⟨Φ⟩T // S ⟨⟨M⟩Q⟩T = S ⟨M⟩ [Q T]

.

Proposition 8.1.13. If a cell Φ is an adjunction, so is the cell ⟨J ,Φ⟩.

Proof. Since the operation ⟨J ,−⟩ is functorial, it preserves isomorphisms.

Note. Proposition 4.3.8 allows the following denition (cf. Denition 4.3.16).

Denition 8.1.14. Let E be a category. Given a right slant cell

X

P

M //A

Φ

Y N// B

Q

OO

248

8. Adjunctions

, the right slant cell

[E,X] ⟨E,M⟩ //

[E,P]

[E,A]⟨E,Φ⟩

[E,Y]⟨E,N ⟩

// [E,B][E,Q]OO


⟨E,M⟩ [E,Q] = ⟨E, ⟨M⟩Q⟩ ⟨E,Φ⟩ // ⟨E,P ⟨N ⟩⟩ = [E,P] ⟨E,N ⟩

, postcomposition (see Denition 4.3.12) with Φ ∶ ⟨M⟩Q→ P ⟨N ⟩. Given a left slant cell

YN //

Φ

B

Q

X M//

P

OO

A

, the left slant cell

[E,Y] ⟨E,N ⟩ //

⟨E,Φ⟩

[E,B][E,Q]

[E,X]⟨E,M⟩

//

[E,P]OO

[E,A]


[E,P] ⟨E,N ⟩ = ⟨E,P ⟨N ⟩⟩ ⟨E,Φ⟩ // ⟨E, ⟨M⟩Q⟩ = ⟨E,M⟩ [E,Q]

, postcomposition (see Denition 4.3.12) with Φ ∶ P ⟨N ⟩→ ⟨M⟩Q.

Remark 8.1.15. Given a pair of functors S ∶ E→X and T ∶ E→ B, the right slant cell ⟨E,Φ⟩ sends each cylinder α ∶ S Q T ∶ E M to the cylinder

α Φ ∶ S P T ∶ E N dened by

α Φ = α S ⟨Φ⟩T = α ⋅∏E

S ⟨Φ⟩T

, the image of the frame α ∈∏E S ⟨M⟩ [Q T] under the function

∏E S ⟨M⟩ [Q T] =∏E S ⟨⟨M⟩Q⟩T∏E S⟨Φ⟩T //∏E S ⟨P ⟨N ⟩⟩T =∏E [S P] ⟨N ⟩T

. the left slant cell ⟨E,Φ⟩ sends each cylinder α ∶ S P T ∶ E N to the cylinder α Φ ∶ S

Q T ∶ EM dened by

α Φ = α S ⟨Φ⟩T = α ⋅∏E

S ⟨Φ⟩T

, the image of the frame α ∈∏E [S P] ⟨N ⟩T under the function

∏E [S P] ⟨N ⟩T =∏E S ⟨P ⟨N ⟩⟩T∏E S⟨Φ⟩T //∏E S ⟨⟨M⟩Q⟩T =∏E S ⟨M⟩ [Q T]

.

Proposition 8.1.16. If a cell Φ is an adjunction, so is the cell ⟨E,Φ⟩.Proof. Since the operation ⟨E,−⟩ is functorial, it preserves isomorphisms.

249

8. Adjunctions

8.2. Adjunctions for categories

Denition 8.2.1. Given a pair of functors G ∶A→X and F ∶X→A, an adjunction Υ ∶ G F ∶XA, written diagrammatically as

XF

//Υ AGoo

, is dened by a module isomorphism

Υ ∶ ⟨X⟩G→ F ⟨A⟩ ∶XA

. If Υ ∶ G F ∶ X A is an adjunction, then the pair (G,Υ), or the functor G itself, iscalled a right adjoint of F.

an adjunction Υ ∶ F G ∶AX, written diagrammatically as

AG //Υ XF

oo

, is dened by a module isomorphism

Υ ∶ F ⟨A⟩→ ⟨X⟩G ∶XA

. If Υ ∶ F G ∶ A X is an adjunction, then the pair (F,Υ), or the functor F itself, iscalled a left adjoint of G.

Remark 8.2.2.(1) An adjunction for categories X and A is a special case of an adjunction dened in De-

nition 8.1.7 where M and N are given by the hom of X and the hom of A; adjunctionsΥ ∶ G F ∶ X A and Υ ∶ F G ∶ A X for categories X and A are the same thing asadjunctions

X

F

⟨X⟩ //X A⟨A⟩ //

Υ

A

G

A⟨A⟩

//

Υ

A

G

OO

X⟨X⟩

//

F

OO

X

for the modules ⟨X⟩ and ⟨A⟩. The two forms of an adjunctions, X A and A X, arereferred to as the right and left forms. If Υ is the left form of an adjunction, then theinverse Υ−1 gives the right form of the adjunction. We mainly deal with the right form foreach adjunction.

(2) If a pair (G,Υ) is a right adjoint of F, then the pair (G,Υ−1) is a corepresentation of the

representable module F ⟨A⟩. Conversely if a pair (G,Υ) is a corepresentation of F ⟨A⟩,then the pair (G,Υ−1) is a right adjoint of F.

If a pair (F,Υ) is a left adjoint of G, then the pair (F,Υ−1) is a representation of thecorepresentable module ⟨X⟩G. Conversely if a pair (F,Υ) is a representation of ⟨X⟩G,then the pair (F,Υ−1) is a left adjoint of G.

Note. Theorem5.5.1 justies the following denition.

Denition 8.2.3.

If Υ ∶ G F ∶XA is an adjunction, then

250

8. Adjunctions

the counit of Υ is the natural transformation ε ∶ G F→ 1A such that Xε = Υ; the unit of Υ is the natural transformation η ∶ 1X → G F such that ηA = Υ−1.

If Υ ∶ F G ∶AX is an adjunction, then the unit of Υ is the natural transformation η ∶ 1X → G F such that ηA = Υ. the counit of Υ is the natural transformation ε ∶ G F→ 1A such that Xε = Υ−1.

Remark 8.2.4.(1) Theorem5.5.1 shows how the counit and unit of an adjunction are obtained and how an

adjunction is recovered from its counit and unit.(2)

An adjunction Υ ∶ G F ∶ X A is also written as (η, ε) ∶ G F ∶ X A, ordiagrammatically as

XF

//(η,ε) AGoo

, showing its unit and counit. An adjunction Υ ∶ F G ∶ A X is also written as (ε, η) ∶ F G ∶ A X, or

diagrammatically as

AG //

(ε,η) XF

oo

, showing its counit and unit.

Proposition 8.2.5. Consider a pair of functors as in

XF

//AGoo

For a natural transformation ε ∶ G F→ 1A, the following conditions are equivalent:(1) (G,Xε) forms a right adjoint of F; that is, ε is the counit of an adjunction G F ∶

XA;(2) the module morphism Xε ∶ ⟨X⟩G→ F ⟨A⟩ ∶XA is iso;(3) ε regarded as a right cylinder

XF⟨A⟩

//ε AGoo

forms a counit of the representable module F ⟨A⟩;(4) ε regarded as a two-sided cylinder

AG

1

##X

F⟨A⟩//

ε

A

forms a pointwise lift of the identity 1A inverse along the representable module F ⟨A⟩.(5) ε forms a pointwise right Kan lift of the identity 1A along F;(6) each component εa ∶ a ⋅G ⋅ F→ a is universal from F to a;

For a natural transformation η ∶ 1X → G F, the following conditions are equivalent:(1) (F, ηA) forms a left adjoint of G; that is, η is the unit of an adjunction F G ∶AX;(2) the module morphism ηA ∶ F ⟨A⟩→ ⟨X⟩G ∶XA is iso;

251

8. Adjunctions

(3) η regarded as a left cylinder

X⟨X⟩G //

F//η A

forms a unit of the corepresentable module ⟨X⟩G;(4) η regarded as a two-sided cylinder

X1

F

##X

⟨X⟩G//

η

A

forms a pointwise lift of the identity 1X direct along the corepresentable module ⟨X⟩G.(5) η forms a pointwise left Kan lift of the identity 1X along G;(6) each component ηx ∶ x→ G ⋅ F ⋅ x is universal from x to G;

Proof. By denition, (1)⇔ (2)⇔ (3) and (4)⇔ (5)⇔ (6). The equivalence (3)⇔ (4) isstated in Proposition 6.4.5.

Note. By Remark 8.2.2(2), Theorem8.2.6 and Corollary 8.2.7 are special cases of Theorem6.3.10and Corollary 6.3.11 whereM is given by a representable (resp. corepresentable) module.

Theorem 8.2.6.

Given a functor F ∶X→A, suppose that there is a family of A-arrows εa ∶ ra ⋅F→ a, one foreach object a ∈ ∥A∥, universal from F to a. Then there is a unique functor G ∶ A → X witha ⋅G = ra such that ε ∶= (εa)a∈∥A∥ forms a natural transformation ε ∶ G F→ 1A; moreover, Gis a right adjoint of F with ε the counit of the adjunction.

Given a functor G ∶A→X, suppose that there is a family of X-arrow ηx ∶ x→ G ⋅rx, one foreach object x ∈ ∥X∥, universal from x to G. Then there is a unique functor F ∶X →A withrx = F ⋅ x such that η ∶= (ηx)x∈∥X∥ forms a natural transformation η ∶ 1X → G F; moreover,F is a left adjoint of G with η the unit of the adjunction.

Proof. As noted above, this follows as a special case of Theorem6.3.10 with M given by therepresentable module F ⟨A⟩ (resp. the corepresentable module ⟨X⟩G).

Corollary 8.2.7.

The following conditions are equivalent for a functor F ∶X→A:(1) F has a right adjoint;(2) for every object a ∈ ∥A∥, the right module F ⟨A⟩a ∶X ∗ is representable;(3) for every object a ∈ ∥A∥, there is an object ra ∈ ∥X∥ and an A-arrow εa ∶ ra ⋅ F → a

universal from F to a. The following conditions are equivalent for a functor G ∶A→X:

(1) G has a left adjoint;(2) for every object x ∈ ∥X∥, the left module x ⟨X⟩G ∶ ∗A is representable;(3) for every object x ∈ ∥X∥, there is an object rx ∈ ∥A∥ and an X-arrow ηx ∶ x → G ⋅ rx

universal from x to G.

Proof. Since F has a right adjoint i F ⟨A⟩ is corepresentable i F ⟨A⟩ has a counit, the assertionis a special case of Corollary 6.3.11 whereM is given by the representable module of F.

252

8. Adjunctions

Theorem 8.2.8. Given functors

XF

//AGoo

, a pair of natural transformations η ∶ 1X → G F and ε ∶ G F → 1A form an adjunction(η, ε) ∶ G F ∶XA if and only if the pasting compositions

X1

F

##η

AG

1

##ε

X

F ##

AGoo

1

X F //

1 ##

A

GA

ε

X

η

yield the identity natural transformations F→ F and G→ G; that is, if and only if the triangles

F

η F &&

1 // F G

G η &&

1 // G

F G FF ε

88

G F Gε G

88

commute.

Proof. By Corollary 5.3.21, the triangles

F ⟨A⟩

ηA &&

[[η F] [F ε]]⟨A⟩ // F ⟨A⟩ ⟨X⟩G

Xε &&

⟨X⟩[[G η] [ε G]] // ⟨X⟩G

⟨X⟩GXε

88

F ⟨A⟩ηA

88

commute. Since the general Yoneda functor is fully faithful, [[η F] [F ε]] ⟨A⟩ (resp.⟨X⟩ [[G η] [ε G]]) is the identity i so is [η F] [F ε] (resp. [G η] [ε G]). Hence theidentities [η F] [F ε] = 1F and [G η] [ε G] = 1G hold i ηA and Xε are the inverseof each other.

Theorem 8.2.9. Given an adjunction (η, ε) ∶ G F ∶XA, the following conditions are equivalent:

(1) the functor G ∶A→X is fully faithful;(2) the counit ε ∶ G F→ 1A is a natural isomorphism.

the following conditions are equivalent:(1) the functor F ∶X→A is fully faithful;(2) the unit η ∶ 1X → F G is a natural isomorphism.

Proof. Since G and ε form a counit of the representable module F ⟨A⟩ (see Proposition 8.2.5),by Theorem6.3.13, G is fully faithful i for each a ∈ ∥A∥ the F ⟨A⟩-arrow εa ∶ a ⋅G a is directuniversal. ε is a natural isomorphism i each component εa ∶ a ⋅ G ⋅ F → a is an isomorphism,and by Corollary 6.2.13, this is the case i the F ⟨A⟩-arrow εa ∶ a ⋅ G a is direct universal.Hence the two conditions are equivalent.

Theorem 8.2.10. Suppose that a module M ∶XA has a counit and a unit as depicted in

XM

//ρ AGoo

XM //

F//λ A

253

8. Adjunctions

. Then there is an adjunction Υ ∶ G F ∶ X A with the module isomorphism Υ ∶ X ⟨G⟩ →F ⟨A⟩ ∶XA dened by the composition

X ⟨G⟩ Xρ //M ⟨λA⟩−1

// F ⟨A⟩

, and the counit ε ∶ G F → 1A and the unit η ∶ 1X → F G of the adjunction are given by thecompositions

AG

1

##

X1

F

##X

F

M //

ρ

A

1

X

1

M //

λ

A

G

A⟨A⟩

//

⟨λA⟩−1

A X⟨X⟩

//

⟨Xρ⟩−1

X

; that is, the component of ε at a ∈ ∥A∥ is given by the adjunct of ρa along λ(a ⋅G) and thecomponent of η at x ∈ ∥X∥ is given by the adjunct of λx along ρ(F ⋅ x) as indicated in

a ⋅Gλ(a ⋅ G) //

ρa((

F ⋅G ⋅ aεa

xηx

λx

((a x ⋅ F ⋅G ρ(F ⋅ x)

// F ⋅ x

.

Proof. The rst assertion is obvious since Xρ and λA are isomorphisms by the denition ofunits. The counit and unit are given by

ε =XΥ η = Υ−1A=X⟨⟨Xρ⟩ ⟨λA⟩−1⟩ = ⟨⟨λA⟩ ⟨Xρ⟩−1⟩A= [X⟨Xρ⟩] ⟨λA⟩−1 = [⟨λA⟩A] ⟨Xρ⟩−1 ∗1

= ρ ⟨λA⟩−1 = λ ⟨Xρ⟩−1

(∗1 by Corollary 5.3.19).

8.3. Transformations of adjoints

Denition 8.3.1. Given a pair of categories X and A, the module

⟨XUA⟩ ∶ [A,X] [X,A]∼


[A,X] XA // [X ∶A] ⟨X∶A⟩ // [X ∶A] [X,A]∼XAoo

, where XA is the right general Yoneda functor for the functor category [A,X] and XAis the left general Yoneda functor for the functor category [X,A].

the module⟨AQX⟩ ∶ [X,A]∼ [A,X]

254

8. Adjunctions


[X,A]∼ XA // [X ∶A] ⟨X∶A⟩ // [X ∶A] [A,X]XAoo

, where XA is the left general Yoneda functor for the functor category [X,A] and XAis the right general Yoneda functor for the functor category [A,X].

Remark 8.3.2.(1) The module ⟨XUA⟩ (resp. ⟨AQX⟩) is a special instance of a module dened in Deni-

tion 8.1.5 whereM and N are given by the hom of X and the hom of A; that is,

⟨XUA⟩ = ⟨⟨X⟩U⟨A⟩⟩ ⟨AQX⟩ = ⟨⟨A⟩Q⟨X⟩⟩

. For any categories X and A,⟨AQX⟩ ≅ ⟨A∼ UX∼⟩

.(2) For each pair of functors

XF

//AGoo

the set (G) ⟨XUA⟩ (F) consists of all module morphisms ⟨X⟩G → F ⟨A⟩ ∶ X A; thatis,

(G) ⟨XUA⟩ (F) = (⟨X⟩G) ⟨X ∶A⟩ (F ⟨A⟩).

the set (F) ⟨AQX⟩ (G) consists of all module morphisms F ⟨A⟩ → ⟨X⟩G ∶ X A; thatis,

(F) ⟨AQX⟩ (G) = (F ⟨A⟩) ⟨X ∶A⟩ (⟨X⟩G).

(3) The bijections in Theorem5.5.1 are now written as

(G F) ⟨A,A⟩ (1A) ≅ (G) ⟨XUA⟩ (F) (1X) ⟨X,X⟩ (G F) ≅ (G) ⟨AQX⟩ (F)

.

Proposition 8.3.3. Let X and A be categories. The right form Υ ∶ G F ∶XA of an adjunction is a two-way universal ⟨XUA⟩-arrow. The left form Υ ∶ F G ∶AX of an adjunction is a two-way universal ⟨AQX⟩-arrow.

Proof. Since the Yoneda functors are fully faithful and Υ is an isomorphism in [X ∶A], theassertion follows from Theorem6.2.16.

Denition 8.3.4.

Given two adjunctions Υ ∶ G F ∶ X A and Υ′ ∶ G′ F′ ∶ X A, a pair of naturaltransformations τ ∶ G→ G′ ∶A→X and σ ∶ F′ → F ∶X→A are called conjugate if the square

G

τ

⟨X⟩GΥ //

⟨X⟩τ

F ⟨A⟩σ⟨A⟩

F

G′ ⟨X⟩G′Υ′

// F′ ⟨A⟩ F′

σ

OO

255

8. Adjunctions

commutes; that is, if the square

GΥ //

τ

F

G′Υ′

// F′σ

OO

is a two-way conjugation along the module ⟨XUA⟩ in the sense of Denition 6.2.24. Given two adjunctions Υ ∶ F G ∶ A X and Υ′ ∶ F′ G′ ∶ A X, a pair of natural

transformations τ ∶ G→ G′ ∶A→X and σ ∶ F′ → F ∶X→A are called conjugate if the square

F F ⟨A⟩ Υ //

σ⟨A⟩

⟨X⟩G

⟨X⟩τ

G

τ

F′

σ

OO

F′ ⟨A⟩Υ′

// ⟨X⟩G′ G′

commutes; that is, if the square

FΥ // G

τ

F′Υ′

//

σ

OO

G′

is a two-way conjugation along the module ⟨AQX⟩ in the sense of Denition 6.2.24.

Proposition 8.3.5. Consider adjunctions and natural transformations as in Denition 8.3.4and let ε and ε′ (resp. η and η′) be the counits (resp. units) of Υ and Υ′.(1)

τ and σ are conjugate if and only if the square

G F′

G σ

τ F′ // G′ F′

ε′

G F ε// 1A

commutes. τ and σ are conjugate if and only if the square

1X

η′

η // G F

τ F

G′ F′G′ σ

// G′ F

commutes.(2)

Given τ , its conjugate σ is obtained by the composition

F′

η F′

σ // F

F G F′F τ F′

// F G′ F′F ε′OO

; that is, by the pasting composition

X1

yy

F

%%

η

X

F′ %%

AG′

ooGooτ

1yyAε′

256

8. Adjunctions

. Given σ, its conjugate τ is obtained by the composition

G

G η′

τ // G′

G F′ G′G σ G′

// G F G′ε G′OO

; that is, by the pasting composition

AG

yy

1

%%ε

XF′

//F //σ

1 %%

A

G′yyXη′

.

Proof.(1) Since the bijection (G F) ⟨A,A⟩ (1A) ≅ (G) ⟨XUA⟩ (F) (see Remark 8.3.2(3)) is natural in

G and F, the square

G

τ

Υ // F

G′Υ′

// F′σ

OO


G F′

G σ

τ F′ // G′ F′

ε′

G F ε// 1A

commutes.(2) The commutativity of the square above, i.e.

τ F′ ε′ = G σ ε

, says that the two pasting compositions

X

F′ %%

AGoo

G′oo τ

1yy

X

F′ %%

F

%%σ

AGoo

1yyAε′

A

ε

yield the same natural transformation G F′ → 1A. Hence the two pasting compositions

X1

yy

F

%%

η

X1

yy

F

%%η

X

F′ %%

AGoo

G′oo τ

1yy

X

F′ %%

F

%%σ

AGoo

1yyAε′

A

ε

yield the same natural transformation, which is σ by Theorem8.2.8.

257

8. Adjunctions

Denition 8.3.6. Given a pair of bifunctors F ∶ X × E∼ → A and G ∶ A × E → X, an E-parameterized adjunction Υ ∶ G F ∶XA is dened by a cylinder

EG⊣

yyF⊣

%%[A,X]

⟨XUA⟩//

Υ

[X,A]∼

such that each component forms an adjunction Υe ∶ G⊣ (e) F⊣ (e) ∶XA.

Remark 8.3.7.(1) If Υ ∶ G F ∶ X A is an E-parameterized adjunction, then, by Proposition 8.3.3, the

cylinder Υ ∶ G⊣ F⊣ ∶ E ⟨XUA⟩ is two-way pointwise universal.(2) Υ forms an E-parameterized adjunction if and only if the square

G⊣ (e)G⊣(h)

⟨X⟩ [G⊣ (e)]⟨X⟩[G⊣(h)]

Υe // [F⊣ (e)] ⟨A⟩[F⊣(h)]⟨A⟩

F⊣ (e)

G⊣ (e′) ⟨X⟩ [G⊣ (e′)]Υe′

// [F⊣ (e′)] ⟨A⟩ F⊣ (e′)F⊣(h)OO

commutes (i.e. G⊣ (h) and F⊣ (h) are conjugate) for every E-arrow h ∶ e→ e′.

Theorem 8.3.8.

Given a bifunctor F ∶ X ×E∼ → A, suppose that there is a family of adjunctions Υe ∶ Ge F⊣ (e) ∶XA, one for each object e ∈ ∥E∥. Then there is a unique bifunctor G ∶A×E→Xwith G⊣ (e) = Ge such that Υ ∶= (Υe)e∈∥E∥ forms a cylinder G⊣ F⊣ ∶ E ⟨XUA⟩, and Υ isan E-parameterized adjunction.

Given a bifunctor G ∶A×E→X, suppose that there is a family of adjunctions Υe ∶ G⊣ (e)Fe ∶ X A, one for each object e ∈ ∥E∥. Then there is a unique bifunctor F ∶ X ×E∼ → Awith F⊣ (e) = Fe such that Υ ∶= (Υe)e∈∥E∥ forms a cylinder G⊣ F⊣ ∶ E ⟨XUA⟩, and Υ isan E-parameterized adjunction.

Proof. Since an adjunctionXA is a two-way universal ⟨XUA⟩-arrow (see Proposition 8.3.3),this is an instance of Theorem6.4.10 whereM is given by the module ⟨XUA⟩.

8.4. Adjoints of a cell

Denition 8.4.1. Given a cell

XM //

P

A

Q

Y N//

Φ

B

a right adjoint (R,Υ) of the collage functor ⌊Φ⌋ (see Denition 3.1.9) along the left hom (seeDenition 5.2.1) ofM and the left hom of N as depicted in

⌊M⌋⌊Φ⌋

⟨MA⟩ //A

⌊N ⌋⟨N B⟩

//

Υ

B

R

OO

is called a right adjoint of Φ.

258

8. Adjunctions

a left adjoint (R,Υ) of the collage functor ⌊Φ⌋ (see Denition 3.1.9) along the right hom (seeDenition 5.2.1) ofM and the right hom of N as depicted in

X⟨XM⟩ // ⌊M⌋

⌊Φ⌋

Y

R

OO

⟨YN ⟩// ⌊N ⌋

Υ

is called a left adjoint of Φ.

Remark 8.4.2.(1)

If (R,Υ) is a right adjoint of Φ, then the restrictions of Υ toX andA yield right adjoints

X

P

M //A A

Q

⟨A⟩ //A

Y N//

X⟨Υ⟩

B

R

OO

B⟨B⟩

//

A⟨Υ⟩

B

R

OO

of P and Q. Conversely, a pair of right adjoints

X

P

M //A A

Q

⟨A⟩ //A

Y N//

Υ0

B

R

OO

B⟨B⟩

//

Υ1

B

R

OO

form a right adjoint of Φ if they satisfy the following naturality condition for each objectb ∈ ∥B∥ and eachM-arrow m ∶ x a: for any A-arrow f ∶ a→ R ⋅ b as in

xm //

m f &&

a

fR ⋅ b

, the triangle

x ⋅ P m ⋅Φ //

(m f) ⋅Υ0 ''

Q ⋅ af ⋅Υ1b

commutes. If (R,Υ) is a left adjoint of Φ, then the restrictions of Υ to X and A yield left adjoints

X⟨X⟩ //X

P

XM //A

Q

Y

R

OO

⟨Y⟩//Y

⟨Υ⟩X

Y

R

OO

N// B

⟨Υ⟩A

of P and Q. Conversely, a pair of left adjoints

X⟨X⟩ //X

P

XM //A

Q

Y

R

OO

⟨Y⟩//Y

Υ0

Y

R

OO

N// B

Υ1

259

8. Adjunctions

form a left adjoint of Φ if they satisfy the following naturality condition for each objecty ∈ ∥Y∥ and eachM-arrow m ∶ x a: for any X-arrow f ∶ y ⋅ R→ x as in

y ⋅ Rf

f m

&&x m

// a

, the triangley

f ⋅Υ0

(f m) ⋅Υ1

''x ⋅ P

m ⋅Φ// Q ⋅ a

commutes.(2) A right (resp. left) adjoint of a functor H is the same thing as a right (resp. left) adjoint

of the hom cell ⟨H⟩.

Theorem 8.4.3.

If a cell has a left adjoint, then it preserves inverse universals. If a cell has a right adjoint, then it preserves direct universals.

Proof. A left adjoint (R,Υ) of Φ in Denition 8.4.1 is depicted more elaborately as

X⟨XM⟩

((Y

⟨YN ⟩ &&

R

77

Υ

R⟨XM⟩,,

⟨YN ⟩⌊Φ⌋22 ⌊M⌋

⌊Φ⌋ww⌊N ⌋

, and the right exponential transposition yields a natural isomorphism Υ as in

[X ∶][R∶]

ww[Y ∶] ⌊M⌋

⌊Φ⌋ww

⟨XM⟩gg

[R∶] [⟨XM⟩]Υqq

[⟨YN ⟩] ⌊Φ⌋mm

⌊N ⌋⟨YN ⟩

gg

by Proposition 2.1.6 and Proposition 2.1.7. Let u ∶ r a be an inverse universal M-arrow.Then [⟨XM⟩] ⋅ u is an isomorphism by the very denition of inverse universality, and sois [R ∶] ⋅ [⟨XM⟩] ⋅ u because any functor preserves isomorphisms. Since Υ is a naturalisomorphism and [R ∶] ⋅ [⟨XM⟩] ⋅u is an isomorphism as we have just seen, [⟨YN ⟩] ⋅ ⌊Φ⌋ ⋅uis an isomorphism as well; that is, ⌊Φ⌋ ⋅ u = Φ ⋅ u is inverse universal.

Theorem 8.4.4. If a cell Φ has a right (resp. left) adjoint, so does the postcomposition cell⟨J ,Φ⟩ (see Denition 1.2.23) for any module J .

Proof. Consider a cell Φ and a postcomposition cell ⟨J ,Φ⟩ as in Denition 1.2.23, and supposethat a cell Φ has a right adjoint (R,Υ). The postcompositions with the adjunction X ⟨Υ⟩ and

260

8. Adjunctions

A ⟨Υ⟩ in Remark 8.4.2(1) yield adjunctions

[E,X][E,P]

⟨J ,M⟩ // [D,A] [D,A] ⟨D,A⟩ //

[D,Q]

[D,A]

[E,Y]⟨J ,N ⟩

//

⟨J ,X⟨Υ⟩⟩

[D,B][D,R]OO

[D,B]⟨D,B⟩

//

⟨D,A⟨Υ⟩⟩

[D,B][D,R]OO

by Proposition 8.1.13 and Proposition 8.1.16. The proof will be complete if we show thatthis pair or right adjoints satisfy the naturality condition in Remark 8.4.2(1) for each functorK ∶D→ B and each cell Θ ∶ S T ∶ J M, i.e. if we show that for any natural transformationτ ∶ T→ R K ∶D→A as in

SΘ //

Θ τ &&

T

τ

R K

, the triangle

S PΘ Φ //

(Θ τ) Υ ''

Q T

τ Υ

K

commutes. But this is reduced to the commutativity of the triangle

e ⋅ [S P] j ⋅ [Θ Φ] //

j ⋅ [(Θ τ) Υ] ))

[Q T] ⋅ d[τ Υ]d

K ⋅ d

, i.e.

e ⋅ S ⋅ P j ⋅Θ ⋅Φ //

((j ⋅Θ) τd) ⋅Υ ((

Q ⋅T ⋅ dτd ⋅Υ

K ⋅ d(see Remark 1.2.9(2) and Remark 4.3.15(1)) for each J -arrow j ∶ e d, i.e. to the naturalityof Υ.

Corollary 8.4.5. If a cell Φ has a right (resp. left) adjoint, each of the following postcom-position cells (see Denition 4.5.7, Denition 4.3.16, and Denition 4.6.15) has a right (resp.left) adjoint as well. ⟨F Φ⟩ and ⟨F Φ⟩, ⟨E,Φ⟩, ⟨∗E,Φ⟩ and ⟨E∗,Φ⟩.

Proof. By the isomorphism in Theorem5.5.2, the cell ⟨F Φ⟩ is identied with the cell ⟨⟨E⟩F,Φ⟩.Hence ⟨F Φ⟩ has an adjoint by Theorem8.4.4. Since ⟨E,Φ⟩ = ⟨1E Φ⟩ (see Remark 4.5.8(2))and ⟨∗E,Φ⟩ = ⟨∆E Φ⟩ (see Corollary 4.6.20), ⟨E,Φ⟩ and ⟨∗E,Φ⟩ have an adjoint as well. Thesame holds for ⟨F Φ⟩ and ⟨E∗,Φ⟩ by duality.

Corollary 8.4.6. Let Φ be a cell from a module M to a module N . Suppose that Φ ∶M N has a left adjoint. Then it preserves inverse lifts; that is, each

inverse universal cylinder µ ∶ E M along M yields by composition with Φ an inverseuniversal cylinder µ Φ ∶ E N along N .

261

8. Adjunctions

Suppose that Φ ∶M N has a right adjoint. Then it preserves direct lifts; that is, eachdirect universal cylinder µ ∶ EM alongM yields by composition with Φ a direct universalcylinder µ Φ ∶ E N along N .

Proof. Since an inverse universal cylinder µ ∶ EM is the same thing as an inverse universal⟨E,M⟩-arrow, the assertion is equivalent to saying that if a cell Φ has a left adjoint, then thepostcomposition cell ⟨E,Φ⟩ preserves inverse universals. But since ⟨E,Φ⟩ has a left adjoint aswell by Corollary 8.4.5, this follows from Theorem8.4.3.

Corollary 8.4.7.

If a cell has a left adjoint, then it preserves limits. If a cell has a right adjoint, then it preserves colimits.

Proof. By denition (see Denition 7.1.8), the assertion is equivalent to saying that if a cell Φhas a left adjoint, then the postcomposition cell ⟨∗E,Φ⟩ preserves inverse universals. But since⟨∗E,Φ⟩ has a left adjoint as well by Corollary 8.4.5, this follows from Theorem8.4.3.

Note. The following is a special case of Corollary 8.4.7 where a cell is given by the hom of afunctor.

Corollary 8.4.8.

If a functor has a left adjoint, then it preserves limits. If a functor has a right adjoint, then it preserves colimits.

Proof. By denition (see Denition 7.3.9), this is equivalent to saying that if a functor H hasa left adjoint, then the postcomposition cell ⟨∗E,H⟩ preserves inverse universals. Since anadjoint of H is the same thing as an adjoint of its hom ⟨H⟩, ⟨∗E,H⟩ = ⟨∗E, ⟨H⟩⟩ has a leftadjoint as well by Corollary 8.4.5. Hence the assertion follows from Theorem8.4.3.

8.5. Equivalence of categories

Denition 8.5.1. A functor H ∶C→ B is called an equivalence provided that it is fully faithfuland essentially surjective.

Proposition 8.5.2. If a functor H ∶ C → B is an equivalence, so is any functor C → Bisomorphic to H.

Proof. Evident.

Denition 8.5.3. A moduleM ∶XA is called an equivalence provided that for each object a ∈ ∥A∥ there exist an object x ∈ ∥X∥ and a two-way universal M-arrow

u ∶ x a, and for each object x ∈ ∥X∥ there exist an object a ∈ ∥A∥ and a two-way universal M-arrow

u ∶ x a.

Proposition 8.5.4. If a module M ∶ X A is an equivalence, so is any module X Aisomorphic to M.

Proof. Evident since any module isomorphism preserves universals.

Proposition 8.5.5.

If a functor G ∶ A → X is an equivalence, then its corepresentable module ⟨X⟩G ∶ X A isan equivalence.

262

8. Adjunctions

If a functor F ∶X→A is an equivalence, then its representable module F ⟨A⟩ ∶XA is anequivalence.

Proof. By Theorem6.2.16, for each a ∈ ∥A∥, the identity X-arrow a ⋅G→ G ⋅ a gives a two-wayuniversal ⟨X⟩G-arrow a ⋅ G a, and for each x ∈ ∥X∥, an iso X-arrow u ∶ x → G ⋅ a gives atwo-way universal ⟨X⟩G-arrow u ∶ x a.

Remark 8.5.6. We will see that the converse also holds in Corollary 8.5.9.


Theorem 8.5.7. Any equivalence module M ∶XA has a counit ρ ∶ GM with G ∶A→X an equivalence functor. a unit λ ∶M F with F ∶X→A an equivalence functor.

Proof. Choose a two-way universalM-arrow ρa ∶ ra a for each a ∈ ∥A∥. By Theorem6.3.10,there is a functor G ∶A→X such that ρ ∶= (ρa)a∈∥A∥ forms a counit ρ ∶ GM. We claim thatG is an equivalence functor. By Theorem6.3.13, G is fully faithful. It remains to show thatG is essentially surjective. Let x ∈ ∥X∥. SinceM is an equivalence, there exist a ∈ ∥A∥ and atwo-way universal M-arrow u ∶ x a. Since x and G ⋅ a are universals of a inverse along M,x ≅ G ⋅ a by Corollary 6.2.8.

Corollary 8.5.8.

Suppose that a module M ∶XA has a counit

XM

//ρ AGoo

. ThenM is an equivalence if and only if G is an equivalence. Moreover, if these equivalentconditions hold, then each component of ρ is two-way universal.

Suppose that a module M ∶XA has a unit

XM //

F//λ A

. ThenM is an equivalence if and only if F is an equivalence. Moreover, if these equivalentconditions hold, then each component of λ is two-way universal.

Proof. Suppose thatM is an equivalence. Then, by Theorem8.5.7, there is a counit ρ′ ∶ G′ Mwith G′ an equivalence. Since G ≅ G′ by Theorem6.3.8, G is an equivalence by Proposition 8.5.2.Conversely, suppose that G is an equivalence. Then the corepresentable module ⟨X⟩G isan equivalence by Proposition 8.5.5. Since ρ yields M ≅ ⟨X⟩G (see Remark 6.3.2), M is anequivalence by Proposition 8.5.4. The second assertion follows from Theorem6.3.13.

Corollary 8.5.9.

Suppose that a moduleM ∶XA is corepresented by a functor G ∶A→X. ThenM is anequivalence if and only if G is an equivalence.

Suppose that a module M ∶ X A is represented by a functor F ∶ X → A. Then M is anequivalence if and only if F is an equivalence.

Proof. Since a corepresenting functor has a counit associated with it (see Remark 6.3.2), thisis immediate from Corollary 8.5.8.

263

8. Adjunctions

Denition 8.5.10. An equivalence of categories X and A, written (F,G, η, ε) ∶X ≃A, consistsof a pair of functors

XF

//AGoo

and a pair of natural isomorphisms

η ∶ 1X → G F ∶X→X ε ∶ G F→ 1A ∶A→A

. In this situation, the functors F and G are said to be quasi-inverse to each other. Twocategories X and A are called equivalent, written X ≃ A, when there is an equivalence ofcategories between them.

Proposition 8.5.11. If (F,G, η, ε) ∶ X ≃ A is an equivalence of categories, then F and G areequivalences.

Proof. Since εa ∶ a ⋅G ⋅ F→ a (resp. ηx ∶ x→ G ⋅ F ⋅ x) is an isomorphism for each a ∈ ∥A∥ (resp.x ∈ ∥X∥), F (resp. G) is essentially surjective. Hence the proof is complete if we show that Fand G are fully faithful. For any arbitrary x,y ∈ ∥X∥ and a,b ∈ ∥A∥, consider the commutativediagrams

x ⟨X⟩y g↦F ⋅ g //

g↦G ⋅ F ⋅ g

(x ⋅ F) ⟨A⟩ (F ⋅ y)

q↦G ⋅ qtt

(a ⋅G) ⟨X⟩ (G ⋅ b)

p↦p ⋅ F **

a ⟨A⟩bf↦f ⋅Goo

f↦f ⋅G ⋅ F

(x ⋅ F ⋅G) ⟨X⟩ (G ⋅ F ⋅ y) (a ⋅G ⋅ F) ⟨A⟩ (F ⋅G ⋅ b)

. For any X-arrow g ∶ x→ y and any A-arrow f ∶ a→ b, the squares

xηx //

g

G ⋅ F ⋅ xG ⋅ F ⋅ g

a ⋅G ⋅ Ff ⋅G ⋅ F

εa // a

f

y ηy// G ⋅ F ⋅ y b ⋅G ⋅ F εb

// b

commutes by the naturality of η and ε. Since ηx and ηy are isomorphisms, g ↦ G ⋅ F ⋅ g isbijective; hence g ↦ F ⋅ g is injective and q↦ G ⋅ q is surjective. Symmetrically, since εa and εbare isomorphisms, f ↦ f ⋅G ⋅ F is bijective; hence f ↦ f ⋅G is injective and p↦ p ⋅ F is surjective.Since x, y, a, and b are arbitrary, it follows that q ↦ G ⋅ q and p ↦ p ⋅ F are both injectiveand surjective, i.e. bijective. The bijectivity of g ↦ F ⋅ g and f ↦ f ⋅ G now follows from thebijectivities of the other edges, proving the fully faithfulness of G and F.

Theorem 8.5.12. If (F,G, η, ε) ∶X ≃A is an equivalence of categories, then for any categoryE the pair of postcomposition functors

[E,X][E,F]

// [E,A][E,G]oo

form an equivalence of categories with the pair of postcomposition natural transformations[E, η] and [E, ε] (see Preliminary 9).

264

8. Adjunctions

the pair of precomposition functors

[A,E][F,E]

// [X,E][G,E]oo

form an equivalence of categories with the pair of precomposition natural transformations[η,E] and [ε,E] (see Preliminary 9).

Proof. First note that the postcomposition natural transformations

[E, η] ∶ [E,1X]→ [E,G F] [E, ε] ∶ [E,G F]→ [E,1A]

yield[E, η] ∶ 1[E,X] → [E,G] [E,F] [E, ε] ∶ [E,G] [E,F]→ 1[E,A]

by the functoriality of the operation [E,−]. Since (like any other functor) the functors [E,−] ∶[X,A]→ [[E,X] , [E,A]] and [E,−] ∶ [A,X]→ [[E,A] , [E,X]] preserve isomorphisms, [E, η]and [E, ε] are natural isomorphisms.

Remark 8.5.13. Theorem8.5.12 says that the functor [E,−] preserves equivalences of categories.In fact, this is immediate if we use the notion of 2-categories and dene [E,−] as a 2-functor.

Theorem 8.5.14. For any adjunction

XF

//(η,ε) AGoo

, the following conditions are equivalent:(1) η and ε are natural isomorphisms, i.e. (F,G, η, ε) forms an equivalence of categories X ≃A.(2) G and F are fully faithful;(3) G and F are equivalences;(4) G (resp. F) is an equivalence.

Proof.(1)⇔ (2) Immediate from Theorem8.2.9.(1)⇒ (3) This is Proposition 8.5.11.(3)⇒ (2) Immediate by denition.(3)⇒ (4) A well known tautology.(4)⇒ (3) By Corollary 8.5.9, this is equivalent to showing that if the corepresentable module

⟨X⟩G (resp. the representable module F ⟨A⟩) is an equivalence, then ⟨X⟩G and F ⟨A⟩ areequivalences. But since ⟨X⟩G ≅ F ⟨A⟩ by the denition of an adjunction, this follows fromProposition 8.5.4.

Denition 8.5.15. An adjoint equivalence of categories X and A is an adjunction (η, ε) ∶ GF ∶XA satisfying the equivalent conditions in Theorem8.5.14.

Theorem 8.5.16. Given a functor G ∶A→X (resp. F ∶X→A), the following conditions areequivalent:(1) G (resp. F) is an equivalence;(2) G (resp. F) is a part of an adjoint equivalence (η, ε) ∶ G F ∶XA;(3) G (resp. F) is a part of an equivalence of categories (F,G, η, ε) ∶X ≃A.

265

8. Adjunctions

Proof.(1)⇒ (2) Since G is an equivalence, its corepresentable module ⟨X⟩G is an equivalence by

Proposition 8.5.5, and thus has a unit η ∶ ⟨X⟩G F by Theorem8.5.7. Hence, by theequivalence of (1) and (3) in Proposition 8.2.5, G is a part of an adjunction (η, ε) ∶ G F ∶XA, in fact an adjoint equivalence because G is an equivalence.

(2)⇒ (3) Immediate by denition.(3)⇒ (1) This is Proposition 8.5.11.

8.6. Equivalence of modules


XM //

P

A

Q

Y N//

Φ

B

is called an equivalence if P and Q are equivalence functors and Φ is fully faithful.

Remark 8.6.2. A functor F ∶ X → A is an equivalence if and only if the hom cell ⟨F⟩ is anequivalence.

Proposition 8.6.3. A cell Φ ∶ M N is an equivalence if and only if the collage functor⌊Φ⌋ ∶ ⌊M⌋→ ⌊N ⌋ (see Denition 3.1.9) is an equivalence.

Proof. By the construction of ⌊Φ⌋ it is easily seen that:(1) ⌊Φ⌋ is fully faithful i so are P, Q, and Φ;(2) ⌊Φ⌋ is essentially surjective i so are P and Q.

Theorem 8.6.4. Any equivalence cell preserves and reects inverse (resp. direct) universals.

Proof. Suppose that a cell

XM //

P

A

Q

Y N//

Φ

B

is an equivalence and let u ∶ r a be anM-arrow. We need to show that u is inverse universali so is Φ ⋅ u; that is, Xu is iso i Y(Φ ⋅ u) is iso. By Example 5.2.9(1), the diagram

⟨X⟩ r X(Φ ⋅ u) //

⟨P⟩r

⟨P ⟨N ⟩Q⟩a = P ⟨N ⟩ (Q ⋅ a)

⟨P ⟨Y⟩P⟩ r = P ⟨Y⟩ (P ⋅ r)P⟨Y(Φ ⋅ u)⟩

33

commutes. Since P is fully faithful, ⟨P⟩ r is iso. Hence, by the commutativity of the diagramabove, X(Φ ⋅ u) is iso i P ⟨Y(Φ ⋅ u)⟩ is iso. Since Φ is fully faithful (i.e. Φ ∶M → P ⟨N ⟩Qis iso), Xu is iso i X (Φ ⋅ u) is iso. Since P is essentially surjective, by Theorem1.1.25,Y(Φ ⋅ u) is iso i P ⟨Y(Φ ⋅ u)⟩ is iso. Hence Xu is iso i Y(Φ ⋅ u) is iso as required.

266

8. Adjunctions

Denition 8.6.5. An equivalence of modulesM and N , written (Φ,Ψ, η, ε) ∶M ≃ N , consistsof a pair of cells

MΦ

// NΨoo

and a pair of cell isomorphisms

η ∶ 1M → Ψ Φ ∶MM ε ∶ Ψ Φ→ 1N ∶ N N

. In this situation, the cells Φ and Ψ are said to be quasi-inverse to each other. Two modulesM and N are called equivalent, writtenM ≃ N , if there is an equivalence of modules betweenthem.

Proposition 8.6.6. Consider cells and cell morphisms as in Denition 8.6.5. The followingconditions are equivalent:(1) (Φ,Ψ, η, ε) forms an equivalence of modules M ≃ N ;(2) (⌊Φ⌋ , ⌊Ψ⌋ , ⌊η⌋ , ⌊ε⌋) forms an equivalence of categories ⌊M⌋ ≃ ⌊N ⌋;(3) (Φ0,Ψ0, η0, ε0) forms an equivalence of categories M0 ≃ N0 and (Φ1,Ψ1, η1, ε1) forms an

equivalence of categories M1 ≃ N1.

Proof. The equivalence (1)⇔ (3) is evident by Proposition 1.3.4. The equivalence (1)⇔ (2)is also evident by the construction of collages in Denition 3.1.9.

Theorem 8.6.7. If (Φ,Ψ, η, ε) ∶M ≃ N is an equivalence of modules, then(1) for any module J , the pair of postcomposition cells

⟨J ,M⟩⟨J ,Φ⟩

// ⟨J ,N ⟩⟨J ,Ψ⟩oo

(see Denition 1.2.23) form an equivalence of modules with the pair of postcomposition cellmorphisms ⟨J , η⟩ and ⟨J , ε⟩ (see Denition 1.3.7).

(2) for any category E, the pair of postcomposition cells

⟨E,M⟩⟨E,Φ⟩

// ⟨E,N ⟩⟨E,Ψ⟩oo

(see Denition 4.3.16) form an equivalence of modules with the pair of postcomposition cellmorphisms ⟨E, η⟩ and ⟨E, ε⟩ (see Denition 4.3.22).

(3) for any category E, each pair of postcomposition cells

⟨∗E,M⟩⟨∗E,Φ⟩

// ⟨∗E,N ⟩⟨∗E,Ψ⟩oo ⟨E∗,M⟩

⟨E∗,Φ⟩// ⟨E∗,N ⟩

⟨E∗,Ψ⟩oo

(see Denition 4.6.15) form an equivalence of modules with the pair of postcomposition cellmorphisms ⟨∗E, η⟩ and ⟨∗E, ε⟩ (resp. ⟨E∗, η⟩ and ⟨E∗, ε⟩) (see Denition 4.6.22).

Proof. All are proved in the same way as Theorem8.5.12; we give a proof of the rst. Firstnote that the postcomposition cell morphisms

⟨J , η⟩ ∶ ⟨J ,1M⟩→ ⟨J ,Ψ Φ⟩ ⟨J , ε⟩ ∶ ⟨J ,Ψ Φ⟩→ ⟨J ,1N ⟩

267

8. Adjunctions

yield⟨J , η⟩ ∶ 1⟨J ,M⟩ → ⟨J ,Ψ⟩ ⟨J ,Φ⟩ ⟨J , ε⟩ ∶ ⟨J ,Ψ⟩ ⟨J ,Φ⟩→ 1⟨J ,N ⟩

by the functoriality of the operation ⟨J ,−⟩ (see Proposition 1.2.25). Since (like any other func-tor) the functors ⟨J ,−⟩ ∶ [M ∶ N ]→ [⟨J ,M⟩ ∶ ⟨J ,N ⟩] and ⟨J ,−⟩ ∶ [N ∶M]→ [⟨J ,N ⟩ ∶ ⟨J ,M⟩](see Remark 1.3.8(2)) preserve isomorphisms, ⟨J , η⟩ and ⟨J , η⟩ are cell isomorphisms.

Remark 8.6.8. Theorem8.6.7 says that the functor ⟨J ,−⟩ (resp. ⟨E,−⟩, ⟨∗E,−⟩, ⟨E∗,−⟩)preserves equivalences of modules (cf. Remark 8.5.13).


Theorem 8.6.9. Given a cell as in Denition 8.6.1, the following conditions are equivalent:(1) Φ is an equivalence;(2) Φ is a part of an equivalence of modules (Φ,Ψ, η, ε) ∶M ≃ N .

Proof.(1)⇒ (2) First note that if Φ is an equivalence, then P, Q, and ⌊Φ⌋ are equivalences by

denition and Proposition 8.6.3. Since P and Q are essentially surjective, we may choose,for each object y ∈ ∥Y∥, an object ry ∈ ∥X∥ and an iso Y-arrow εy ∶ ry ⋅P→ y, and for eachobject b ∈ ∥B∥, an object rb ∈ ∥A∥ and an iso B-arrow εb ∶ rb ⋅ Q → b. Since ⌊Φ⌋ is fullyfaithful, by Corollary 6.2.15, each εy is universal from ⌊Φ⌋ to y and each εb is universal from⌊Φ⌋ to b. Hence, by Theorem8.2.6, ⌊Φ⌋ has a right adjoint G ∶ ⌊N ⌋→ ⌊M⌋ with y ⋅G = ry andb ⋅G = rb; clearly, G denes a cell Ψ ∶ N M such that G = ⌊Ψ⌋. Let ⌊η⌋ ∶ 1⌊M⌋ → ⌊Ψ⌋ ⌊Φ⌋and ⌊ε⌋ ∶ ⌊Ψ⌋ ⌊Φ⌋→ 1⌊N ⌋ be the collage natural transformations forming the unit and counitof the adjunction. Since ⌊Φ⌋ is an equivalence, (⌊Φ⌋ , ⌊Ψ⌋ , ⌊η⌋ , ⌊ε⌋) forms an equivalence ofcategories ⌊M⌋ ≃ ⌊N ⌋ (see Theorem8.5.14). Hence, by the equivalence of (1) and (2) inProposition 8.6.6, (Φ,Ψ, η, ε) forms an equivalence of modulesM ≃ N .

(2)⇒ (1) By the equivalence of (1) and (2) in Proposition 8.6.6, (⌊Φ⌋ , ⌊Ψ⌋ , ⌊η⌋ , ⌊ε⌋) is anequivalence of categories ⌊M⌋ ≃ ⌊N ⌋. Hence, by Theorem8.5.16, ⌊Φ⌋ is an equivalence. Φis thus an equivalence by Proposition 8.6.3.

Theorem 8.6.10. If a cell Φ is an equivalence, so are the postcomposition cells ⟨J ,Φ⟩, ⟨E,Φ⟩,⟨∗E,Φ⟩, and ⟨E∗,Φ⟩.

Proof. By the equivalence of (1) and (2) in Theorem8.6.9, this follows from Theorem8.6.7.

Corollary 8.6.11. Any equivalence cell preserves and reects limits (resp. colimits).

Proof. By denition (see Denition 7.1.8), the assertion is equivalent to saying that if a cellΦ is an equivalence, then for any category E the postcomposition cell ⟨∗E,Φ⟩ preserves andreects inverse universals. But since ⟨∗E,Φ⟩ is an equivalence as well by Theorem8.6.10, thisfollows from Theorem8.6.4.

Note. The following is a special case of Corollary 8.6.11 where a cell is given by the hom of afunctor.

Corollary 8.6.12. Any equivalence functor preserves and reects limits (resp. colimits).

Proof. By denition (see Denition 7.3.9), this is equivalent to saying that if a functor H isan equivalence, then for any category E the postcomposition cell ⟨∗E,H⟩ preserves and re-ects inverse universals. Since H is an equivalence i its hom ⟨H⟩ is an equivalence, ⟨∗E,H⟩ =⟨∗E, ⟨H⟩⟩ is an equivalence as well by Theorem8.6.10. Hence the assertion follows from Theo-rem8.6.4.

268

8. Adjunctions

Theorem 8.6.13. In the situation of Theorem8.2.10, suppose that the following equivalentconditions (see Corollary 8.5.8) hold:(1) M is an equivalence;(2) G and F are equivalences.Then the module morphism Xλ ∶ ⟨X⟩ → ⟨M⟩F ∶ X X (see Example 5.3.5(1)) and the inverse

of the module morphism Xρ ∶ ⟨X⟩G→M ∶XA yield equivalence cells

X⟨X⟩ //

1

X

F

XM //

1

A

G

X M//

Xλ

A X⟨X⟩

//

⟨Xρ⟩−1

X

quasi-inverse to each other, with the cell isomorphisms

(1X, η) ∶ 1⟨X⟩ ⟨Xρ⟩−1 ⟨Xλ⟩ (1X, ε) ∶ ⟨Xρ⟩−1 ⟨Xλ⟩ 1M

given by the unit and counit of the adjunction in Theorem8.2.10. the module morphism ρA ∶ ⟨A⟩ → G ⟨M⟩ ∶ A A (see Example 5.3.5(1)) and the inverse

of the module morphism λA ∶ F ⟨A⟩→M ∶XA yield equivalence cells

A

G

⟨A⟩ //A

1

X

F

M //A

1

X M//

ρA

A A⟨A⟩

//

⟨λA⟩−1

A

quasi-inverse to each other, with the cell isomorphisms

(η,1A) ∶ 1M ⟨ρA⟩ ⟨λA⟩−1 (ε,1A) ∶ ⟨ρA⟩ ⟨λA⟩−1 1⟨A⟩

given by the unit and counit of the adjunction in Theorem8.2.10.

Proof. Once we verify the claim below, it remains to show that (1X, η) and (1X, ε) are cellisomorphisms. For this, we just need to show that η ∶ 1X → G F and ε ∶ G F→ 1A are naturalisomorphisms. But since G and F are equivalences, this follows from Theorem8.5.14.

Claim.(1) The pair (1X, η) forms a cell morphism 1⟨X⟩ ⟨Xρ⟩−1 ⟨Xλ⟩; that is, the diagram

y

1y

f // x

ηx

yf ⋅ ⟨Xλ⟩ ⋅ ⟨Xρ⟩−1

// G ⋅ F ⋅ x

commutes for every X-arrow f ∶ y → x.(2) The pair (1X, ε) forms a cell morphism ⟨Xρ⟩−1 ⟨Xλ⟩ 1M; that is, the diagram

x

1x

m ⋅ ⟨Xρ⟩−1 ⋅ ⟨Xλ⟩ // F ⋅G ⋅ aεa

x m// a

commutes for everyM-arrow m ∶ x a.

269

8. Adjunctions

Proof.(1) By Theorem8.2.10, the diagram

y

f

f λx((

xηx

λx // F ⋅ x

x ⋅ F ⋅Gρ(F ⋅ x)

66

commutes; that is,f λx = f ηx ρ(F ⋅ x)

. Butf λx = f ⋅ ⟨Xλ⟩ and f ηx ρ(F ⋅ x) = (f ηx) ⋅ ⟨Xρ⟩

. Hencef ⋅ ⟨Xλ⟩ = (f ηx) ⋅ ⟨Xρ⟩

, i.e.f ⋅ ⟨Xλ⟩ ⋅ ⟨Xρ⟩−1 = f ηx

, as required.(2) By Theorem8.2.10, the diagram

a ⋅Gλ(a ⋅ G) //

ρa((

F ⋅G ⋅ aεa

x

m/ρaOO

m// a

commutes, where m/ρa is the adjunct of m along ρa. Since

m/ρa λ(a ⋅G) = (m ⋅ ⟨Xρ⟩−1) λ(a ⋅G) = m ⋅ ⟨Xρ⟩−1 ⋅ ⟨Xλ⟩

, we havem = m/ρa λ(a ⋅G) εa = (m ⋅ ⟨Xρ⟩−1 ⋅ ⟨Xλ⟩) εa

as required.

270

9. Adjoint Functor Theorem

9.1. Conality

Denition 9.1.1. A right (resp. left) moduleM is said to be connected if the comma category⌈M↓⌉ is non-empty and connected.

Remark 9.1.2. A right moduleM ∶ E ∗ is connected if and only if there exists at least oneM-arrow and any twoM-arrows m ∶ e ∗ and m′ ∶ e′ ∗ are connected by a nite sequenceof commutative diagrams as in

e

m

f0 // e1

m1

e2

m2

f1oo enfn //

mn

e′

m′∗ ∗ ∗ ∗ ∗

(the direction of each E-arrow fi is arbitrary).

Proposition 9.1.3. A representable left (resp. right) module is connected.

Proof. If a left moduleM is representable, then the comma category ⌈M↓⌉ has an initial object(see Proposition 6.1.4). Clearly, a category with an initial object is connected.

Denition 9.1.4.

(1) A functor H ∶D→ E is called left (or downward) conal1 if for every object e ∈ ∥E∥ the right module H ⟨E⟩e ∶D ∗

is connected. right (or upward) conal if for every object e ∈ ∥E∥ the left module e ⟨E⟩H ∶ ∗ D is

connected.(2) A subcategory D of a category E is called

left conal in E if the inclusion D→ E is left conal. right conal in E if the inclusion D→ E is right conal.

(3) A set D of objects in a category E is called left conal in E if the full subcategory of E generated by D is left conal. right conal in E if the full subcategory of E generated by D is right conal.

Remark 9.1.5. A functor H ∶D→ E is conal if and only if its image is conal in E.

Proposition 9.1.6.

If d is an initial object of a category E, then the set d is left conal in E. If d is a terminal object of a category E, then the set d is right conal in E.

Proof. Evident.

Proposition 9.1.7. If a functor H ∶D→ E has a right (resp. left) adjoint, then H is left (resp.right) conal.

1Some authors use the term nal instead of conal.

271


Proof. Since H has a right adjoint i for every object e ∈ ∥E∥ the right module H ⟨E⟩e ∶D ∗is representable (see Corollary 8.2.7), the assertion follows from Proposition 9.1.3.

Remark 9.1.8. As an immediate consequence, ifD is a coreective (resp. reective) subcategoryof a category E, then D is left (resp. right) conal in E.

Proposition 9.1.9. An equivalence functor is both left and right conal.

Proof. Since an equivalence functor has both left and right adjoints (see Theorem8.5.16), theassertion follows from Proposition 9.1.7.

Theorem 9.1.10.

LetM ∶ ∗ E be a left module. If H ∶D→ E is a left conal functor, then for any frame αof the composite left module ⟨M⟩H ∶ ∗ D there exists a unique frame α′ of M such thatα = α′ H.

LetM ∶ E ∗ be a right module. If H ∶D→ E is a right conal functor, then for any frameα of the composite right module H ⟨M⟩ ∶ D ∗ there exists a unique frame α′ of M suchthat α = H α′.

Proof. Let α be a frame of ⟨M⟩H. Given e ∈ ∥E∥, choose d ∈ ∥D∥ and an E-arrow f ∶ H ⋅d→ e(the left conality of H allows such choices), and dene anM-arrow α′e ∶ ∗ e by α′e = αd f.We claim that(1) the denition of α′e is independent of the choices of d and f;(2) α′ ∶= (α′e)e∈∥E∥ forms a frame ofM;(3) α′ is the only frame ofM such that α = α′ H.Let d′ and f ′ be alternative choices for dening α′e. Then the diagram

∗αd

∗αd′

H ⋅ df

H ⋅ d′f′

e e

commutes because, by the left conality of H, the interior of the diagram is divided into a nitenumber of commutative diagrams as in

∗αd

∗αd1

∗αd2

∗αdn

∗αd′

H ⋅ df

H ⋅ g0 // H ⋅ d1

f1

H ⋅ d2H ⋅ g1oo

f2

H ⋅ dnfn

H ⋅ gn // H ⋅ d′f′

e e e e e′

dg0 // d1 d2

g1oo dngn // d′

. Hence the denition of α′e is independent of the choices of d and f. Let α′e = αd f andα′e′ = αd′ f ′ be two M-arrows dened as above and let h ∶ e → e′ be an E-arrow. Then thediagram

∗α′e

α′e′

##e

h// e′

272


i.e.∗

αd

∗αd′

H ⋅ df

H ⋅ d′f′

eh

// e′

commutes by the rst claim. α′ ∶= (α′e)e∈∥E∥ thus forms a frame ofM. By setting e = H ⋅d andf = 1H ⋅ d, we have αd = α′H ⋅ d for every d ∈ ∥D∥; hence α = α′ H. It is clear that this is the onlypossible construction of α′ with α = α′ H if α′ is to satisfy the naturality condition.

Corollary 9.1.11. Let H ∶D→ E be a functor and M ∶XA be a module. If H is left conal, then the precomposition cell

X⟨∗E,M⟩ //

1

[E,A][H,A]

X⟨∗D,M⟩

//

⟨∗H,M⟩

[D,A]

(see Denition 4.6.26) is fully faithful. If H is right conal, then the precomposition cell

[E,X] ⟨E∗,M⟩ //

[H,A]

A

1

[D,X]⟨D∗,M⟩

//

⟨H∗,M⟩

A

(see Denition 4.6.26) is fully faithful.

Proof. Let x be an object in X and K be a functor E → A. By Theorem9.1.10, for a coneα ∶ x K H ∶ ∗DM, i.e. a frame α of the left module x ⟨M⟩ [K H] = ⟨x ⟨M⟩K⟩H ∶ ∗D,there is a unique frame α′ of the left module x ⟨M⟩K ∶ ∗ E, i.e. a unique cone α′ ∶ x K ∶∗EM, such that α = α′ H.

Remark 9.1.12.(1) As a consequence, by Proposition 6.2.21,

the precomposition cell ⟨∗H,M⟩ preserves, reects, and creates inverse universals; thatis:a) a cone π ∶ r K ∶ ∗E M is universal if and only if the composite cone π H ∶ r

K H ∶ ∗DM is universal;b) given a functor K ∶ E → A, if the composite functor K H ∶ D → A has a limit

π ∶ r K H ∶ ∗DM, then the unique cone π′ ∶ r K ∶ ∗EM with π = π′ H isa limit of K.

the precomposition cell ⟨H∗,M⟩ preserves, reects, and creates direct universals; thatis:a) a cone π ∶ K r ∶ E∗ M is universal if and only if the composite cone H π ∶

H K r ∶D∗M is universal;b) given a functor K ∶ E → X, if the composite functor H K ∶ D → X has a colimit

π ∶ H K r ∶D∗M, then the unique cone π′ ∶ K r ∶ E∗M with π = H π′ isa colimit of K.

273


(2) By Proposition 9.1.9 and Theorem8.5.12, if H is an equivalence, so are the precompositioncells ⟨∗H,M⟩ and ⟨H∗,M⟩.

Theorem 9.1.13.

Let M ∶ ∗ E be a left module. If D is a left conal subcategory of E, then a frame of theleft module ⟨M⟩D ∶ ∗D (the restriction of M to D) uniquely extends to a frame of M.

Let M ∶ E ∗ be a right module. If D is a right conal subcategory of E, then a frame ofthe right module D ⟨M⟩ ∶D ∗ (the restriction ofM to D) uniquely extends to a frame ofM.

Proof. This is a special case of Theorem9.1.10 where H ∶D→ E is the inclusion.

Corollary 9.1.14. Let M ∶XA be a module and let D be a subcategory of a category E. If D is left conal in E, then the cell

X⟨∗E,M⟩ //

1

[E,A][D,A]

X⟨∗D,M⟩

//

⟨∗D,M⟩

[D,A]

, restriction to D (see Example 4.6.28(1)), is fully faithful. If D is right conal in E, then the cell

[E,X] ⟨E∗,M⟩ //

[D,X]

A

1

[D,X]⟨D∗,M⟩

//

⟨D∗,M⟩

A

, restriction to D (see Example 4.6.28(1)), is fully faithful.

Proof. Let x be an object in X and K be a functor E → A. By Theorem9.1.13, a coneα ∶ x K D ∶ ∗E M, i.e. a frame α of the left module x ⟨M⟩ [K D] = ⟨x ⟨M⟩K⟩D ∶∗ D, uniquely extends to a frame α′ of the left module x ⟨M⟩K ∶ ∗ E, i.e. a coneα′ ∶ x K ∶ ∗EM.


the cell ⟨∗D,M⟩ preserves, reects, and creates inverse universals; that is:a) a cone π ∶ r K ∶ ∗EM is universal if and only if its restriction π D ∶ r K D ∶

∗DM to D is universal;b) if the restriction of a functor K ∶ E → C to D has a limit π ∶ r K D ∶ ∗D M,

then its unique extension π′ ∶ r K ∶ ∗EM is a limit of K. the cell ⟨D∗,M⟩ preserves, reects, and creates direct universals; that is:

a) a cone π ∶ K r ∶ E∗ M is universal if and only if its restriction D π ∶ D K r ∶D∗M to D is universal;

b) if the restriction of a functor K ∶ E→C to D has a colimit π ∶D K r ∶D∗M,then its unique extension π′ ∶ K r ∶ E∗M is a colimit of K.

(2) As a special case whereM is given by the hom of a category of C,

274


if D is left conal in E, the cell

C⟨∗E,C⟩ //

1

[E,C][D,C]

C⟨∗D,C⟩

//

⟨∗D,C⟩

[D,C]

(see Example 4.9.9(1)) is fully faithful, and thus preserves, reects, and creates inverseuniversals; that is:a) a cone π ∶ r K ∶ ∗E→C is universal if and only if its restriction π D ∶ r K D ∶

∗D→C to D is universal;b) if the restriction of a functor K ∶ E → C to D has a limit π ∶ r K D ∶ ∗D → C,

then its unique extension π′ ∶ r K ∶ ∗E→C is a limit of K. if D is right conal in E, the cell

[E,C] ⟨E∗,C⟩ //

[D,C]

C

1

[D,C]⟨D∗,C⟩

//

⟨D∗,C⟩

C

(see Example 4.9.9(1)) is fully faithful, and thus preserves, reects, and creates directuniversals; that is:a) a cone π ∶ K r ∶ E∗→C is universal if and only if its restriction D π ∶D K r ∶

D∗→C to D is universal;b) if the restriction of a functor K ∶ E → C to D has a colimit π ∶D K r ∶D∗ → C,

then its unique extension π′ ∶ K r ∶ E∗→C is a colimit of K.

Theorem 9.1.16.

Let M ∶ ∗ E be a left module. If d is an initial object of E, then for any M-arrowm ∶ ∗ d there is a unique frame α ofM with αd = m. Specically, α is given by the familyof M-arrows αe = m υe, one for each e ∈ ∥E∥, with υe the unique E-arrow d→ e.

Let M ∶ E ∗ be a right module. If d is a terminal object of E, then for any M-arrowm ∶ d ∗ there is a unique frame α ofM with αd = m. Specically, α is given by the familyof M-arrows αe = υe m, one for each e ∈ ∥E∥, with υe the unique E-arrow e→ d.

Proof. Evident.

Remark 9.1.17. If d is an initial object of E, the functor d ∶ ∗ → E is left conal by Proposi-tion 9.1.6. Dually, if d is a terminal object of E, the functor d ∶ ∗ → E is right conal. HenceTheorem9.1.16 is seen a special case of Theorem9.1.10 where D is the terminal category.

Corollary 9.1.18. Let M ∶XA be a module and E be a category. If d is an initial object of E, then the cell

X⟨∗E,M⟩ //

1

[E,A][d,A]

X M//

⟨∗d,M⟩

A

, evaluation at d (see Example 4.6.28(2)), is fully faithful. Specically, for an M-arrowm ∶ x K ⋅ d with K a functor E → A, there is a unique cone α ∶ x K ∶ ∗E M withαd = m, given by the family of M-arrows αe = m (K ⋅ υe), one for each e ∈ ∥E∥, with υethe unique E-arrow d→ e.

275


If d is a terminal object of E, then the cell

[E,X] ⟨E∗,M⟩ //

[d,X]

A

1

X M//

⟨d∗,M⟩

A

, evaluation at d (see Example 4.6.28(2)), is fully faithful. Specically, for an M-arrowm ∶ d ⋅ K a with K a functor E → X, there is a unique cone α ∶ K a ∶ E∗ M withαd = m, given by the family of M-arrows αe = (υe ⋅K) m, one for each e ∈ ∥E∥, with υethe unique E-arrow e→ d.

Proof. Let x be an object in X and K be a functor E→A. By Theorem9.1.16, for anM-arrowm ∶ x K ⋅ d, i.e. an arrow m ∶ ∗ d in the left module x ⟨M⟩K, there is a unique frame αof x ⟨M⟩K, i.e. a unique cone α ∶ x K ∶ ∗E M, with αd = m, and the component of α ate ∈ ∥E∥ is given by the x ⟨M⟩K-arrow αe = m υe, i.e. theM-arrow αe = m (K ⋅ υe).


the cell ⟨∗d,M⟩ preserves, reects, and creates inverse universals; that is:a) a cone π ∶ r K ∶ ∗E M is universal if and only if the M-arrow πd ∶ r K ⋅ d is

inverse universal;b) given a functor K ∶ E → A, if its value at d has an inverse universal u ∶ r K ⋅ d

alongM, then the unique cone π ∶ r K ∶ ∗EM with πd = u is a limit of K. the cell ⟨d∗,M⟩ preserves, reects, and creates direct universals; that is:

a) a cone π ∶ K r ∶ E∗ M is universal if and only if the M-arrow πd ∶ d ⋅ K r isdirect universal;

b) given a functor K ∶ E→X, if its value at d has a direct universal u ∶ d ⋅K r alongM, then the unique cone π ∶ K r ∶ E∗M with πd = u is a colimit of K.

(2) As a special case whereM is given by the hom of a category of C, if d is an initial object of E, the cell

C⟨∗E,C⟩ //

1

[E,C][d,C]

C⟨C⟩

//

⟨∗d,C⟩

C

(see Example 4.9.9(2)) is fully faithful, and thus preserves, reects, and creates inverseuniversals; that is:a) a cone π ∶ r K ∶ ∗E→C is universal if and only if the C-arrow πd ∶ r→ K ⋅d is iso

(see Proposition 6.2.5);b) for a functor K ∶ E → C and an iso C-arrow u ∶ r → K ⋅ d, there is a unique cone

π ∶ r K ∶ ∗E → C with πd = u, given by the family of C-arrows πe = u (K ⋅ υe),one for each e ∈ ∥E∥, with υe the unique E-arrow d→ e; moreover, this unique coneis a limit of K.

if d is a terminal object of E, the cell

[E,C] ⟨E∗,C⟩ //

[d,C]

C

1

C⟨C⟩

//

⟨d∗,C⟩

C

276


(see Example 4.9.9(2)) is fully faithful, and thus preserves, reects, and creates directuniversals; that is:a) a cone π ∶ K r ∶ E∗→C is universal if and only if the C-arrow πd ∶ d ⋅K→ r is iso

(see Proposition 6.2.5);b) for a functor K ∶ E → C and an iso C-arrow u ∶ d ⋅ K → r, there is a unique cone

π ∶ K r ∶ E∗ → C with πd = u, given by the family of C-arrows πe = (υe ⋅K) u,one for each e ∈ ∥E∥, with υe the unique E-arrow e→ d; moreover, this unique coneis a colimit of K.

Corollary 9.1.20.

If a category E has an initial object d, then any functor K ∶ E → C has a limit π ∶ d ⋅ K K ∶ ∗E → C, given by the family of C-arrows πe = K ⋅ υe, one for each e ∈ ∥E∥, with υe theunique E-arrow d→ e.

If a category E has a terminal object d, then any functor K ∶ E → C has a colimit π ∶ K K ⋅ d ∶ E∗ → C, given by the family of C-arrows πe = υe ⋅ K, one for each e ∈ ∥E∥, with υethe unique E-arrow e→ d.

Proof. The cell ⟨∗d,C⟩ in Remark 9.1.19(2) creates a limit of K from the identity d ⋅ K →K ⋅ d.

Theorem 9.1.21.

If E has an initial object d, then the identity functor E→ E has a limit υ ∶ d E, given bythe unique E-arrow υe ∶ d → e for each e ∈ ∥E∥. Conversely, if the identity functor E → Ehas a limit υ ∶ d E, then d is an initial object.

If E has a terminal object d, then the identity functor E→ E has a colimit υ ∶ E d, givenby the unique E-arrow υe ∶ e→ d for each e ∈ ∥E∥. Conversely, if the identity functor E→ Ehas a colimit υ ∶ E d, then d is a terminal object.

Proof. The rst assertion follows from Corollary 9.1.20. Now suppose that the identity functorE → E has a limit υ ∶ d E. We just need to show that f = υe for any E-arrow f ∶ d → e. Bythe naturality υ, the triangle

d

f

d

υd66

υe (( e

commutes for every E-arrow f ∶ d→ e; in particular, the triangle

d

υe

d

υd66

υe (( e

commutes for every e ∈ ∥E∥. The commutativity of the second triangle shows that υd υ = υ,and we have υd = 1d by the uniqueness of factorization. The desired identity now follows fromthe commutativity of the rst triangle.

9.2. General adjoint functor theorem

Denition 9.2.1. Given a category E, a set D of the objects in E is called

277


[jointly] weakly initial in E if to every object e ∈ ∥E∥ there is an E-arrow d→ e with d ∈ D. [jointly] weakly terminal in E if to every object e ∈ ∥E∥ there is an E-arrow e → d with

d ∈ D.

Proposition 9.2.2. Let D be a set of objects in a category E. If D is left conal, then D is weakly initial. The converse holds if E is complete. If D is right conal, then D is weakly terminal. The converse holds if E is cocomplete.

Proof. The rst assertion is obvious. Now suppose that E is complete and D ⊆ ∥E∥ is weaklyinitial. Given e ∈ ∥E∥, we need to show that the right module D ⟨E⟩e ∶ D ∗ is connected,where D is the full subcategory of E generated by D. Let f0 ∶ d0 → e and f1 ∶ d1 → e be twoE-arrows with d0,d1 ∈ D. Since E is complete, there is a pullback

d0 f0

((ru0OO

u1 e

d1f1

66

. Then, since D is weakly initial, there is an arrow h ∶ d→ r with d ∈ D, yielding a commutativediagram

d0 f0

((d

h u066

h u1 ((

h // ru0OO

u1 e

d1f1

66

. Hence f0 and f1 are connected in D ⟨E⟩e.

Theorem 9.2.3.

A complete category E has an initial object if and only if it has a small weakly initial set. A cocomplete category E has a terminal object if and only if it has a small weakly terminal

set.

Proof. If E has an initial object d, then it has a small weakly initial set d. Conversely,suppose that E has a small weakly initial set D and letD be the full subcategory of E generatedby D. Then D is left conal in E by Proposition 9.2.2. Since E is complete and D is small,the inclusion D → E has a limit π ∶ r D, and this limit extends to a limit π ∶ r E of theidentity E→ E (see Remark 9.1.15(2)). Hence E has an initial object by Theorem9.1.21.

Denition 9.2.4.

Given a left moduleM ∶ ∗ A, a weakly initial set in the comma category ⌈M↓⌉ is calleda solution set ofM; that is, a solution set ofM is a set si ∶ ∗ ai ofM-arrows such thateveryM-arrow m ∶ ∗ a factors through some si along an A-arrow h ∶ ai → a as indicatedin

∗ si //

m%%

aiha

. Given a right module M, a weakly terminal set in the comma category ⌈M↓⌉ is called a

solution set ofM; that is, a solution set ofM is a set si ∶ xi ∗ ofM-arrows such that

278


everyM-arrow m ∶ x ∗ factors through some si along an X-arrow h ∶ x → xi as indicatedin

x

h

m

%%xi si

// ∗.

Theorem 9.2.5.

A left module M ∶ ∗ A over a complete category A is representable if and only if M iscontinuous and has a small solution set.

A right module M ∶ X ∗ over a cocomplete category X is representable if and only if Mis cocontinuous and has a small solution set.

Proof. Suppose that M is representable. Then M is continuous by Corollary 7.4.18, and aunit u ofM yields a small solution set u. Conversely, suppose thatM is continuous and hasa small solution set. Then the comma category ⌈M↓⌉ is complete by Corollary 7.4.23, and hasan initial object by Theorem9.2.3. HenceM is representable by Proposition 6.1.4.

Corollary 9.2.6.

A moduleM ∶XA with A complete is representable if and only if for every object x ∈ ∥X∥the left module x ⟨M⟩ ∶ ∗A is continuous and has a small solution set.

A module M ∶ X A with X cocomplete is corepresentable if and only if for every objecta ∈ ∥A∥ the right module ⟨M⟩a ∶X ∗ is cocontinuous and has a small solution set.

Proof. Since a module M ∶ X A is representable i the left module x ⟨M⟩ ∶ ∗ A isrepresentable for every x ∈ ∥X∥ (see Corollary 6.3.11), the assertion follows from Theorem9.2.5.

Theorem 9.2.7. (General Adjoint Functor Theorem). A functor G ∶A→X with A complete has a left adjoint if and only if the following conditions

hold:(1) G is continuous;(2) for each object x ∈ ∥X∥, the left module x ⟨X⟩G ∶ ∗ A has a small solution set, i.e.

a small set si ∶ x→ G ⋅ ai of X-arrows such that every X-arrow f ∶ x → G ⋅ a factorsthrough some si along an A-arrow h ∶ ai → a as indicated in

xsi //

f &&

G ⋅ aiG ⋅ h

ai

h

G ⋅ a a.

A functor F ∶ X → A with X cocomplete has a right adjoint if and only if the followingconditions hold:(1) F is cocontinuous;(2) for each object a ∈ ∥A∥, the right module F ⟨A⟩a ∶ X ∗ has a small solution set, i.e.

a small set si ∶ xi ⋅ F→ a of A-arrows such that every A-arrow f ∶ x ⋅ F → a factorsthrough some si along an X-arrow h ∶ x→ xi as indicated in

x

h

x ⋅ Fh ⋅ F

f

&&xi xi ⋅ F si

// a

.

279


Proof. By Remark 8.2.2(2), a functor G ∶A→X has a left adjoint i the module ⟨X⟩G ∶XAis representable. By Corollary 9.2.6, this is the case i for every x ∈ ∥X∥ the left modulex ⟨X⟩G ∶ ∗A is continuous and has a small solution set. By Corollary 7.4.17, G is continuousi x ⟨X⟩G is continuous for every x ∈ ∥X∥. The assertion now follows.

9.3. Epimorphisms and Monomorphisms

Denition 9.3.1.

(1) An arrow m ∶ ∗ a in a left moduleM ∶ ∗A is called epic (or an epimorphism) if for

any parallel pair of A-arrows h,h′ ∶ a→ b, m h = m h′ implies that h = h′. An arrow m ∶ x ∗ in a right moduleM ∶X ∗ is called monic (or a monomorphism)

if for any parallel pair of X-arrows h,h′ ∶ y → x, h m = h′ m implies that h = h′.(2) An arrow m ∶ x a in a moduleM ∶XA is called

epic (or an epimorphism) if for any parallel pair of A-arrows h,h′ ∶ a→ b, m h = m h′

implies that h = h′. monic (or a monomorphism) if for any parallel pair ofX-arrows h,h′ ∶ y → x, hm = h′ m

implies that h = h′.(3) An arrow m ∶ x→ a in a category C is called

epic (or an epimorphism) if for any parallel pair of C-arrows h,h′ ∶ a→ b, m h = m h′

implies that h = h′. monic (or a monomorphism) if for any parallel pair of C-arrows h,h′ ∶ y → x, hm = h′ m

implies that h = h′.

Remark 9.3.2.(1) An arrow m ∶ x→ a in a category C is epic (resp. monic) if and only if it is an epimorphism

(resp. monomorphism) in the hom endomodule ⟨C⟩ ∶CC.(2) An arrow m ∶ x a in a moduleM ∶XA is

epic if and only if it is an epimorphism in the left module x ⟨M⟩ ∶ ∗A, monic if and only if it is a monomorphism in the right module ⟨M⟩a ∶X ∗,and an arrow m ∶ x→ a in a category C is epic if and only if it is an epimorphism in the left module x ⟨C⟩ ∶ ∗C. monic if and only if it is a monomorphism in the right module ⟨C⟩a ∶C ∗.

Proposition 9.3.3.

Given a left module M ∶ ∗A,(1) anM-arrow m ∶ ∗ a is epic if and only if it is an epimorphism in the collage category

⌊M⌋.(2) an A-arrow f ∶ a → b is epic if and only if it is an epimorphism in the collage category

⌊M⌋. Given a right module M ∶X ∗,

(1) an M-arrow m ∶ x ∗ is monic if and only if it is a monomorphism in the collagecategory ⌊M⌋.

(2) an X-arrow f ∶ y → x is monic if and only if it is a monomorphism in the collagecategory ⌊M⌋.

Proof. Obvious by the construction of a collage.

Remark 9.3.4. By Proposition 9.3.3, the properties that hold for epimorphisms (resp. monomor-phisms) in a category are also enjoyed by epimorphisms (resp. monomorphisms) in a module.

280


Proposition 9.3.5.

Let M ∶ ∗A be a left module. Given an M-arrow m ∶ ∗ a and an A-arrow f ∶ a→ b,(1) if m and f are epic, so is m f;(2) if m f is epic, so is f.

Let M ∶X ∗ be a right module. Given an M-arrow m ∶ x ∗ and an X-arrow f ∶ y → x,(1) if m and f are monic, so is f m;(2) if f m is monic, so is f.

Proof. See [AHS09] 7.34 and 7.41 (see also Remark 9.3.4).

Proposition 9.3.6. Let M ∶XA be a module. If u ∶ x r is a direct universalM-arrow, then anM-arrow m ∶ x a is epic if and only if

its adjunct u/m ∶ r→ a along u (see Remark 6.2.2(3)) is epic. If u ∶ r a is an inverse universal M-arrow, then an M-arrow m ∶ x a is monic if and

only if its adjunct m/u ∶ x→ r along u (see Remark 6.2.2(3)) is monic.

Proof. m ∶ x a is epic if it is an epimorphism in the left module x ⟨M⟩ ∶ ∗ A andu/m ∶ r→ a is epic if it is an epimorphism in the left module r ⟨A⟩ ∶ ∗A. But the left moduleisomorphism u ⟨MA⟩ ∶ r ⟨A⟩→ x ⟨M⟩ ∶ ∗A maps u/m to m.

Note. We list below some properties of monomorphisms in a category.

Proposition 9.3.7.

(1) If a monomorphism m ∶ x → a has a section (i.e. if there exists an arrow h ∶ a → x suchthat h m = 1), then m is an isomorphism.

(2) Monomorphisms are pullback stable.(3) A continuous function preserves monomorphisms.

Proof.(1) See [AHS09] 7.36.(2) See [AHS09] 11.18.(3) See [AHS09] 13.5.

Denition 9.3.8.

An epic arrow m ∶ ∗ a in a left moduleM ∶ ∗A is called extremally epic (or an extremalepimorphism) if it does not factor through a proper monic A-arrow; that is, if it satisesthe following extremal condition: m = m′ u with u a monic A-arrow implies that u is anisomorphism.

A monic arrow m ∶ x ∗ in a right module M ∶ X ∗ is called extremally monic (or anextremal monomorphism) if it does not factor through a proper epic X-arrow; that is, if itsatises the following extremal condition: m = u m′ with u an epic X-arrow implies that uis an isomorphism.

Remark 9.3.9. An extremal epimorphism (resp. monomorphism) in a two sided module andin a category is dened similarly. An arrow m ∶ x a in a moduleM ∶XA is extremally epic if and only if it is an extremal

epimorphism in the left module x ⟨M⟩ ∶ ∗ A, and an arrow m ∶ x → a in a category C isextremally epic if and only if it is an extremal epimorphism in the left module x ⟨C⟩ ∶ ∗C.

An arrow m ∶ x a in a module M ∶ X A is extremally monic if and only if it is anextremal monomorphism in the right module ⟨M⟩a ∶ X ∗, and an arrow m ∶ x → a in acategory C is extremally monic if and only if it is an extremal monomorphism in the rightmodule ⟨C⟩a ∶C ∗.

281


Proposition 9.3.10. A unit of a left (resp. right) module is extremally epic (resp. monic).

Proof. Let u be a unit of a left module M ∶ ∗ A. Clearly u is epic. Now consider afactorization u = v k with k a monic A-arrow and let u/v be the adjunct of v along u asindicated in

u/v

∗

u66

v (( k

OO

. Since u u/v k = v k = u, we have u/v k = 1 by the universality of u. Hence k is anisomorphism by Proposition 9.3.7(1).

Note. The following is a restatement of Proposition 9.3.10 in terms of universal arrows.

Proposition 9.3.11. A direct (resp. inverse) universal arrow is extremally epic (resp. monic).

Proof. Since a direct universal arrow u ∶ x r in a moduleM ∶X A is the same thing as aunit of the left module x ⟨M⟩ ∶ ∗ A (see Remark 6.2.2(2)), and u ∶ x r is extremally epicinM i it is so in x ⟨M⟩ (see Remark 9.3.9), the assertion follows from Proposition 9.3.10.

Proposition 9.3.12. . Given a left module M ∶ ∗A, if A is complete and M is continuous, then any M-arrow

satisfying the extremal condition is epic. Given a right module M ∶ X ∗, if X is cocomplete and M is cocontinuous, then anyM-arrow satisfying the extremal condition is monic.

Proof. Suppose that anM-arrow m satises the extremal condition. Let h and h′ be a parallelpair of A-arrows such that m h = m h′, and let u be an equalizer of (h,h′). Since M iscontinuous, there is a unique M-arrow m/u, the adjunct of m along u (see Remark 7.4.20),making the diagram

u∗

m/u99

m// h //

h′//

commute. Since u is an equalizer, it is monic and thus is an isomorphism because m satisesthe extremal condition; hence h = h′ as required.

9.4. Subobjects

Denition 9.4.1.

A subobject s in a right moduleM ∶X ∗ is a pair s = (s, s) consisting of an object s ∈ ∥X∥and a monicM-arrow s ∶ s ∗.

A quotient object s in a left module M ∶ ∗ A is a pair s = (s, s) consisting of an objects ∈ ∥A∥ and an epicM-arrow s ∶ ∗ s.

Remark 9.4.2.(1) A subobject and its components are denoted using the same letter.(2) We regard a subobject in a right moduleM ∶ X ∗ as an object in the comma category

⌈M↓⌉ and say that

282


a) subobjects s and t are isomorphic when they are so in ⌈M↓⌉; that is, when there is aniso X-arrow u ∶ s→ t making the triangle

s

u

s

(( ∗

tt

66

commute;b) two sets S and T of subobjects are equivalent when any subobject in S is isomorphic

to some subobject in T, and vice versa;c) a set S of subobjects is essentially small when it is equivalent to a small set of subob-

jects.(3) We write s ≅ t if subobjects s and t are isomorphic. Assuming the axiom of choice, the

following conditions are equivalent for a set S of subobjects:a) S is essentially small;b) S has a small subset equivalent to S;c) the quotient set S/ ≅ is small.

(4) A subobject s in a right moduleM ∶ X ∗ is called extremal if theM-arrow s ∶ s ∗ isextremally monic. Dually, a quotient object s in a left moduleM ∶ ∗A is called extremalif theM-arrow s ∶ ∗ s is extremally epic.

Denition 9.4.3. Let C be a category and c be an object in C. A subobject of c is a subobject of the right module ⟨C⟩c ∶C ∗; that is, a subobject of c

is a pair s = (s, s) consisting of an object s ∈ ∥C∥ and a monic C-arrow s ∶ s→ c. A quotient object of c is a quotient object of the left module c ⟨C⟩ ∶ ∗ C; that is, a

quotient object of c is a pair s = (s, s) consisting of an object s ∈ ∥C∥ and an epic C-arrows ∶ c→ s.

Proposition 9.4.4.

A continuous function H ∶ C → B preserves subobjects; that is, H sends each subobject ofc ∈ ∥C∥ to a subobject of c ⋅H ∈ ∥B∥.

A cocontinuous function H ∶C→ B preserves quotient objects; that is, H sends each quotientobject of c ∈ ∥C∥ to a quotient object of c ⋅H ∈ ∥B∥.

Proof. Immediate from Proposition 9.3.7(3).

Denition 9.4.5.

Let S = si ∶ si ∗ be a set of subobjects in a right module M ∶ X ∗. We say that anM-arrow m ∶ x ∗ factors through S if for each si ∈ S there exists a (necessarily unique)X-arrow αi ∶ x→ si making the diagram

xαi

m // ∗

sisi

99

commute. Let S = si ∶ ∗ si be a set of quotient objects in a left module M ∶ ∗ A. We say that

anM-arrow m ∶ ∗ a factors through S if for each si ∈ S there exists a (necessarily unique)A-arrow αi ∶ si → a making the diagram

∗si %%

m // a

si

αi

OO

283


commute.

Remark 9.4.6. If anM-arrow m ∶ x ∗ factors through S = si ∶ si ∗, then it determines aunique discrete cone (αi ∶ x→ si) from m to S in the comma category ⌈M↓⌉.

Denition 9.4.7.

If S is a set of subobjects in a right module M ∶ X ∗, its intersection is an M-arrowr ∶ r ∗ which is terminal among those that factors through S; that is;(1) r factors through S,(2) if anM-arrow m ∶ x ∗ factors through S, then it uniquely factors through r ∶ r ∗.

If S is a set of quotient objects in a left moduleM ∶ ∗A, its union is anM-arrow r ∶ ∗ rwhich is initial among those that factors through S; that is;(1) r factors through S,(2) if anM-arrow m ∶ ∗ a factors through S, then it uniquely factors through r ∶ ∗ r.

Remark 9.4.8.(1) The unique factorization requirement in the denition implies that any intersection r ∶ r ∗

of S is monic, i.e. a subobject inM.(2) By Remark 9.4.6, an intersection of S is identied with a product of S in the comma

category ⌈M↓⌉. For example, an intersection r of subobjects s0 and s1 is given by a productdiagram

s0 s0

((rπ0OO

π1

r // ∗

s1s1

66

in ⌈M↓⌉ (π0 and π1 are also monic by Proposition 9.3.5).(3) Note that, given a pair of subobjects (s0, s1) of an object c in a category C, a commutative

diagram as in

rπ0

r

&&

π1 // s1

s1

s0 s0

// c

is a pullback diagram of s0s0Ð→ c

s1←Ð s1 in C if and only if it is a product diagram of s0s0Ð→ c

and s1s1Ð→ c in the slice category ⌈C↓c⌉. Hence an intersection of s0 and s1 is identied

with a pullback of s0s0Ð→ c

s1←Ð s1. This is generalized for arbitrary set S of subobjects ofc ∈ ∥C∥; an intersection of S is identied with a limit, multiple pullback, of S.

Proposition 9.4.9.

Two equivalent sets of subobjects have same intersections. Two equivalent sets of quotient objects have same unions.

Proof. Let S and T be two sets of subobjects and suppose that they are equivalent. Clearlyan M-arrow m ∶ ∗ a factors through S if and only if it factors through T. Hence S and Thave same intersections.

Proposition 9.4.10.

If a category C is complete, then any small set of subobjects of an object c ∈ ∥C∥ has anintersection. If a functor H ∶ C → B is continuous, then it preserves subobjects and anyintersection of a small set of subobjects.

284


If a category C is cocomplete, then any small set of quotient objects of an object c ∈ ∥C∥has a union. If a functor H ∶ C → B is cocontinuous, then it preserves quotient objects andany union of a small set of quotient objects.

Proof. Immediate since an intersection is given as a limit (see Remark 9.4.8(3)). (We havealready seen in Proposition 9.4.4 that a continuous functor preserves subobjects.)

Denition 9.4.11.

A right moduleM ∶X ∗ is called well-powered (resp. extremally well-powered) if the setof subobjects (resp. extremal subobjects) inM is essentially small.

A left moduleM ∶ ∗ A is called well-copowered (resp. extremally well-copowered) if theset of quotient objects (resp. extremal quotient objects) inM is essentially small.

Remark 9.4.12. If S denotes the set of subobjects (resp. extremal subobjects) ofM, then, byRemark 9.4.2(3),M is well-powered (resp. extremally well-powered) if and only if the quotientset S/ ≅ is small.

Denition 9.4.13. A category C is called well-powered (resp. extremally well-powered) if the set of subobjects (resp. extremal sub-

objects) of any c ∈ ∥C∥ is essentially small. well-copowered (resp. extremally well-copowered) if the set of quotient objects (resp. ex-

tremal quotient objects) of any c ∈ ∥C∥ is essentially small.

Remark 9.4.14. A category C is well-powered (resp. extremally well-powered) if for every c ∈ ∥C∥ the right module ⟨C⟩c ∶

C ∗ is well-powered (resp. extremally well-powered). well-copowered (resp. extremally well-copowered) if for every c ∈ ∥C∥ the left module

c ⟨C⟩ ∶ ∗C is well-copowered (resp. extremally well-copowered).

Proposition 9.4.15.

If a category C is complete and well-powered, then any set of subobjects of an object c ∈ ∥C∥has an intersection. If a functor H ∶ C → B is continuous, then it preserves subobjects andintersections.

If a category C is cocomplete and well-copowered, then any set of quotient objects of anobject c ∈ ∥C∥ has a union. If a functor H ∶ C → B is cocontinuous, then it preservesquotient objects and unions.

Proof. Since C is well-powered, any set S of subobjects of c has a small set S′ of subobjectsof c equivalent to S, and, since any functor preserves isomorphisms, the images of S and S′under H are also equivalent. Now, since equivalent sets of subobjects have same intersections(see Proposition 9.4.9), the assertion is reduced to Proposition 9.4.10.

9.5. Epi-mono factorizations

Denition 9.5.1.

A left module M ∶ ∗ A is said to have (extremal-epi,mono)-factorizations when everyM-arrow m factors as m = p s with p an extremally epicM-arrow and s a monic A-arrow.

A right module M ∶ X ∗ is said to have (epi,extremal-mono)-factorizations when everyM-arrow m factors as m = s p with p an extremally monicM-arrow and s an epic X-arrow.

Proposition 9.5.2.

285


LetM ∶ ∗A be a left module and suppose that A is complete andM is continuous. Then(extremal-epi,mono)-factorizations are essentially unique; that is, if p0 s0 = p1 s1 are two(extremal-epi,mono)-factorizations of anM-arrow; then there exists a (necessarily unique)isomorphism u making the diagram

u

s0

((p1 ((

p066

s1

66

commute. LetM ∶X ∗ be a right module and suppose that X is cocomplete andM is cocontinuous.

Then (epi,extremal-mono)-factorizations are essentially unique; that is, if s0 p0 = s1 p1

are two (epi,extremal-mono)-factorizations of an M-arrow; then there exists a (necessarilyunique) isomorphism u making the diagram

u

p0

((s1 ((

s066

p1

66

commute.

Proof. Let s0

((u0OO

u1

s1

66

be a pullback of (s0, s1) in A. Since M is continuous, there exists a unique M-arrow m, theadjunct of (p0,p1) along (u0,u1) (see Remark 7.4.20), making the diagram

s0

((p1 ((

p066

m // u0OO

u1

s1

66

commute. Since monomorphisms are pullback stable, u0 and u1 are monic, and hence theyare isomorphisms by the extremal epicity of p0 and p1. Now u ∶= u−1

0 u1 gives a desiredisomorphism.

Theorem 9.5.3.

LetM ∶ ∗A be a left module and suppose that A is complete andM is continuous. Underthis condition, if A is well-powered, then M has essentially unique (extremal-epi,mono)-factorizations.

LetM ∶X ∗ be a right module and suppose that X is cocomplete andM is cocontinuous.Under this condition, if X is well-copowered, then M has essentially unique (epi,extremal-mono)-factorizations.

Proof. The essential uniqueness of factorizations follows from Proposition 9.5.2. Now let m ∶∗ a be an M-arrow and consider all the possible factorizations m = mk sk with sk monic.By Proposition 9.4.15, there exists an intersection r ∶ r → a of all sk. Since M is continuous,by the second assertion of Proposition 9.4.15, it preserves this intersection. Hence m factors

286


through r as m = n r for some (necessarily unique)M-arrow n ∶ ∗ r. The proof is completeif we show that n is extremally epic. For this, by Proposition 9.3.12, it suces to show that nsatises the extremal condition. Let n = n′ u be a factorization of n with u monic. We need toshow that u is an isomorphism. Since m = n′ (u r) and u r is monic, there exists a uniqueA-arrow v making the diagram

∗n′ %%

n // rv

r // a

u r

99

commute. Since r = v u r and r is monic, we have 1 = v u. Hence u is an isomorphism byProposition 9.3.7(1) as required.

Denition 9.5.4. A category C is said to have (extremal-epi,mono)-factorizations when every C-arrow m factors as m = p s with p ex-

tremally epic and s monic. (epi,extremal-mono)-factorizations when every C-arrow m factors as m = s p with p ex-

tremally monic and s epic.

Remark 9.5.5. A category C has (extremal-epi,mono)-factorizations if and only if for every c ∈ ∥C∥ the left module c ⟨C⟩ ∶

∗C has (extremal-epi,mono)-factorizations. (epi,extremal-mono)-factorizations if and only if for every c ∈ ∥C∥ the right module ⟨C⟩c ∶

C ∗ has (epi,extremal-mono)-factorizations.

Theorem 9.5.6.

A complete and well-powered category has essentially unique (extremal-epi,mono)-factorizations. A cocomplete and well-copowered category has essentially unique (epi,extremal-mono)-factorizations.

Proof. Let C be a complete and well-powered category. By Remark 9.5.5, it suces to showthat for any c ∈ ∥C∥, the left module c ⟨C⟩ ∶ ∗C has essentially unique (extremal-epi,mono)-factorizations. But this follows from Theorem9.5.3.

9.6. Generators

Denition 9.6.1. Let E be a category andM ∶XA be a module. A cone α ∶ K a ∶ E∗ M is said to be [jointly] epic if for any pair of parallel A-arrows

h,h′ ∶ a→ b, αe h = αe h′ for every e ∈ ∥E∥ implies that h = h′. A cone α ∶ x K ∶ ∗EM is said to be [jointly] monic if for any pair of parallel X-arrows

h,h′ ∶ y → x, h αe = h′ αe for every e ∈ ∥E∥ implies that h = h′.

Remark 9.6.2. A cone α ∶ K a ∶ E∗ M is epic if and only if α is an epic ⟨E∗,M⟩-arrow(see Denition 4.6.5). Dually, a cone α ∶ x K ∶ ∗E M is monic if and only if α is a monic⟨∗E,M⟩-arrow.

Proposition 9.6.3.

If π ∶ K r ∶ E∗ M is a universal cone, then a cone α ∶ K a ∶ E∗ M is epic if andonly if its adjunct π/α ∶ r→ a along π is epic.

If π ∶ r K ∶ ∗E M is a universal cone, then a cone α ∶ x K ∶ ∗E M is monic if andonly if its adjunct α/π ∶ x→ r along π is monic.

Proof. By Remark 9.6.2, this is an instance of Proposition 9.3.6 whereM is given by ⟨E∗,M⟩(resp. ⟨∗E,M⟩).

287


Denition 9.6.4. A set E of objects in a category C is said to generate C if for any parallel pair of C-arrows h,h′ ∶ c → d, h ≠ h′ implies that there is an

x ∈ E and a C-arrow f ∶ x→ c with f h ≠ f h′. cogenerate C if for any parallel pair of C-arrows h,h′ ∶ d→ c, h ≠ h′ implies that there is an

x ∈ E and a C-arrow f ∶ c→ x with h f ≠ h′ f.

Remark 9.6.5. Let C be a category. If c is an object in C and E is a set of objects in C,we denote by [E ,c] the set of all C-arrows f ∶ x → c with x ∈ E (to be precise2, the set of allpairs (x, f) with f ∈ x ⟨C⟩c and x ∈ E). Indexed by itself, the set [E ,c] is seen as a discretecone (see Remark 4.6.4(5)) from the family of objects (dom (f))f∈[E,c] to c. Dually, we denoteby [c,E] the set of all C-arrows f ∶ c → x with x ∈ E ; [c,E] is seen as a discrete cone from cto the family of objects (cod (f))f∈[c,E]. With this notation and the terminology introduced inDenition 9.6.1, Denition 9.6.4 is stated more succinctly as: E generate C if [E ,c] is epic for every c ∈ ∥C∥. E cogenerate C if [c,E] is monic for every c ∈ ∥C∥.

Theorem 9.6.6.

LetM ∶ ∗A be a left module and suppose that A is complete andM is continuous. Underthis condition, if A has a small cogenerating set, then M is extremally well-copowered.

LetM ∶X ∗ be a right module and suppose that X is cocomplete andM is cocontinuous.Under this condition, if X has a small generating set, then M is extremally well-powered.

Proof. We will use the notation in Remark 9.6.5. Let E be a small cogenerating set of A anddenote by [∗,E] the set of all M-arrows m ∶ ∗ a with a ∈ E . Let ℘ [∗,E] denote the setof all subsets of [∗,E]; since E is small, so is ℘ [∗,E]. Given an extremal quotient object sof M, dene [s] ∈ ℘ [∗,E] by [s] = s f ∣ f ∈ [s,E]. By the smallness of ℘ [∗,E] and notingRemark 9.4.12, the proof is complete if we show that, given two extremal quotient objects sand t, [s] = [t] i s ≅ t. If s and t are isomorphic, i.e. if there is an iso A-arrow u ∶ t→ s suchthat s = t u, then, by the bijectiveness of f ↦ u f ∶ [s,E]→ [t,E], we have

[s] = s f ∣ f ∈ [s,E] = t u f ∣ f ∈ [s,E] = t g ∣g ∈ [t,E] = [t]

. Now suppose that [s] = [t] and let I = [s] = [t]. Since s ∶ ∗ s and t ∶ ∗ t are epic, theassignments f ↦ s f ∶ [s,E]→ I and f ↦ t f ∶ [t,E]→ I are bijective. Hence the discrete cones[s,E] and [t,E] are written as σ = (σi ∶ s→ ai)i∈I and τ = (τi ∶ t→ ai)i∈I with s σi = t τi foreach i ∈ I. Let π = (πi ∶ r→ ai)i∈I be a product diagram of the family of object (ai)i∈I, and letm ∶ ∗ r be the adjunct of s σ = t τ along π inM (see Remark 7.4.20), i.e. let m ∶ ∗ r bethe uniqueM-arrow such that m π = s σ = t τ . Since s σ = s σ/π π and t τ = t τ/π π,we have m = s σ/π = t τ/π by the uniqueness of m, as shown in the commutative diagram

sσ/π

σ((∗

s77

t ''

m // r π // (ai)

tτ/πOO

τ

66

. Since σ = [s,E] and τ = [t,E] are monic by the denition of a cogenerating set, so are σ/πand τ/π by Proposition 9.6.3. Hence s σ/π and t τ/π are (extremal-epi,mono)-factorizationsof m, and by the essential uniqueness of such factorizations (see Proposition 9.5.2), s and t areisomorphic as required.

2Recall that we do not require pairwise disjointness of hom-sets.

288


Corollary 9.6.7.

A complete category with a small cogenerating set is extremally well-copowered. A cocomplete category with a small generating set is extremally well-powered.

Proof. Let C be a complete category with a small cogenerating set. By Remark 9.4.14, itsuces to show that for any c ∈ ∥C∥, the left module c ⟨C⟩ ∶ ∗ C is extremally well-copowered. But this follows from Theorem9.6.6.

9.7. Special adjoint functor theorem

Theorem 9.7.1.

If a category A is complete, well-powered and with a small cogenerating set, then a leftmodule M ∶ ∗A is representable if and only if M is continuous.

If a category X is cocomplete, well-copowered and with a small generating set, then a rightmodule M ∶X ∗ is representable if and only if M is cocontinuous.

Proof. If M is representable, then M is continuous by Corollary 7.4.18. Conversely, supposethatM is continuous. ThenM has (extremal-epi,mono)-factorizations by Theorem9.5.3. Thisimplies that the set of extremal quotient objects inM forms a solution set ofM, and this setis essentially small by Theorem9.6.6. HenceM is representable by Theorem9.2.5.

Corollary 9.7.2. Let M ∶XA be a module. If a category A is complete, well-powered and with a small cogenerating set, then M is

representable if and only if for every object x ∈ ∥X∥ the left module x ⟨M⟩ ∶ ∗ A iscontinuous.

If a category X is cocomplete, well-copowered and with a small generating set, then M iscorepresentable if and only if for every object a ∈ ∥A∥ the right module ⟨M⟩a ∶ X ∗ iscocontinuous.

Proof. Since a module M ∶ X A is representable i the left module x ⟨M⟩ ∶ ∗ A isrepresentable for every x ∈ ∥X∥ (see Corollary 6.3.11), the assertion follows from Theorem9.7.1.

Theorem 9.7.3. (Special Adjoint Functor Theorem). If a category A is complete, well-powered and with a small cogenerating set, then a functor

G ∶A→X has a left adjoint if and only if G is continuous. If a category X is cocomplete, well-copowered and with a small generating set, then a functor

F ∶X→A has a right adjoint if and only if F is cocontinuous.

Proof. By Remark 8.2.2(2), a functor G ∶A→X has a left adjoint i the module ⟨X⟩G ∶XAis representable. By Corollary 9.7.2, this is the case i for every object x ∈ ∥X∥ the left modulex ⟨X⟩G ∶ ∗A is continuous. By Corollary 7.4.17, this is the case i G is continuous.

Corollary 9.7.4.

If a category C is complete, well-powered and with a small cogenerating set, then C iscocomplete as well.

If a category C is cocomplete, well-copowered and with a small generating set, then C iscomplete as well.

Proof. Since C is cocomplete i for every small category E the diagonal functor [∆E,C] has aleft adjoint (see Remark 7.3.5), and since [∆E,C] is continuous by Theorem7.3.17, the assertionfollows from Theorem9.7.3.

289


Corollary 9.7.5.

If a category C is complete, well-powered and with a small cogenerating set, then C has aninitial object.

If a category C is cocomplete, well-copowered and with a small generating set, then C hasa terminal object.

Proof. Immediate from Corollary 9.7.4 since any cocomplete category has an initial object.

Remark 9.7.6. Conversely, Theorem9.7.3 follows from Corollary 9.7.5. [ML98] takes this routeto tackle the special adjoint functor theorem.

290

10. Collages and Commas (continued)

10.1. Cylinder modules

Note. The following modules are dened in Denition 10.1.1: ↑CYL ∶CATCLG

↑CYL ∶CATMOD

↓CYL ∶COMCAT

Although ↑CYL ∶ CAT MOD and ↑CYL ∶ CAT CLG are regarded as the same thingunder the identication CLG ≅ MOD, paralleling denitions are presented for the sake ofreference.

Denition 10.1.1.

(1) The module ↑CYL ∶CATCLG is dened in the following way:a) a ↑CYL-arrow from a category E to a collageM ∶XA, written α ∶ S T ∶ EM,

is a triple (S, α,T) consisting of a functor S ∶ E →X, a second functor T ∶ E →A, anda natural transformation

ES

zz

T

$$X M0

// ⌊M⌋α

AM1

oo

from S M0 toM1 T.b) for a functor H ∶D→ E and a ↑CYL-arrow α ∶ S T ∶ EM as in

D

HE

S

zz

T

$$X M0

// ⌊M⌋α

AM1

oo

, their composite is the ↑CYL-arrow H α ∶ H S T H ∶ D M with the naturaltransformation H α ∶ H S M0 →M1 T H ∶D→ ⌊M⌋ given by the usual compositionof a functor and a natural transformation.

c) for a ↑CYL-arrow α ∶ S T ∶ EM and a collage cell Φ ∶ P Q ∶M N as in

ES

zz

T

$$X

P

M0

// ⌊M⌋α

⌊Φ⌋

A

Q

M1

oo

Y N0

// ⌊N ⌋ BN1

oo

291


, their composite is the ↑CYL-arrow α Φ ∶ S P Q T ∶ E N with the naturaltransformation α Φ ∶ S P N0 → N1 Q T ∶ E→ ⌊N ⌋ dened by

α Φ = α ⌊Φ⌋, the usual composite of a natural transformation and a functor.

(2) The module ↑CYL ∶CATMOD is dened in the following way:a) a ↑CYL-arrow from a category E to a moduleM ∶X→A is a cylinder α ∶ S T ∶ EM.

b) for a functor H ∶D→ E and a ↑CYL-arrow α ∶ S T ∶ EM as in

D

HE

S

T

##X M

//

α

A

, their composite is the ↑CYL-arrow H α ∶ H S T H ∶DM, the usual compositeof a functor and a cylinder (see Denition 4.3.24).

c) for a ↑CYL-arrow α ∶ S T ∶ EM and a module cell Φ ∶ P Q ∶M N as in

ES

T

##X

P

M //

α

A

Q

Y N//

Φ

B

, their composite is the ↑CYL-arrow α Φ ∶ S P Q T ∶ E N , the usual compositeof a cylinder and a cell (see Denition 4.3.14).

(3) The module ↓CYL ∶COMCAT is dened in the following way:a) a ↓CYL-arrow from a comma K ∶ X A to a category E, written α ∶ S T ∶ K E,

is a triple (S, α,T) consisting of a functor S ∶X → E, a second functor T ∶A → E, anda natural transformation

X

S $$

⌈K⌉K0oo K1 //A

TE

α

from K0 S to T K1.b) for a comma cell Φ ∶ P Q ∶ J K and a ↓CYL-arrow α ∶ S T ∶ K E as in

Y

P

⌈J⌉⌈Φ⌉

J0oo J1 // B

Q

X

S $$

⌈K⌉K0

ooK1

//A

TE

α

, their composite is the ↓CYL-arrow Φ α ∶ P S T Q ∶ J E with the naturaltransformation Φ α ∶ J0 P S→ T Q J1 ∶ ⌈J⌉→ E dened by

Φ α = ⌈Φ⌉ α, the usual composite of a functor and a natural transformation.

292


c) for a ↓CYL-arrow α ∶ S→ T ∶ K E and a functor H ∶ E→D as in

X

S $$

⌈K⌉K0oo K1 //A

TE

H

α

D

, their composite is the ↓CYL-arrow α H ∶ S H H T ∶ K D with the naturaltransformation α H ∶ K0 S H→ H T K1 ∶ ⌈K⌉→D given by the usual compositionof a natural transformation and a functor.

Remark 10.1.2. By Remark 4.3.4(4), the identity

CAT

1

↑CYL //MOD

♯

CAT ↑CYL//

1

CLG

holds.

Denition 10.1.3.

(1) The unit cylinder of a moduleM ∶XA is the cylinder

⌈M↓⌉M↓

0

zz

M↓1

$$X M

//

1↓M

A

dened by[1↓M]

m= m

for m an arrow ofM.(2) The unit cylinder of a comma K ∶XA is the cylinder

⌈K⌉K0

K1

$$X

K↑//

1↑K

A

dened by[1↑K]k = k

for k an object of ⌈K⌉.

Remark 10.1.4.(1) By Remark 4.3.4(4), the unit cylinders 1↓M and 1↑K are also written as

⌈M↓⌉M↓0

vv

M↓1

((

⌈K⌉K0

vvK1

((X

M0 ((

A

M1vv

X

K↑0((

A

K↑1vv

⌊M⌋

1↓M

⌊K↑⌋

1↑K

using the collages ofM and K↑.

293


(2) The unit cylinder 1↓M forms a ↑CYL-arrow 1↓M ∶M↓0 M↓

1 ∶ ⌈M↓⌉M.(3) The unit cylinder 1↑K forms a ↓CYL-arrow 1↑K ∶ K

↑0 K↑1 ∶ K ⌊K↑⌋.

Proposition 10.1.5.

(1) Given a collage cell Φ ∶ P Q ∶M N , the two compositions

⌈M↓⌉M↓

0

xx

M↓1

&&

X

P

⌈M↓⌉⌈Φ↓⌉

M↓0oo

M↓1 //A

Q

X

P

M0// ⌊M⌋

1↓M

⌊Φ⌋

AM1oo

Q

Y

N0 &&

⌈N ↓⌉N ↓0oo N ↓1 // B

N1xxY N0

// ⌊N ⌋ BN1

oo ⌊N ⌋1↓N

yield the same natural transformations ⌈M↓⌉→ ⌊N ⌋; that is, the square

⌈M↓⌉⌈Φ↓⌉

1↓M // ⌊M⌋⌊Φ⌋

⌈N ↓⌉1↓N

// ⌊N ⌋

commutes.(2) Given a comma cell Φ ∶ P Q ∶ J K, the two compositions

Y

P

⌈J⌉⌈Φ⌉

J0oo J1 // B

Q

⌈J⌉J0

yy

J1

%%X

K↑0 %%

⌈K⌉K0oo K1

//A

K↑1yy

Y

P

J↑0 // ⌊J↑⌋1↑J

⌊Φ↑⌋

BJ↑1oo

Q

⌊K↑⌋1↑K

XK↑0

// ⌊K↑⌋ AK↑1

oo

yield the same natural transformations ⌈J⌉→ ⌊K↑⌋; that is, the square

⌈J⌉⌈Φ⌉

1↑J // ⌊J↑⌋⌊Φ↑⌋

⌈K⌉1↑K

// ⌊K↑⌋

commutes.

Proof. Immediate from the denitions of unit cylinders and the constructions of Φ↓ and Φ↑

(see Denition 3.2.17 and Denition 3.2.21).

Proposition 10.1.6.

(1) For a module M ∶ X A, the unit cylinder 1↓M (see Remark 10.1.4(1)) is given by thecomposition

⌈M↓⌉M↓

0

ww

M↓1

''X

M0 ''

⟨M↓⟩↑0// ⌊⟨M↓⟩↑⌋

⌊εM⌋

1↑M↓

A

M1ww

⟨M↓⟩↑1

oo

⌊M⌋

294


; that is, the triangle

⌈M↓⌉1↑M↓ //

1↓M ''

⌊⟨M↓⟩↑⌋⌊εM⌋

⌊M⌋

commutes, where 1↑M↓ is the unit cylinder of the commaM↓ and εM is the isomorphism inTheorem3.2.25.

(2) For a comma K ∶ X A, the unit cylinder 1↑K (see Remark 10.1.4(1)) is given by thecomposition

⌈K⌉K0

ww⌈ηK⌉

K1

''X

K↑0 ''

⌈⟨K↑⟩↓⌉⟨K↑⟩↓0

oo ⟨K↑⟩↓1

//A

K↑1ww⌊K↑⌋

1↓K↑

; that is, the triangle⌈K⌉

1↑K

((⌈ηK⌉ ⌈⟨K↑⟩↓⌉

1↓K↑

// ⌊K↑⌋

commutes, where ηK is the isomorphism in Theorem3.2.25 and 1↓K↑ is the unit cylinder ofthe module K↑.


Denition 10.1.7.

(1) The comma adjunct of a ↑CYL-arrow α ∶ S T ∶ EM, i.e. a cylinder

ES

T

##X M

//

α

A

, is the functor ⌈α↓⌉ ∶ E→ ⌈M↓⌉ dened by

⌈α↓⌉ ⋅ e = αe ⌈α↓⌉ ⋅ h = (S ⋅ h,T ⋅ h)

for e an object and h an arrow of E.(2) The collage adjunct of a ↓CYL-arrow α ∶ S T ∶ K E, i.e. a natural transformation

X

S $$

⌈K⌉K0oo K1 //A

TE

α

, is the functor ⌊α↑⌋ ∶ ⌊K↑⌋→ E adjunct (see Theorem3.1.17) to the cell

XK↑ //

S

A

T

E⟨E⟩

//

⌊α↑⌋

E

295


dened by⌊α↑⌋ ⋅ k = αk


Proposition 10.1.8.

(1) The assignment α ↦ ⌈α↓⌉ yields a bijection from the set of ↑CYL-arrows EM to the setof functors E→ ⌈M↓⌉, and the triangle

E

⌈α↓⌉

α

&&⌈M↓⌉1↓M

//M

commutes; the unit cylinder 1↓M thus forms an inverse universal ↑CYL-arrow.(2) The assignment α ↦ ⌊α↑⌋ yields a bijection from the set of ↓CYL-arrows K E to the set

of functors ⌊K↑⌋→ E, and the triangle

K1↑K //

α&&

⌊K↑⌋⌊α↑⌋E

commutes; the unit cylinder 1↑K thus forms a direct universal ↓CYL-arrow.

Proof. The commutativity of each triangle is easily veried. It is also easy to verify that

⌈[H 1↓M]↓⌉ = H

for any functor H ∶ E→ ⌈M↓⌉, and verify that

⌊[1↑K H]↑⌋ = H

for any functor H ∶ ⌊K↑⌋→ E.

Theorem 10.1.9.

(1) The functor M ↦ ⌈M ↓⌉ ∶MOD → CAT and the family of unit cylinders 1↓M ∶ ⌈M↓⌉ M,one for each locally small moduleM, form a counit of the module ↑CYL ∶CATMOD.

(2) The functor K ↦ ⌊K↑⌋ ∶COM→CAT and the family of unit cylinders 1↑K ∶ K ⌊K↑⌋, onefor each locally small comma K, form a unit of the module ↓CYL ∶COMCAT.

Proof. The family of unit cylinders 1↓M (resp. 1↑K) satises the naturality condition by Propo-sition 10.1.5, and each unit cylinder is universal as we have seen in Proposition 10.1.8.

Denition 10.1.10. The unit cylinder of a category E is the cylinder

E1

1

##E

⟨E⟩//

[1⟨E⟩]

E

dened by[1⟨E⟩]e = 1e

for e ∈ ∥E∥, i.e. by the identity natural transformation 1E → 1E.

296


Proposition 10.1.11. For any functor H ∶D→ E, the square

D[1⟨D⟩] //

H

⟨D⟩⟨H⟩

E[1⟨E⟩]

// ⟨E⟩

commutes.

Proof. Easily veried.

Proposition 10.1.12. For any locally small category E, the unit cylinder [1⟨E⟩] forms a directuniversal ↑CYL-arrow.Proof. The family of module isomorphisms ΨE

M ∶ ⟨E,M⟩→ ⟨⟨E⟩ ,M⟩ in Corollary 5.5.4, one foreach locally small moduleM, gives a representation of the left slice of ↑CYL at E. The unitcorresponding to this representation is given by the inverse image of identity cell ⟨E⟩ ⟨E⟩under ΨE

⟨E⟩, but it is immediately seen that this is nothing but the unit cylinder [1⟨E⟩] of E.

Theorem 10.1.13. The embedding ⟨−⟩ ∶ CAT →MOD (see Theorem1.2.30) and the familyof unit cylinders [1⟨E⟩] ∶ E ⟨E⟩, one for each locally small category E, form a unit of themodule ↑CYL ∶CATMOD.

Proof. Immediate from Proposition 10.1.11 and Proposition 10.1.12.

Theorem 10.1.14. There is a canonical adjunction between the functor M ↦ ⌈M ↓⌉ ∶MOD→CAT and the embedding ⟨−⟩ ∶CAT→MOD.

Proof. This follows by applying Theorem8.2.10 to the counit and unit of the module ↑CYLgiven in Theorem10.1.9(1) and Theorem10.1.13.

Remark 10.1.15. The functor M ↦ ⌈M ↓⌉ ∶ MOD → CAT is thus a right adjoint of theembedding ⟨−⟩ ∶CAT→MOD (cf. Remark 3.1.18).

10.2. Equivalence COM ≃ CLGNote. The following modules are dened in Denition 10.2.1: CYL ∶COMCLG

CYL ∶COMMOD

Although they are regarded as the same thing under the identication CLG ≅MOD, paral-leling denitions are presented for the sake of reference. After Remark 10.2.2(3) we will dealwith only the module CYL ∶ COM CLG; however, any result in this section stated forCYL ∶COMCLG also holds with CLG changed to MOD.

Denition 10.2.1.

(1) The module CYL ∶COMCLG is dened in the following way:a) a CYL-arrow from a comma K ∶ Y B to a collage M ∶ X A, written α ∶ S

T ∶ K M, is a triple (S, α,T) consisting of a functor S ∶ Y → X, a second functorT ∶ B→A, and a natural transformation

Y

S

⌈K⌉K0oo K1 // B

T

X M0

// ⌊M⌋α

AM1

oo

297


from K0 S M0 toM1 T K1.b) for a comma cell Φ ∶ P Q ∶ J K and a CYL-arrow α ∶ S T ∶ KM as in

Z

P

⌈J⌉⌈Φ⌉

J0oo J1 // C

Q

Y

S

⌈K⌉K0oo K1 // B

T

X M0

// ⌊M⌋α

AM1

oo

, their composite is the CYL-arrow Φ α ∶ P S T Q ∶ J M with the naturaltransformation Φ α ∶ J0 P S M0 →M1 T Q J1 ∶ ⌈J⌉→ ⌊M⌋ dened by

Φ α = ⌈Φ⌉ α

, the usual composite of a functor and a natural transformation.c) for a CYL-arrow α ∶ S T ∶ KM and a collage cell Φ ∶ P Q ∶M N as in

Y

S

⌈K⌉K0oo K1 // B

T

X

P

M0

// ⌊M⌋α

⌊Φ⌋

A

Q

M1

oo

Z N0

// ⌊N ⌋ CN1

oo

, their composite is the CYL-arrow α Φ ∶ S P Q T ∶ K N with the naturaltransformation α Φ ∶ K0 S P N0 → N1 Q T K1 ∶ ⌈K⌉→ ⌊N ⌋ dened by

α Φ = α ⌊Φ⌋

, the usual composite of a natural transformation and a functor.(2) The module CYL ∶COMMOD is dened in the following way:

a) a CYL-arrow from a comma K ∶ Y B to a module M ∶ X A, written α ∶ S T ∶ K M, is a triple (S, α,T) consisting of a functor S ∶ Y → X, a second functorT ∶ B→A, and a cylinder

Y

S

⌈K⌉K0oo K1 // B

T

X M//

α

A

from K0 S to T K1 alongM.b) for a comma cell Φ ∶ P Q ∶ J K and a CYL-arrow α ∶ S T ∶ KM as in

Z

P

⌈J⌉⌈Φ⌉

J0oo J1 // C

Q

Y

S

⌈K⌉K0oo K1 // B

T

X M//

α

A

298


, their composite is the CYL-arrow Φ α ∶ P S T Q ∶ J M with the cylinderΦ α ∶ J0 P S T Q J1 ∶ ⌈J⌉M dened by

Φ α = ⌈Φ⌉ α, the usual composite of a functor and a cylinder (see Denition 4.3.24).

c) for a CYL-arrow α ∶ S T ∶ KM and a module cell Φ ∶ P Q ∶M N as in

Y

S

⌈K⌉K0oo K1 // B

T

X

P

M //

α

A

Q

Z N//

Φ

C

, their composite is the CYL-arrow α Φ ∶ S P Q T ∶ K N with the cylinderα Φ ∶ K0 S P Q T K1 ∶ ⌈K⌉ N given by the usual composition of a cylinderand a cell (see Denition 4.3.14).

Remark 10.2.2.(1) The unit cylinder (see Denition 10.1.3(1)) of a module (resp. collage) M ∶ X A forms

a CYL-arrow 1↓M ∶ 1X 1A ∶M↓ M.(2) The unit cylinder (see Denition 10.1.3(2)) of a comma K ∶ X A forms a CYL-arrow

1↑K ∶ 1X 1A ∶ K K↑.(3) Since a cylinder

Y

S

⌈K⌉K0oo K1 // B

T

X M//

α

A

is the same thing as a natural transformation

Y

S

⌈K⌉K0oo K1 // B

T

X M0

// ⌊M⌋α

AM1

oo

(see Remark 4.3.4(4)), the identity

COM

1

CYL //MOD

♯

COM CYL//

1

CLG

holds.(4) Each CYL-arrow α ∶ S T ∶ KM gives the following arrows:

↑CYL-arrow α ∶ K0 S T K1 ∶ ⌈K⌉M ↓CYL-arrow α ∶ S M0 M1 T ∶ K ⌊M⌋, dening the cells below:

COM

⌈−⌉

CYL // CLG

1

COM

1

CYL // CLG

⌊−⌋

CAT ↑CYL//

−

CLG COM ↓CYL//

−

CAT

299


Denition 10.2.3. Given a CYL-arrow α ∶ S T ∶ KM, i.e. a natural transformation

Y

S

⌈K⌉K0oo K1 // B

T

X M0

// ⌊M⌋α

AM1

oo

(1) the comma cell α↓ ∶ S T ∶ KM↓, depicted in

Y

S

⌈K⌉⌈α↓⌉

K0oo K1 // B

T

X ⌈M↓⌉M↓

0

ooM↓

1

//A

, is dened by the comma adjunct ⌈α↓⌉ ∶ ⌈K⌉ → ⌈M↓⌉ (see Denition 10.1.7(1)) of the↑CYL-arrow α ∶ K0 S T K1 ∶ ⌈K⌉M.

(2) the collage cell α↑ ∶ S T ∶ K↑ M, depicted in

Y

S

K↑0 // ⌊K↑⌋⌊α↑⌋

B

T

K↑1oo

X M0

// ⌊M⌋ AM1

oo

, is dened by the collage adjunct ⌊α↑⌋ ∶ ⌊K↑⌋ → ⌊M⌋ (see Denition 10.1.7(2)) of the↓CYL-arrow α ∶ S M0 M1 T ∶ K ⌊M⌋.

Proposition 10.2.4.

(1) The assignment α ↦ α↓ yields a bijection from the set of CYL-arrows K M to the setof comma cells KM↓, and the triangle

Kα↓

α

&&M↓1↓M

//M

commutes; the unit cylinder 1↓M thus forms an inverse universal CYL-arrow.(2) The assignment α ↦ α↑ yields a bijection from the set of CYL-arrows K M to the set

of collage cells K↑ M, and the triangle

K1↑K //

α %%

K↑

α↑M

commutes; the unit cylinder 1↑K thus forms a direct universal CYL-arrow.

Proof. By the denitions of α↓ and α↑, this is reduced to Proposition 10.1.8.

Proposition 10.2.5.

(1) The functor COM↓←Ð CLG (see Remark 3.2.26(2)) and the family of unit cylinders 1↓M ∶

M↓ M, one for each collage M, form a counit of the module CYL;

300


(2) The functor COM↑Ð→ CLG (see Remark 3.2.26(2)) and the family of unit cylinders 1↑K ∶

K K↑, one for each comma K, form a unit of the module CYL.

Proof. We have seen in Proposition 10.2.4 the universality of each unit cylinder. It remains toshow that the family of unit cylinders 1↓M (resp. 1↑K) satises the naturality condition. Butthis follows immediately from Proposition 10.1.5.

Remark 10.2.6. The counit and unit in Proposition 10.2.5 are depicted as

COMCYL

//1↓ CLG↓oo

COM

CYL //

↑//1↑ CLG

.

Theorem 10.2.7. There exists an adjoint equivalence

COM

↑//(η,ε) CLG

↓oo

with the unit η and the counit ε given by the isomorphisms in Theorem3.2.25.

Proof. This follows by applying Theorem8.2.10 to the counit and unit of CYL in Remark 10.2.6,and observing that the isomorphism εM (resp. ηK) in Theorem3.2.25 gives the adjunct of 1↓Malong 1↑M↓ (resp. 1↑K along 1↓K↑) by the commutativity of the triangle in Proposition 10.1.6.

Corollary 10.2.8. The module CYL is an equivalence and each unit cylinder is a two-wayuniversal CYL-arrow.

Proof. Since the functors COM↑Ð→CLG and COM

↓←ÐCLG are equivalences as we have justseen in Theorem10.2.7, the assertion follows by applying Corollary 8.5.8 to the counit and unitof CYL in Remark 10.2.6.

10.3. Equivalence [X ↓A] ≃ [X ↑A]Note. Given a pair of categoriesX andA, the following modules are dened in Denition 10.3.1: ⟨X A⟩ ∶ [X ↓A] [X ↑A] ⟨X A⟩ ∶ [X ↓A] [X ∶A]Although they are regarded as the same thing under the identication [X ↑A] ≅ [X ∶A],paralleling denitions are presented for the sake of reference. After Remark 10.3.2 (3) we willdeal with only the module ⟨X A⟩ ∶ [X ↓A] [X ↑A]; however, any result in this sectionstated for ⟨X A⟩ ∶ [X ↓A] [X ↑A] also holds with [X ↑A] changed to [X ∶A]. Thissection is analogous to the previous section; all denitions and results given for [X ↓A] and[X ↑A] in this section are the reections of those given for COM and CLG in the previoussection along the embedding [X ↓A] COM and [X ↑A] CLG (see Remark 3.2.14(3)and Remark 3.1.6(3)).

Denition 10.3.1. Let X and A be categories.(1) The module ⟨X A⟩ ∶ [X ↓A] [X ↑A] is dened in the following way:

301


a) an ⟨X A⟩-arrow α ∶ K M from a comma K ∶ X A to a collage M ∶ X A isgiven by a natural transformation

⌈K⌉K0

vv

K1

((X

M0 ((

A

M1vv⌊M⌋

α

from K0 M0 toM1 K1.b) for a comma morphism Φ ∶ J→ K ∶XA and an ⟨X A⟩-arrow α ∶ KM as in

⌈J⌉⌈Φ⌉

J0

yy

J1

%%X

M0 %%

⌈K⌉K0oo K1

//A

M1yy⌊M⌋α

, their composite is the ⟨X A⟩-arrow Φ α ∶ J M with the natural transformationΦ α ∶ J0 M0 →M1 J1 ∶ ⌈J⌉→ ⌊M⌋ dened by

Φ α = ⌈Φ⌉ α

, the usual composite of a functor and a natural transformation.c) for an ⟨X A⟩-arrow α ∶ KM and a collage morphism Φ ∶M→ N ∶XA as in

⌈K⌉K0

yy

K1

%%X M0

//

N0 %%

⌊M⌋⌊Φ⌋

α

AM1oo

N1yy⌊N ⌋

, their composite is the ⟨X A⟩-arrow α Φ ∶ K N with the natural transformationα Φ ∶ K0 N0 → N1 K1 ∶ ⌈K⌉→ ⌊N ⌋ dened by

α Φ = α ⌊Φ⌋

, the usual composite of a natural transformation and a functor.(2) The module ⟨X A⟩ ∶ [X ↓A] [X ∶A] is dened in the following way:

a) an ⟨X A⟩-arrow α ∶ K M from a comma K ∶ X A to a module M ∶ X A isgiven by a cylinder

⌈K⌉K0

K1

$$X M

//

α

A

.

302


b) for a comma morphism Φ ∶ J→ K ∶XA and an ⟨X A⟩-arrow α ∶ KM as in

⌈J⌉⌈Φ⌉J0

J1

⌈K⌉

K0yy K1 %%X M

//

α

A

, their composite is the ⟨X A⟩-arrow Φ α ∶ JM with the cylinder Φ α ∶ J0 J1 ∶⌈J⌉M dened by

Φ α = ⌈Φ⌉ α, the usual composite of a functor and a cylinder (see Denition 4.3.24).

c) for an ⟨X A⟩-arrow α ∶ KM and a module morphism Φ ∶M→ N ∶XA as in

⌈K⌉K0

K1

$$X

M //

NΦ

55

α

A

, their composite is the ⟨X A⟩-arrow α Φ ∶ K N with the cylinder α Φ ∶ K0 K1 ∶ ⌈K⌉ N given by the usual composition of a cylinder and a module morphism(see Denition 4.3.10).

Remark 10.3.2.(1) The unit cylinder (see Denition 10.1.3(1)) of a module (resp. collage) M ∶ X A forms

an ⟨X A⟩-arrow 1↓M ∶M↓ M.(2) The unit cylinder (see Denition 10.1.3(2)) of a comma K ∶XA forms an ⟨X A⟩-arrow

1↑K ∶ K K↑.(3) Since a cylinder

⌈K⌉K0

K1

$$X M

//

α

A

is the same thing as a natural transformation

⌈K⌉K0

vv

K1

((X

M0 ((

A

M1vv⌊M⌋

α

(see Remark 4.3.4(4)), the identity

[X ↓A]1

⟨XA⟩ // [X ∶A]♯

[X ↓A]⟨XA⟩

//

1

[X ↑A]

holds.

303


(4) Each ⟨X A⟩-arrow α ∶ KM gives the following arrows: CYL-arrow α ∶ 1X 1A ∶ KM ↑CYL-arrow α ∶ K0 → K1 ∶ ⌈K⌉M ↓CYL-arrow α ∶M0 →M1 ∶ K ⌊M⌋, dening the faithful cells below:

[X ↓A]

⟨XA⟩ // [X ↑A]

[X ↓A]⌈−⌉

⟨XA⟩ // [X ↑A]

[X ↓A]

⟨XA⟩ // [X ↑A]⌊−⌋

COM CYL//

−

CLG CAT ↑CYL//

−

CLG COM ↓CYL//

−

CAT

Denition 10.3.3. Given an ⟨X A⟩-arrow α ∶ KM, i.e. a natural transformation

⌈K⌉K0

vv

K1

((X

M0 ((

A

M1vv⌊M⌋

α

(1) the comma morphism α↓ ∶ K→M↓ ∶XA, depicted in

⌈K⌉

⌈α↓⌉

K0

vv

K1

((X A

⌈M↓⌉M↓0

hh

M↓1

66

, is dened by the comma adjunct ⌈α↓⌉ ∶ ⌈K⌉ → ⌈M↓⌉ (see Denition 10.1.7(1)) of the↑CYL-arrow α ∶ K0 K1 ∶ ⌈K⌉M.

(2) the collage morphism α↑ ∶ K↑ →M ∶XA, depicted in

⌊K↑⌋

⌊α↑⌋

X

K↑0 66

M0 ((

A

K↑1hh

M1vv⌊M⌋

, is dened by the collage adjunct ⌊α↑⌋ ∶ ⌊K↑⌋ → ⌊M⌋ (see Denition 10.1.7(2)) of the↓CYL-arrow α ∶M0 M1 ∶ K ⌊M⌋.

Proposition 10.3.4.

(1) The assignment α ↦ α↓ yields a bijection from the set of ⟨X A⟩-arrows K M to theset of comma morphisms K→M↓, and the triangle

Kα↓

α

&&M↓1↓M

//M

commutes; the unit cylinder 1↓M thus forms an inverse universal ⟨X A⟩-arrow.

304


(2) The assignment α ↦ α↑ yields a bijection from the set of ⟨X A⟩-arrows K M to theset of collage morphisms K↑ →M, and the triangle

K1↑K //

α %%

K↑

α↑M

commutes; the unit cylinder 1↑K thus forms a direct universal ⟨X A⟩-arrow.

Proof. By the denitions of α↓ and α↑, this is reduced to Proposition 10.1.8.

Proposition 10.3.5.

(1) The functor [X ↓A] ↓←Ð [X ↑A] (see Remark 3.2.26(2)) and the family of unit cylinders1↓M ∶M↓ M, one for each collage M ∶XA, form a counit of the module ⟨X A⟩;

(2) The functor [X ↓A] ↑Ð→ [X ↑A] (see Remark 3.2.26(2)) and the family of unit cylinders1↑K ∶ K K↑, one for each comma K ∶XA, form a unit of the module ⟨X A⟩.

Proof. We have seen in Proposition 10.3.4 the universality of each unit cylinder. It remains toshow that the family of unit cylinders 1↓M (resp. 1↑K) satises the naturality condition. Butthis follows immediately from Proposition 10.1.5.


[X ↓A]⟨XA⟩

//1↓ [X ↑A]↓oo

[X ↓A]⟨XA⟩ //

↑//1↑ [X ↑A]

.

Theorem 10.3.7. Given a pair of categories X and A, there exists an adjoint equivalence

[X ↓A]↑

//(η,ε) [X ↑A]↓oo

with the unit η and the counit ε given by the isomorphisms in Theorem3.2.25.

Proof. This follows by applying Theorem8.2.10 to the counit and unit of ⟨X A⟩ in Re-mark 10.3.6, and observing that the isomorphism εM (resp. ηK) in Theorem3.2.25 gives theadjunct of 1↓M along 1↑M↓ (resp. 1↑K along 1↓K↑) by the commutativity of the triangle in Propo-sition 10.1.6.

Corollary 10.3.8. Given a pair of categories X and A, the module ⟨X A⟩ is an equivalenceand each unit cylinder is a two-way universal ⟨X A⟩-arrow.

Proof. Since the functors [X ↓A] ↑Ð→ [X ↑A] and [X ↓A] ↓←Ð [X ↑A] are equivalences as wehave just seen in Theorem10.3.7, the assertion follows by applying Corollary 8.5.8 to the counitand unit of ⟨X A⟩ in Remark 10.3.6.

305


10.4. Equivalences [X ↓] ≃ [X ∶] and [↓A] ≃ [∶A]Note. The following denition is regarded as a special case of Denition 10.1.3 where A (resp.X) is the terminal category.

Denition 10.4.1.

(1) The unit cone of a right moduleM ∶X ∗ is the cone

⌈M↓⌉M↓

zz

∆

$$X M

//

1↓M

∗

dened by[1↓M]

m= m

for m an arrow ofM. The unit cone of a left moduleM ∶ ∗A is the cone

⌈M↓⌉∆

zz

M↓

$$∗ M//

1↓M

A

dened by[1↓M]

m= m

for m an arrow ofM.(2)

The unit cone of a right comma K ∶X ∗ is the cone

⌈K⌉K

∆

##X

K↑//

1↑K

∗


for k an object of ⌈K⌉. The unit cone of a left comma K ∶ ∗A is the cone

⌈K⌉∆

K

$$∗

K↑//

1↑K

A



Note. The following denition is regarded as a special case of Denition 10.3.1 where A (resp.X) is the terminal category.

306


Denition 10.4.2.

Given a category X, the module ⟨X ⟩ ∶ [X ↓] [X ∶] is dened in the following way:(1) an ⟨X ⟩-arrow α ∶ KM from a right comma K ∶X ∗ to a right moduleM ∶X ∗

is given by a cone⌈K⌉

K

∆

##X M

//

α

∗.

(2) for a right comma morphism Φ ∶ J→ K ∶X ∗ and an ⟨X ⟩-arrow α ∶ KM as in

⌈J⌉⌈Φ⌉J

∆

⌈K⌉

Kyy ∆ %%X M

//

α

∗

, their composite is the ⟨X ⟩-arrow Φ α ∶ JM with the cone Φ α ∶ J ∗ ∶ ⌈J⌉∗Mdened by

Φ α = ⌈Φ⌉ α, the usual composite of a functor and a cone.

(3) for an ⟨X ⟩-arrow α ∶ KM and a right module morphism Φ ∶M→ N ∶X ∗ as in

⌈K⌉K

∆

##X

M //

NΦ

66

α

∗

, their composite is the ⟨X ⟩-arrow αΦ ∶ K N with the cone αΦ ∶ K ∗ ∶ ⌈K⌉∗ Ngiven by the usual composition of a cone and a module morphism (see Denition 4.6.9).

Given a category A, the module ⟨A⟩ ∶ [↓A] [∶A] is dened in the following way:(1) an ⟨A⟩-arrow α ∶ K M from a left comma K ∶ ∗ A to a left moduleM ∶ ∗ A is

given by a cone⌈K⌉

∆

K

$$∗ M

//

α

A

.(2) for a left comma morphism Φ ∶ J→ K ∶ ∗A and an ⟨A⟩-arrow α ∶ KM as in

⌈J⌉⌈Φ⌉

∆

J

⌈K⌉

∆yy K %%∗ M//

α

A

, their composite is the ⟨A⟩-arrow Φ α ∶ JM with the cone Φ α ∶ ∗ J ∶ ∗ ⌈J⌉Mdened by

Φ α = ⌈Φ⌉ α

307


, the usual composite of a functor and a cone.(3) for an ⟨A⟩-arrow α ∶ KM and a left module morphism Φ ∶M→ N ∶ ∗A as in

⌈K⌉∆

K

$$∗ M //

NΦ

66

α

A

, their composite is the ⟨A⟩-arrow αΦ ∶ K N with the cone αΦ ∶ ∗ K ∶ ∗ ⌈K⌉ Ngiven by the usual composition of a cone and a module morphism (see Denition 4.6.9).

Remark 10.4.3.(1) The unit cone of a right moduleM ∶ X ∗ forms an ⟨X ⟩-arrow 1↓M ∶M↓ M. Dually,

the unit cone of a left moduleM ∶ ∗A forms an ⟨A⟩-arrow 1↓M ∶M↓ M.(2) The unit cone of a right comma K ∶X ∗ forms an ⟨X ⟩-arrow 1↑K ∶ K K↑. Dually, the

unit cone of a left comma K ∶ ∗A forms an ⟨A⟩-arrow 1↑K ∶ K K↑.(3) The following identities hold

[X ↓] ⟨X⟩ //

≅

[X ∶]≅

[↓A] ⟨A⟩ //

≅

[∶A]≅

[X ↓ ∗]⟨X∗⟩

//

1

[X ∶ ∗] [∗ ↓A]⟨∗A⟩

//

1

[∗ ∶A]

(≅ denotes the canonical isomorphisms), giving canonical isomorphisms

⟨X ⟩ ≅ ⟨X ∗⟩ and ⟨A⟩ ≅ ⟨∗ A⟩.

Proposition 10.4.4.

(1)

The functor [X ↓] ↓←Ð [X ∶] (see Remark 3.2.20) and the family of unit cones 1↓M ∶M↓ M, one for each right module M ∶X ∗, form a counit of the module ⟨X ⟩;

The functor [↓A] ↓←Ð [∶A] (see Remark 3.2.20) and the family of unit cones 1↓M ∶M↓ M, one for each left module M ∶ ∗A, form a counit of the module ⟨A⟩;

(2)

The functor [X ↓] ↑Ð→ [X ∶] (see Remark 3.2.24) and the family of unit cones 1↑K ∶ K K↑,one for each right comma K ∶X ∗, form a unit of the module ⟨X ⟩.

The functor [↓A] ↑Ð→ [∶A] (see Remark 3.2.24) and the family of unit cones 1↑K ∶ K K↑,one for each left comma K ∶ ∗A, form a unit of the module ⟨A⟩.

Proof. This is a special case of Proposition 10.3.5 where A (resp. X) is the terminal category.


[X ↓]⟨X⟩

//1↓ [X ∶]↓oo

[X ↓]⟨X⟩ //

↑//1↑ [X ∶]

[↓A]⟨A⟩

//1↓ [∶A]↓oo

[↓A]⟨A⟩ //

↑//1↑ [∶A]

.

308


Theorem 10.4.6.

Given a category X, there exists an adjoint equivalence

[X ↓]↑

//(η,ε) [X ∶]↓oo

. Given a category A, there exists an adjoint equivalence

[↓A]↑

//(η,ε) [∶A]↓oo

.

Proof. This is a special case of Theorem10.3.7 where A (resp. X) is the terminal category.

Corollary 10.4.7.

Given a category X, the module ⟨X ⟩ is an equivalence and each unit cone is a two-wayuniversal ⟨X ⟩-arrow.

Given a category A, the module ⟨A⟩ is an equivalence and each unit cone is a two-wayuniversal ⟨A⟩-arrow.

Proof. This is a special case of Corollary 10.3.8 where A (resp. X) is the terminal category.

Note. Since the module ⟨X ⟩ (resp. ⟨A⟩) is an equivalence, Theorem8.6.13 allows the fol-lowing denition.

Denition 10.4.8. Let X and A be categories.(1)

The equivalence cells

[X ↓]1

⟨X↓⟩ // [X ↓]↑

[X ↓]1

⟨X⟩ // [X ∶]↓

[X ↓]⟨X⟩

//

↑

[X ∶] [X ↓]⟨X↓⟩

//

↓

[X ↓]

quasi-inverse to each other are dened by

⟨↑⟩ = [X ↓][1↑] ⟨↓⟩ = ⟨[X ↓][1↓]⟩−1

, where [X ↓][1↑] and [X ↓][1↓] are the module morphisms generated by [X ↓] directalong the unit and counit of ⟨X ⟩ (see Remark 10.4.5).


[↓A]1

⟨↓A⟩ // [↓A]↑

[↓A]1

⟨A⟩ // [∶A]↓

[↓A]⟨A⟩

//

↑

[∶A] [↓A]⟨↓A⟩

//

↓

[↓A]


⟨↑⟩ = [↓A][1↑] ⟨↓⟩ = ⟨[↓A][1↓]⟩−1

, where [↓A][1↑] and [↓A][1↓] are the module morphisms generated by [↓A] directalong the unit and counit of ⟨A⟩ (see Remark 10.4.5).

309


(2) The equivalence cells

[X ∶]↓

⟨X∶⟩ // [X ∶]1

[X ↓]↑

⟨X⟩ // [X ∶]1

[X ↓]⟨X⟩

//

↓

[X ∶] [X ∶]⟨X∶⟩

//

↑

[X ∶]


⟨↓⟩ = [1↓][X ∶] ⟨↑⟩ = ⟨[1↑][X ∶]⟩−1

, where [1↓][X ∶] and [1↑][X ∶] are the module morphisms generated by [X ∶] inversealong the counit and unit of ⟨X ⟩ (see Remark 10.4.5).


[∶A]↓

⟨∶A⟩ // [∶A]1

[↓A]↑

⟨A⟩ // [∶A]1

[↓A]⟨A⟩

//

↓

[∶A] [∶A]⟨∶A⟩

//

↑

[∶A]


⟨↓⟩ = [1↓][∶A] ⟨↑⟩ = ⟨[1↑][∶A]⟩−1

, where [1↓][∶A] and [1↑][∶A] are the module morphisms generated by [∶A] inversealong the counit and unit of ⟨A⟩ (see Remark 10.4.5).

Remark 10.4.9.(1)

The cell ⟨X ↓⟩ ↑ // ⟨X ⟩ sends each right comma morphism Φ ∶ J → K ∶ X ∗ to the

⟨X ⟩-arrow Φ↑ ∶ J K↑, the collage transpose of Φ, given by postcomposition withthe unit cone of K as indicated in

JΦ

Φ↑

%%K

1↑K

// K↑

; the cone Φ↑ ∶ J ∗ ∶ ⌈J⌉∗ K↑ is dened by the composition

⌈J⌉⌈Φ⌉J

∆

⌈K⌉

Kyy ∆ %%X

K↑//

1↑K

∗

, i.e. by[Φ↑]

j= ⌈Φ⌉ ⋅ j

for j an object of ⌈J⌉.

310


The cell ⟨↓A⟩ ↑ // ⟨A⟩ sends each left comma morphism Φ ∶ J → K ∶ ∗ A to the

⟨A⟩-arrow Φ↑ ∶ J K↑, the collage transpose of Φ, given by postcomposition withthe unit cone of K as indicated in

JΦ

Φ↑

%%K

1↑K

// K↑

; the cone Φ↑ ∶ ∗ J ∶ ∗ ⌈J⌉ K↑ is dened by the composition

⌈J⌉⌈Φ⌉

∆

J

⌈K⌉

∆yy K %%∗K↑

//

1↑K

A

, i.e. by[Φ↑]

j= ⌈Φ⌉ ⋅ j

for j an object of ⌈J⌉.(2)

The cell ⟨X ⟩ ↓ // ⟨X ↓⟩ sends each ⟨X ⟩-arrow α ∶ K M to the right comma mor-

phism α↓ ∶ K →M↓ ∶ X ∗, the adjunct of α along the unit cone of M as indicatedin

Kα↓

α

&&M↓1↓M

//M

; α↓ is given by the functor ⌈α↓⌉ ∶ ⌈K⌉→ ⌈M↓⌉ dened by

⌈α↓⌉ ⋅ k = αk ⌈α↓⌉ ⋅ h = K ⋅ h

for k an object and h an arrow of ⌈K⌉ (cf. Denition 10.3.3(1)). The cell ⟨A⟩ ↓ // ⟨↓A⟩ sends each ⟨A⟩-arrow α ∶ KM to the left comma morphism

α↓ ∶ K→M↓ ∶ ∗A, the adjunct of α along the unit cone ofM as indicated in

Kα↓

α

&&M↓1↓M

//M

; α↓ is given by the functor ⌈α↓⌉ ∶ ⌈K⌉→ ⌈M↓⌉ dened by

⌈α↓⌉ ⋅ k = αk ⌈α↓⌉ ⋅ h = K ⋅ h

for k an object and h an arrow of ⌈K⌉ (cf. Denition 10.3.3(1)).(3)

311


The cell ⟨X ∶⟩ ↓ // ⟨X ⟩ sends each right module morphism Φ ∶M → N ∶ X ∗ to the

⟨X ⟩-arrow Φ↓ ∶M↓ N , the comma transpose of Φ, given by precomposition withthe unit cone ofM as indicated in

M↓ 1↓M //

Φ↓ &&

MΦN

; the cone Φ↓ ∶M↓ ∗ ∶ ⌈M↓⌉∗ N is dened by the composition

⌈M↓⌉M↓

zz

∆

$$X

M //

NΦ

55

1↓M

∗

, i.e. by[Φ↓]

m= Φ ⋅m

for m an arrow ofM.

The cell ⟨∶A⟩ ↓ // ⟨A⟩ sends each left module morphism Φ ∶M → N ∶ ∗ A to the

⟨A⟩-arrow Φ↓ ∶M↓ N , the comma transpose of Φ, given by precomposition withthe unit cone ofM as indicated in

M↓ 1↓M //

Φ↓ &&

MΦN

; the cone Φ↓ ∶ ∗M↓ ∶ ∗ ⌈M↓⌉ N is dened by the composition

⌈M↓⌉∆

zz

M↓

$$∗ M //

NΦ

55

1↓M

A

, i.e. by[Φ↓]

m= Φ ⋅m

for m an arrow ofM.(4)

The cell ⟨X ⟩ ↑ // ⟨X ∶⟩ sends each ⟨X ⟩-arrow α ∶ K M to the right module mor-

phism α↑ ∶ K↑ →M ∶ X ∗, the adjunct of α along the unit cone of K as indicatedin

K1↑K //

α %%

K↑

α↑M

; α↑ is dened byα↑ ⋅ k = αk

for k an object of ⌈K⌉ (cf. Denition 10.3.3(2)).

312


The cell ⟨A⟩ ↑ // ⟨∶A⟩ sends each ⟨A⟩-arrow α ∶ KM to the left module morphism

α↑ ∶ K↑ →M ∶ ∗A, the adjunct of α along the unit cone of K as indicated in

K1↑K //

α %%

K↑

α↑M

; α↑ is dened byα↑ ⋅ k = αk

for k an object of ⌈K⌉ (cf. Denition 10.3.3(2)).

313

11. Extensions

11.1. coYoneda lemma

Denition 11.1.1. LetM ∶XA be a module. The corepresentable module of the right exponential transpose of M is denoted by M;

that is, the module⟨M⟩ ∶ [X ∶]A


[X ∶] ⟨X∶⟩ // [X ∶] AMoo

. The representable module of the left exponential transpose ofM is denoted by M; that

is, the module⟨M⟩ ∶X [∶A]∼


XM // [∶A]∼ ⟨∶A⟩∼ // [∶A]∼

.

Remark 11.1.2.(1)

Given a right module J ∶X ∗ and an object a ∈ ∥A∥, an ⟨M⟩-arrow Φ ∶ J a is aright module morphism Φ ∶ J → ⟨M⟩a ∶X ∗, i.e. a conical cell

X

1

J // ∗a

X M//

Φ

A

. This cell is denoted by Φ ∶ J a ∶M, depicted also as

∗a

##X M

//

J;;

Φ

A

, and called a conical cell from J to a alongM. Given a left module J ∶ ∗ A and an object x ∈ ∥X∥, an ⟨M⟩-arrow Φ ∶ x J is a

left module morphism Φ ∶ J → x ⟨M⟩ ∶ ∗A, i.e. a conical cell

∗x

J //A

1

X M//

Φ

A

314

11. Extensions

. This cell is denoted by Φ ∶ x J ∶M, depicted also as

∗x

J##

X M//

Φ

A

, and called a conical cell from x to J alongM.(2)

A conical cell

X

P

M // ∗b

Y N//

Φ

B

from P to b along ⟨M,N ⟩ can be also depicted as a conical cell

∗b

##X

P⟨N ⟩//

M;;

Φ

B

from M to b along P ⟨N ⟩, both being dened by a right module morphism Φ ∶M →P ⟨N ⟩b ∶X ∗.

A conical cell

∗y

M //A

Q

Y N//

Φ

B

from y to Q along ⟨M,N ⟩ can be also depicted as a conical cell

∗y

M##

Y⟨N ⟩Q

//Φ

A

from y to M along ⟨N ⟩Q, both being dened by a left module morphism Φ ∶ M →y ⟨N ⟩Q ∶ ∗A.

(3) With the notation introduced above, the right and left Yoneda morphisms for M aredepicted as

X

X

M //A

1

X

1

M //A

A

[X ∶] M//

⟨XM⟩

A X M//

⟨MA⟩

[∶A]∼

so that they send eachM-arrows m ∶ s t to the conical cells

∗t

##

∗s

t⟨A⟩

##X M

//

⟨X⟩s;;Xm

A X M//

mA

A

. With this depiction of Yoneda morphisms, Theorem5.2.11 is restated as follows:

315

11. Extensions

for any pair of objects s ∈ ∥X∥ and t ∈ ∥A∥, the assignment m↦Xm yields a bijection

s ⟨M⟩ t ≅ (⟨X⟩ s) ⟨M⟩ (t)

from the set ofM-arrows s t to the set of conical cells ⟨X⟩ s t alongM; moreover,the bijection is natural in s and t.

for any pair of objects s ∈ ∥X∥ and t ∈ ∥A∥, the assignment m↦ mA yields a bijection

s ⟨M⟩ t ≅ (s) ⟨M⟩ (t ⟨A⟩)

from the set ofM-arrows s t to the set of conical cells s t ⟨A⟩ alongM; moreover,the bijection is natural in s and t.

(4) For any moduleM,⟨M⟩∼ = ⟨M∼⟩

.


Denition 11.1.3.

The corepresentable module of the right Yoneda functor for a category X is denoted byX; that is, the module

⟨X⟩ ∶ [X ∶]X


[X ∶] ⟨X∶⟩ // [X ∶] XXoo

. The representable module of the left Yoneda functor for a category A is denoted by A;

that is, the module⟨A⟩ ∶A [∶A]∼


AA // [∶A]∼ ⟨∶A⟩∼ // [∶A]∼

.

Remark 11.1.4.(1) The identities

⟨X⟩ = ⟨⟨X⟩⟩ ⟨A⟩ = ⟨⟨A⟩⟩follow from the identities

[X] = [⟨X⟩] [A] = [⟨A⟩]

(see Denition 2.3.1). Hence the module X (resp. A) is a special instance of a moregeneral module in Denition 11.1.1 whereM is given by the hom of X (resp. A).

(2)

316

11. Extensions

Given a right module M ∶ X ∗ and an object x ∈ ∥X∥, an ⟨X⟩-arrow Φ ∶M xis a right module morphism Φ ∶M → ⟨X⟩x ∶ X ∗. This right module morphism isdenoted by Φ ∶M x ∶X, depicted variously as

X

1

M // ∗x

∗x

##X

⟨X⟩//

Φ

X X⟨X⟩

//

M;;

Φ

X XM //Φ ∗x

oo

, and called a conical cell fromM to x. Given a left moduleM ∶ ∗ A and an object a ∈ ∥A∥, an ⟨A⟩-arrow Φ ∶ a M is a

left module morphism Φ ∶M → a ⟨A⟩ ∶ ∗ A. This left module morphism is denotedby Φ ∶ aM ∶A, depicted variously as

∗a

M //A

1

∗a

M##

A⟨A⟩

//

Φ

A A⟨A⟩

//

Φ

A ∗a //

M//Φ A

, and called a conical cell from a toM.

Denition 11.1.5. LetM ∶XA be a module. The module

⟨M⟩ ∶ [X ↓]A


[X ↓] ⟨X⟩ // [X ∶] AMoo

(see Denition 10.4.2 for ⟨X ⟩). The module

⟨M⟩ ∶X [↓A]∼


XM // [∶A]∼ ⟨A⟩∼ // [↓A]∼

(see Denition 10.4.2 for ⟨A⟩).

Remark 11.1.6.(1)

Given a right comma K ∶X ∗ and an object a ∈ ∥A∥, an ⟨M⟩-arrow α ∶ K a is acone

⌈K⌉K

∆ // ∗a

X M//

α

A

from the right comma bration K ∶ ⌈K⌉→X to a alongM.

317

11. Extensions

Given a left comma K ∶ ∗ A and an object x ∈ ∥X∥, an ⟨M⟩-arrow α ∶ x K is acone

∗x

⌈K⌉∆oo

K

X M//

α

A

from x to the left comma bration K ∶ ⌈K⌉→A alongM.(2)

The module ⟨M⟩ ∶ [X ↓] A is corepresented by the functor [M] ∶ A → [X ↓],the right comma exponential transpose of M (see Denition 3.2.27); indeed, a moduleisomorphism

⟨M⟩ ≅ ⟨X ↓⟩ [M]is obtained from the equivalence cell in Denition 10.4.8(1) by the pasting composition

[X ↓]1

⟨X⟩ // [X ∶]↓

A

1

Moo

[X ↓]⟨X↓⟩

//

↓

[X ↓] AMoo

(see Remark 1.2.32). The module ⟨M⟩ ∶ X [↓A]∼ is represented by the functor [M] ∶ X → [↓A]∼,

the left comma exponential transpose of M (see Denition 3.2.27); indeed, a moduleisomorphism

⟨M⟩ ≅ [M] ⟨↓A⟩∼

is obtained from the equivalence cell in Denition 10.4.8(1) by the pasting composition

XM //

1

[∶A]∼

↓

⟨A⟩∼ // [↓A]∼

1

X M// [↓A]∼

⟨↓A⟩∼//

↓

[↓A]∼

(see Remark 1.2.32).(3)

The modules ⟨M⟩ ∶ [X ∶] A and ⟨M⟩ ∶ [X ↓] A are equivalent. Indeed, a pairof equivalence cells

[X ∶]↓

M //A

1

[X ↓]↑

M //A

1

[X ↓] M//

↓M

A [X ∶] M//

↑M

A

quasi-inverse to each other is obtained from the pair of equivalence cells in Deni-tion 10.4.8(2) by the pasting compositions

[X ∶]↓

⟨X∶⟩ // [X ∶]1

A

1

Moo [X ↓]↑

⟨X⟩ // [X ∶]1

A

1

Moo

[X ↓]⟨X⟩

//

↓

[X ∶] AMoo [X ∶]

⟨X∶⟩//

↑

[X ∶] AMoo

318

11. Extensions

. By this construction, we see that the equivalence ⟨M⟩ ≃ ⟨M⟩ is natural in M.The cell ↓M sends each conical cell

∗a

##X M

//

J;;

Φ

A

to its comma transpose, i.e. the cone

⌈J ↓⌉J ↓

zz

∆a

$$X M

//

Φ↓

A

(cf. Remark 10.4.9(3)). The modules ⟨M⟩ ∶ X [∶A]∼ and ⟨M⟩ ∶ X [↓A]∼ are equivalent. Indeed, a

pair of equivalence cells

X

1

M // [∶A]∼

↓

X

1

M // [↓A]∼

↑

X M//

↓M

[↓A]∼ X M//

↑M

[∶A]∼

quasi-inverse to each other is obtained from the pair of equivalence cells in Deni-tion 10.4.8(2) by the pasting compositions

XM //

1

[∶A]∼ ⟨∶A⟩∼ //

1

[∶A]∼

↓

XM //

1

[∶A]∼ ⟨A⟩∼ //

1

[↓A]∼

↑

X M// [∶A]∼

⟨A⟩∼//

↓

[↓A]∼ X M// [∶A]∼

⟨∶A⟩∼//

↑

[∶A]∼

. By this construction, we see that the equivalence ⟨M⟩ ≃ ⟨M⟩ is natural in M.The cell ↓M sends each conical cell

∗x

J##

X M//

Φ

A

to its comma transpose, i.e. the cone

⌈J ↓⌉∆x

zz

J ↓

$$X M

//

Φ↓

A

(cf. Remark 10.4.9(3)).(4) The equivalence in (3) above (to be precise, a special case whereM is given by the hom of

a category) is called the coYoneda lemma in [ML98]. The composition of the equivalencecell ↓M∶ ⟨M⟩ ⟨M⟩ and the right Yoneda morphism in Remark 11.1.2(3) yields afully faithful cell

X

X

M //A

1

[X ↓] M//

⟨XM⟩

A

319

11. Extensions

(see Denition 3.2.29 for X) making the diagram

M⟨XM⟩yy

⟨XM⟩%%

M↓M //M↑M

oo

commute up to isomorphism. The cell ⟨X M⟩ establishes a bijection from the set ofM-arrows s t to the set of cones [X↓s] t along M. It may be more appropriateto call ⟨X M⟩ coYoneda; however, it is the equivalence ⟨M⟩ ≃ ⟨M⟩ that playsa critical role in Section 11.2 providing a way to transform a weighted limit to a conicallimit.

Denition 11.1.7. Let E be a category andM ∶XA be a module. The corepresentable module of the right action ofM on [E,A] is denoted byME; that

is the module⟨ME⟩ ∶ [X ∶ E] [E,A]


[X ∶ E] ⟨X∶E⟩ // [X ∶ E] [E,A]MEoo

. The representable module of the left action of M on [E,X] is denoted by EM; that is

the module⟨EM⟩ ∶ [E,X] [E ∶A]∼


[E,X] EM // [E ∶A]∼ ⟨E∶A⟩∼ // [E ∶A]∼

.

Remark 11.1.8.(1)

Given a module J ∶X E and a functor K ∶ E →A, an ⟨ME⟩-arrow Φ ∶ J K is amodule morphism Φ ∶ J → ⟨M⟩K ∶X E, i.e. a cell

X

1

J // E

K

X M//

Φ

A

. This cell is denoted by Φ ∶ J K ∶ME, depicted also as

EK

##X M

//

J;;

Φ

A

, and called a cell from J to K alongM.

320

11. Extensions

Given a module J ∶ E A and a functor K ∶ E →X, an ⟨EM⟩-arrow Φ ∶ K J is amodule morphism Φ ∶ J → K ⟨M⟩ ∶ EA, i.e. a cell

E

K

J //A

1

X M//

Φ

A

. This cell is denoted by Φ ∶ K J ∶ EM, depicted also as

EK

J##

X M//

Φ

A

, and called a cell from K to J alongM.(2) A cell

X

P

M //A

Q

Y N//

Φ

B

from P to Q along ⟨M,N ⟩, i.e. a module morphism Φ ∶M→ P ⟨N ⟩Q ∶XA, can be alsodepicted as a cell

AQ

##X

P⟨N ⟩//

M;;

Φ

B

fromM to Q along P ⟨N ⟩. a cell

XP

M##

Y⟨N ⟩Q

//

Φ

A

from P toM along ⟨N ⟩Q.(3)

The right exponential transpose of a cell

EK

##X M

//

J;;

Φ

A

, i.e. a module morphism Φ ∶ J → ⟨M⟩K ∶X E, is a natural transformation

EJzz

K

""[X ∶]

Φ

AMoo

321

11. Extensions

from J to [M] K (see Proposition 2.1.6), i.e. a cylinder

EJzz

K

""[X ∶] M

//

Φ

A

. The slice of Φ at e ∈ ∥E∥ is the conical cell

∗K ⋅ e##

X

⟨J ⟩e;;

M//

⟨Φ⟩e

A

given by the component of the cylinder Φ at e. The right exponential transpositionΦ↦ Φ yields an iso cell

[X ∶ E]

⟨ME⟩ // [E,A]1

[E, [X ∶]]⟨E,⟨M⟩⟩

//

[E,A]

giving a canonical isomorphism

⟨ME⟩ ≅ ⟨E, ⟨M⟩⟩

. The left exponential transpose of a cell

EK

J##

X M//

Φ

A

, i.e. a module morphism Φ ∶ J → K ⟨M⟩ ∶ EA, is a natural transformation

EK

||

J%%

X M//

Φ

[∶A]∼

from K [M] to J (see Proposition 2.1.6), i.e. a cylinder

EK

||

J%%

X M//

Φ

[∶A]∼

. The slice of Φ at e ∈ ∥E∥ is the conical cell

∗e ⋅K

e⟨J ⟩

##X M

//

e⟨Φ⟩

A

322

11. Extensions

given by the component of the cylinder Φ at e. The left exponential transpositionΦ↦Φ yields an iso cell

[E,X]1

⟨EM⟩ // [E ∶A]∼

[E,X]⟨E,⟨M⟩⟩

//

[E, [∶A]∼]

giving a canonical isomorphism

⟨EM⟩ ≅ ⟨E, ⟨M⟩⟩

.(4) For any category E and any moduleM,

⟨ME⟩∼ ≅ ⟨E∼M∼⟩

.(5) The module ⟨M⟩ ∶ [X ∶] A in Denition 11.1.1 is identied with ⟨M∗⟩ ∶ [X ∶ ∗]

[∗,A]; that is, the module ⟨M⟩ is regarded as a special instance of a more general⟨ME⟩ where E is the terminal category. Dually, the module ⟨M⟩ ∶ X [∶A]∼ isidentied with ⟨∗M⟩ ∶ [∗,X] [∗ ∶A]∼.


Denition 11.1.9.

Given categories A and X, the corepresentable module of the right general Yoneda functorfor [A,X] is denoted by XA; that is the module

⟨XA⟩ ∶ [X ∶A] [A,X]


[X ∶A] ⟨X∶A⟩ // [X ∶A] [A,X]XAoo

. Given categories X and A, the representable module of the left general Yoneda functor for

[X,A] is denoted by XA; that is the module

⟨XA⟩ ∶ [X,A] [X ∶A]∼


[X,A] XA // [X ∶A]∼ ⟨X∶A⟩∼ // [X ∶A]∼

.

Remark 11.1.10.

323

11. Extensions

(1) The identities⟨XA⟩ = ⟨⟨X⟩A⟩ ⟨XA⟩ = ⟨X⟨A⟩⟩

follow from the identities

[XA] = [⟨X⟩A] [XA] = [X⟨A⟩](see Denition 2.3.5). Hence the module XA (resp. XA) is a special instance of amore general module in Denition 11.1.7 whereM is given by the hom of X (resp. A).

(2) Given a module M ∶ X A and a functor K ∶ A → X, an ⟨XA⟩-arrow Φ ∶M K

is a module morphism Φ ∶M → ⟨X⟩K ∶ X A. This module morphism is denoted byΦ ∶M K ∶XA, depicted variously as

X

1

M //A

K

AK

##X

⟨X⟩//

Φ

X X⟨X⟩

//

M;;

Φ

X XM //Φ AK

oo

, and called a cell fromM to K. Given a module M ∶ X A and a functor K ∶ X → A, an ⟨XA⟩-arrow Φ ∶ K M

is a module morphism Φ ∶M → K ⟨A⟩ ∶ X A. This module morphism is denoted byΦ ∶ KM ∶XA, depicted variously as

X

K

M //A

1

XK

M##

A⟨A⟩

//

Φ

A A⟨A⟩

//

Φ

A XK //

M//Φ A

, and called a cell from K toM.

11.2. Weighted limits

Note. Denition 11.2.1 and Denition 11.2.9 give two denitions of universal conical cells. Wewill see that they are equivalent in Theorem11.2.11.

Denition 11.2.1.

A conical cell∗

r

##X M

//

J;;

Υ

A

is called universal if it is a direct universal ⟨M⟩-arrow (see Denition 11.1.1). Given aright module J ∶ X ∗, a universal conical cell Υ ∶ J r ∶M or the pair (r,Υ), or theobject r itself, is called a colimit of J alongM.

A conical cell∗

r

J##

X M//

Υ

A

is called universal if it is an inverse universal ⟨M⟩-arrow (see Denition 11.1.1). Given aleft module J ∶ ∗ A, a universal conical cell Υ ∶ r J ∶M or the pair (r,Υ), or theobject r itself, is called a limit of J alongM.

324

11. Extensions

Remark 11.2.2. A conical cell Φ ∶ J r ∶M is universal if and only if to every conical cellΦ ∶ J a ∶M there is a unique A-arrow Υ/Φ ∶ r → a such that Φ = Υ Υ/Φ. Dually, aconical cell Φ ∶ r J ∶M is universal if and only if to every conical cell Φ ∶ x J ∶Mthere is a unique X-arrow Φ/Υ ∶ x→ r such that Φ = Φ/Υ Υ.

Proposition 11.2.3.

A conical cell∗

r

##X M

//

J;;

Υ

A

is universal if and only if its comma transpose

⌈J ↓⌉J ↓

zz

∆r

$$X M

//

Υ↓

A

(see Remark 11.1.6(3)) is a universal cone. A conical cell

∗r

J##

X M//

Υ

A


⌈J ↓⌉∆r

zz

J ↓

$$X M

//

Υ↓

A

(see Remark 11.1.6(3)) is a universal cone.

Proof. By Theorem8.6.4, the equivalence cell ↓M∶ ⟨M⟩ ⟨M⟩ in Remark 11.1.6(3) pre-serves and reects direct universals.

Theorem 11.2.4. Let M ∶XA be a module. An M-arrow u ∶ x r is direct universal if and only if the conical cell

∗r

##X M

//

⟨X⟩x;;Xu

A

(see Remark 11.1.2(3)) is universal. An M-arrow u ∶ r a is inverse universal if and only if the conical cell

∗r

a⟨A⟩

##X M

//

uA

A

(see Remark 11.1.2(3)) is universal.

325

11. Extensions

Proof. Apply Proposition 6.2.21 to the fully faithful cell in Remark 11.1.2(3).


Denition 11.2.5.

A conical cell∗

r

##X

⟨X⟩//

M;;

Υ

X

is called universal if it is a direct universal ⟨X⟩-arrow (see Denition 11.1.3). Given aright moduleM ∶X ∗, a universal conical cell Υ ∶M r ∶X or the pair (r,Υ), or theobject r itself, is called a colimit ofM in X.

A conical cell∗

r

M##

A⟨A⟩

//Υ

A

is called universal if it is an inverse universal ⟨A⟩-arrow (see Denition 11.1.3). Given aleft module M ∶ ∗ A, a universal conical cell Υ ∶ r M ∶A or the pair (r,Υ), or theobject r itself, is called a limit ofM in A.

Remark 11.2.6. A colimit of a right moduleM ∶X ∗ in X is the same thing as a colimit ofM along the hom endomodule ⟨X⟩. Dually, a limit of a left module M ∶ ∗ A in A is thesame thing as a limit ofM along the hom endomodule ⟨A⟩.Proposition 11.2.7.

A representation (r,Υ) of a right module M ∶X ∗ gives a colimit

∗r

##X

⟨X⟩//

M;;

Υ

X

of M in X (cf. Remark 2.3.4(2)). A representation (r,Υ) of a left module M ∶ ∗A gives a limit

∗r

M##

A⟨A⟩

//Υ

A

of M in A (cf. Remark 2.3.4(2)).

Proof. Since Υ ∶M → ⟨X⟩ r ∶ X ∗ is an isomorphism and the Yoneda functor X is fullyfaithful, Υ is a direct universal ⟨X⟩-arrow by Corollary 6.2.15.

Remark 11.2.8. The converse does not hold in general. For example, consider a conical cell

∗1

##∗⟨∗⟩

//

0,1 ;;∆

∗

given by the unique function ∆ from the set 0,1 to the hom ⟨∗⟩ of the terminal category.The cell, being the only cell 0,1 ⟨∗⟩, forms a colimit of 0,1 but not a representation.

326

11. Extensions

Denition 11.2.9.

A conical cell

EJ //

K

∗r

X M//

Υ

A

is called universal if it is a direct universal ⟨J ,M⟩-arrow (see Denition 1.2.8). Given afunctor K ∶ E → X, a universal conical cell Υ ∶ K r ∶ J M or the pair (r,Υ), or theobject r itself, is called a colimit of K weighted by J (or J -weighted colimit of K) alongM, with the object r denoted by ∐J K.

A conical cell

∗ J //

r

E

K

X M//

Υ

A

is called universal if it is an inverse universal ⟨J ,M⟩-arrow (see Denition 1.2.8). Given afunctor K ∶ E → A, a universal conical cell Υ ∶ r K ∶ J M or the pair (r,Υ), or theobject r itself, is called a limit of K weighted by J (or J -weighted limit of K) along M,with the object r denoted by ∏J K.

Remark 11.2.10.(1) A conical cell Φ ∶ K r ∶ J M is universal if and only if to every conical cell Φ ∶ K a ∶J M there is a unique A-arrow Υ/Φ ∶ r → a such that Φ = Υ Υ/Φ. Dually, a conicalcell Φ ∶ r K ∶ J M is universal if and only if to every conical cell Φ ∶ x K ∶ J Mthere is a unique X-arrow Φ/Υ ∶ x→ r such that Φ = Φ/Υ Υ.

(2) As a special case whereM is given by the hom of a category C, a universal conical cell

EJ //

K

∗r

C⟨C⟩

//

Υ

C

or the pair (r,Υ), or the object r itself, is called a J -weighted colimit of K in C. a universal conical cell

∗ J //

r

E

K

C⟨C⟩

//

Υ

C

or the pair (r,Υ), or the object r itself, is called a J -weighted limit of K in C.

Theorem 11.2.11.

A conical cell

EJ //

K

∗r

X M//

Υ

A

from K to r along ⟨J ,M⟩ is universal if and only if the conical cell

∗r

##E

K⟨M⟩//

J;;

Υ

A

327

11. Extensions

from J to r along K ⟨M⟩ is universal; hence a J -weighted colimit of K alongM is the samething as a colimit of J along K ⟨M⟩. Conversely, a conical cell

∗r

##X M

//

J;;

Υ

A

from J to r along M is universal if and only if the conical cell

XJ //

1

∗r

X M//

Υ

A

from the identity X → X to r along ⟨J ,M⟩ is universal; hence a colimit of J along M isthe same thing as a J -weighted colimit of the identity X→X along M.

A conical cell

∗ J //

r

E

K

X M//

Υ

A

from r to K along ⟨J ,M⟩ is universal if and only if the conical cell

∗r

J##

X⟨M⟩K

//Υ

E

from r to J along ⟨M⟩K is universal; hence a J -weighted limit of K along M is the samething as a limit of J along ⟨M⟩K. Conversely, a conical cell

∗r

J##

X M//

Υ

A

from r to J along M is universal if and only if the conical cell

∗ J //

r

A

1

X M//

Υ

A

from r to the identity A→A along ⟨J ,M⟩ is universal; hence a limit of J alongM is thesame thing as a J -weighted limit of the identity A→A along M.

Proof. The left slice of the module ⟨J ,M⟩ at K and the left slice of the module ⟨K ⟨M⟩⟩ atJ are the same left module over A given by

a↦ (J ) ⟨E ∶⟩ (K ⟨M⟩a)

. Hence Υ ∶ K r is a direct universal ⟨J ,M⟩-arrow i Υ ∶ J r is a direct universal⟨⟨K ⟨M⟩⟩⟩-arrow.

328

11. Extensions

The left slice of the module M at J and the left slice of the module ⟨J ,M⟩ at 1X arethe same left module over A given by

a↦ (J ) ⟨X ∶⟩ (⟨M⟩a)

. Hence Υ ∶ J r is a direct universal ⟨M⟩-arrow i Υ ∶ 1X r is a direct universal⟨J ,M⟩-arrow.

Theorem 11.2.12.

A conical cell

EJ //

K

∗r

X M//

Υ

A


E

K

⌈J ↓⌉J ↓oo

∆r

X M//

Υ↓

A

is a universal cone. A conical cell

∗ J //

r

E

K

X M//

Υ

A


⌈J ↓⌉ J ↓ //

∆r

E

K

X M//

Υ↓

A

is a universal cone.

Proof. Since a conical cell Υ ∶ K r ∶ J M is universal i the conical cell

∗r

##E

K⟨M⟩//

J;;

Υ

A

is universal (see Theorem11.2.11), and since the cone Υ↓ ∶ J ↓ K r ∶ ⌈J ↓⌉∗M is universali the cone

⌈J ↓⌉J ↓

zz

∆r

$$X

K⟨M⟩//

Υ↓

A

is universal (see Theorem7.1.3), the assertion is reduced to Proposition 11.2.3.

329

11. Extensions

Remark 11.2.13. By Theorem11.2.12 and by the bijectiveness of comma transposition, we seethat a J -weighted colimit of K alongM is the same thing as a colimit of J ↓ K alongM. a J -weighted limit of K alongM is the same thing as a limit of K J ↓ alongM.Limits thus subsume weighted limits. We will see that the converse is also the case in Theo-rem11.2.15.

Corollary 11.2.14. Let J andM be modules and K be a functor as in Theorem11.2.12. IfMis cocomplete (resp. complete) and J is small, then K has a J -weighted colimit (resp. limit)along M.

Proof. By Remark 11.2.13, it suces to show that J ↓ K ∶ ⌈J ↓⌉ → X has a colimit along M.But since the smallness of J guarantees the smallness of the comma category ⌈J ↓⌉, this holdsby the cocompleteness ofM.

Theorem 11.2.15.

A cone

E

K

∆E // ∗r

X M//

µ

A

is universal if and only if the conical cell

E∆E∗ //

K

∗r

X M//

⟨µ⟩

A

(see Corollary 5.5.6) is universal. A cone

∗r

E

K

∆Eoo

X M//

µ

A


∗r

∆∗E // E

K

X M//

⟨µ⟩

A

(see Corollary 5.5.6) is universal.

Proof. Immediate by the isomorphism in Corollary 5.5.6.

Remark 11.2.16. By Theorem11.2.15 and by the bijectiveness of the assignment µ ↦ ⟨µ⟩, wesee that given a moduleM ∶XA, a colimit of a functor K ∶ E → X along M is the same thing as an ⟨∆E∗⟩-weighted colimit

of K alongM. a limit of a functor K ∶ E → A alongM is the same thing as an ⟨∆∗E⟩-weighted limit of K

alongM.

330

11. Extensions

Weighted limits thus subsume limits, and vice versa as we saw in Remark 11.2.13.


XM //

P

A

Q

Y N//

Φ

B

is said to preserve (reect, create) colimits weighted by a right module J ∶ E ∗ if the postcomposi-

tion cell

[E,X] ⟨J ,M⟩ //

[E,P]

[∗,A][∗,Q]

[E,Y]⟨J ,N ⟩

//

⟨J ,Φ⟩

[∗,B]

(see Denition 1.2.23) preserves (reects, creates) direct universals. preserve (reect, create) limits weighted by a left module J ∶ ∗ E if the postcomposition

cell

[∗,X] ⟨J ,M⟩ //

[∗,P]

[E,A][E,Q]

[∗,Y]⟨J ,N ⟩

//

⟨J ,Φ⟩

[E,B]

(see Denition 1.2.23) preserves (reects, creates) inverse universals.


a) preserve J -weighted colimits if each universal conical cell Υ ∶ K r ∶ J M yields by

composition with Φ a universal conical cell Υ Φ ∶ K P Q ⋅ r ∶ J N . J -weighted limits if each universal conical cell Υ ∶ r K ∶ J M yields by

composition with Φ a universal conical cell Υ Φ ∶ r ⋅ P Q K ∶ J N .b) reect

J -weighted colimits if a conical cell Υ ∶ K r ∶ J M is universal whenever theconical cell Υ Φ ∶ K P Q ⋅ r ∶ J N is universal.

J -weighted limits if a conical cell Υ ∶ r K ∶ J M is universal whenever theconical cell Υ Φ ∶ r ⋅ P Q K ∶ J N is universal.

c) create J -weighted colimits if for every functor K ∶ E → X and for every universal conical

cell Θ ∶ K P s ∶ J N there is exactly one conical cell Υ ∶ K r ∶ J M withΥ Φ = Θ, and if this Υ is universal.

J -weighted limits if for every functor K ∶ E→A and for every universal conical cellΘ ∶ s Q K ∶ J N there is exactly one conical cell Υ ∶ r K ∶ J M withΥ Φ = Θ, and if this Υ is universal.

(2) A cell Φ is said to preserve (reect, create) weighted colimits if it preserves (reects, creates) J -weighted

colimits for any right module J . preserve (reect, create) weighted limits if it preserves (reects, creates) J -weighted

limits for any left module J .

331

11. Extensions

Proposition 11.2.19. A cell Φ preserves (reects, creates) weighted colimits (resp. weightedlimits) if and only if Φ preserves (reects, creates) colimits (resp. limits).

Proof.(⇒) By Theorem11.2.15 and since the isomorphism in Corollary 5.5.6 is natural inM, for any

category E, if a cell Φ ∶M N preserves (reects, creates) colimits weighted by ∆E∗, thenit preserves (reects, creates) colimits over E.

(⇐) By Theorem11.2.12 and since the equivalence ⟨M⟩ ≃ ⟨M⟩ is natural in M (seeRemark 11.1.6(3)), for any right module J , if a cell Φ ∶M N preserves (reects, creates)colimits over ⌈J ↓⌉, then it preserves (reects, creates) colimits weighted by J .

Theorem 11.2.20.

For any right module M ∶X ∗, the following conditions are equivalent:(1) M is representable;(2) M preserves colimits and M has a colimit (in the sense of Denition 11.2.5) in X;(3) M preserves colimits and the right comma brationM↓ ∶ ⌈M↓⌉→X has a colimit in X.When these conditions hold, a conical cell

∗r

##X

⟨X⟩//

M;;

Υ

X

is universal if and only if the right module morphism Υ ∶M → ⟨X⟩ r is iso; that is, if andonly if the pair (r,Υ) forms a representation of M.

For any left module M ∶ ∗A, the following conditions are equivalent:(1) M is representable;(2) M preserves limits and M has a limit (in the sense of Denition 11.2.5) in A;(3) M preserves limits and the left comma bration M↓ ∶ ⌈M↓⌉→A has a limit in A.When these conditions hold, a conical cell

∗r

M##

A⟨A⟩

//Υ

A

is universal if and only if the left module morphism Υ ∶M → r ⟨A⟩ is iso; that is, if andonly if the pair (r,Υ) forms a representation of M.

Proof.(2)⇔ (3) By Proposition 11.2.3.(1)⇒ (3) M preserves colimits by Corollary 7.4.18. Since ⌈M↓⌉ has a terminal object by

Proposition 6.1.4,M↓ ∶ ⌈M↓⌉→X has a colimit by Corollary 9.1.20.(3)⇒ (1) M↓ ∶ ⌈M↓⌉ → X creates a colimit by Corollary 7.4.22; in particular, M↓ creates a

colimit of the identity ⌈M↓⌉ → ⌈M↓⌉ from its own colimit. Hence ⌈M↓⌉ has a terminalobject by Theorem9.1.21, andM is representable by Proposition 6.1.4.

The if part of the second assertion is already seen in Proposition 11.2.7. Since a universalconical cell Υ ∶M r ∶X is the same thing as a direct universal ⟨X⟩-arrow, the only ifpart follows by applying Theorem6.2.9 to the right Yoneda functor X.

332

11. Extensions

11.3. Extensions

Note. Denition 11.3.1 and Denition 11.3.12 give two denitions of universal cells. We willsee that they are equivalent in Theorem11.3.14.

Denition 11.3.1.

A cellE

R

##X M

//

J;;

Υ

A

alongM is called(1) universal if it is a direct universal ⟨ME⟩-arrow (see Denition 11.1.7);(2) pointwise universal if each slice

∗R ⋅ e##

X

⟨J ⟩e;;

M//

⟨Υ⟩e

A

(see Remark 11.1.8(3)) is a universal conical cell (see Denition 11.2.1), i.e. a directuniversal ⟨M⟩-arrow.

Given a module J ∶ X E, a universal (resp. pointwise universal) cell Υ ∶ J R ∶MEor the pair (R,Υ), or the functor R itself, is called an extension (resp. pointwise extension)of J direct alongM.

A cellE

R

J##

X M//

Υ

A

alongM is called(1) universal if it is an inverse universal ⟨EM⟩-arrow (see Denition 11.1.7);(2) pointwise universal if each slice

∗e ⋅ R

e⟨J ⟩

##X M

//

e⟨Υ⟩

A

(see Remark 11.1.8(3)) is a universal conical cell (see Denition 11.2.1), i.e. an inverseuniversal ⟨M⟩-arrow.

Given a module J ∶ E A, a universal (resp. pointwise universal) cell Υ ∶ R J ∶ EMor the pair (R,Υ), or the functor R itself, is called an extension (resp. pointwise extension)of J inverse alongM.

Remark 11.3.2. A cell Φ ∶ J R ∶ME is universal if and only if to every cell Φ ∶ J K ∶ME there is a unique natural transformation Υ/Φ ∶ R → K such that Φ = Υ Υ/Φ. Dually,a cell Φ ∶ R J ∶ EM is universal if and only if to every cell Φ ∶ K J ∶ EM there is aunique natural transformation Φ/Υ ∶ K→ R such that Φ = Φ/Υ Υ.

Proposition 11.3.3.

333

11. Extensions

A cellE

R

##X M

//

J;;

Υ

A

is universal (resp. pointwise universal) if and only if its right exponential transpose

EJzz

R

""[X ∶]

Υ

AMoo

forms a left Kan lift (resp. pointwise left Kan lift) of J alongM, i.e. if and only if thecylinder

EJzz

R

""[X ∶] M

//

Υ

A

is direct universal (resp. pointwise direct universal). A cell

ER

J##

X M//

Υ

A

is universal (resp. pointwise universal) if and only if its left exponential transpose

ER

||

J%%

X M//

Υ

[∶A]∼

forms a right Kan lift (resp. pointwise right Kan lift) of J along M, i.e. if and only ifthe cylinder

ER

||

J%%

X M//

Υ

[∶A]∼

is inverse universal (resp. pointwise inverse universal).

Proof. By the isomorphism in Remark 11.1.8(3), Υ is universal i the cylinder Υ is directuniversal. Since the slice of Υ at each e ∈ ∥E∥ is given by the component of Υ at e, Υ ispointwise universal i the cylinder Υ is pointwise direct universal.

Remark 11.3.4. By Proposition 11.3.3 and by the bijectiveness of exponential transposition, wesee that an extension (resp. pointwise extension) of J direct along M is the same thing as a left

Kan lift (resp. pointwise left Kan lift) of J alongM, i.e. a lift (resp. pointwise lift) ofJ direct alongM.

an extension (resp. pointwise extension) of J inverse alongM is the same thing as a rightKan lift (resp. pointwise right Kan lift) of J along M, i.e. a lift (resp. pointwise lift)of J inverse along M.

334

11. Extensions

Hence lifts (resp. pointwise lifts), in fact Kan lifts (resp. pointwise Kan lifts), subsumeextensions (resp. pointwise extensions). We will see in Corollary 11.3.18 that the converse isalso the case.

Proposition 11.3.5. A pointwise universal cell is universal.

Proof. This is reduced to Proposition 6.4.8 by the equivalence of conditions in Proposition 11.3.3.

Theorem 11.3.6. Let J and M be modules as in Denition 11.3.1. If there is a family of universal conical cells

∗re

##X

⟨J ⟩e;;

M//

Υe

A

, one for each object e ∈ ∥E∥, then there is a unique cell

ER

##X M

//

J;;

Υ

A

such that re = R ⋅ e and ⟨Υ⟩e = Υe, and Υ is pointwise universal. If there is a family of universal conical cells

∗re

e⟨J ⟩

##X M

//

Υe

A


ER

J##

X M//

Υ

A

such that e ⋅ R = re and e ⟨Υ⟩ = Υe, and Υ is pointwise universal.

Proof. By the equivalence of conditions in Proposition 11.3.3, this is reduced to an instance ofTheorem6.4.10 whereM is given by the moduleM and K is given by the functor J.

Theorem 11.3.7. Consider a cell Υ as in Denition 11.3.1 and let D be an essentially widesubcategory of E. Then Υ is pointwise universal if and only if its right (resp. left) slice at eachd ∈ ∥D∥ is universal.

Proof. Since the slice of Υ at each e ∈ ∥E∥ is given by the component of Υ at e (see Re-mark 11.1.8(3)), the assertion is reduced to Theorem6.4.18.


Denition 11.3.8.

335

11. Extensions

A cellA

R

##X

⟨X⟩//

M;;

Υ

X

is called(1) universal if it is a direct universal ⟨XA⟩-arrow (see Denition 11.1.9);(2) pointwise universal if each slice

∗R ⋅ a##

X

⟨M⟩a;;

⟨X⟩//

⟨Υ⟩a

X

is a universal conical cell (see Denition 11.2.5), i.e. a direct universal ⟨X⟩-arrow.Given a moduleM ∶XA, a universal (resp. pointwise universal) cell Υ ∶ J R ∶MEor the pair (R,Υ), or the functor R itself, is called a left extension (resp. pointwise leftextension) ofM.

A cellX

R

M##

A⟨A⟩

//

Υ

A

is called(1) universal if it is an inverse universal ⟨XA⟩-arrow (see Denition 11.1.9);(2) pointwise universal if each slice

∗x ⋅ R

x⟨M⟩

##A

⟨A⟩//

x⟨Υ⟩

A

is a universal conical cell (see Denition 11.2.5), i.e. an inverse universal ⟨A⟩-arrow.Given a moduleM ∶XA, a universal (resp. pointwise universal) cell Υ ∶ RM ∶XAor the pair (R,Υ), or the functor R itself, is called a right extension (resp. pointwise rightextension) ofM.

Remark 11.3.9. A left extension (resp. pointwise left extension) of a moduleM ∶XA is thesame thing as an extension (resp. pointwise extension) ofM direct along the hom endomodule⟨X⟩. Dually, a right extension (resp. pointwise right extension) of a module M ∶ X Ais the same thing as an extension (resp. pointwise extension) of M inverse along the homendomodule ⟨A⟩.

Proposition 11.3.10. Let M ∶XA be a module. A corepresentation (R,Υ) of M gives a pointwise left extension

AR

##X

⟨X⟩//

M;;

Υ

X

of M (cf. Remark 2.3.9(3)).

336

11. Extensions

A representation (R,Υ) of M gives a pointwise right extension

XR

M##

A⟨A⟩

//

Υ

A

of M (cf. Remark 2.3.9(3)).

Proof. By denition, Υ is a pointwise left extension ofM i the slice

∗R ⋅ a##

X⟨X⟩

//

⟨M⟩a;;⟨Υ⟩a

X

at each a ∈ ∥A∥ is a universal conical cell, and by Proposition 2.3.10, (R,Υ) is a corepresentationof M i for each a ∈ ∥A∥, (R ⋅ a, ⟨Υ⟩a) is a representation of ⟨M⟩a. The assertion is thusreduced to Proposition 11.2.7.

Remark 11.3.11. The converse does not hold as we saw in Remark 11.2.8.

Denition 11.3.12.

A cell

D

K

J // E

R

X M//

Υ

A

along ⟨J ,M⟩ is called(1) direct universal if it is a direct universal ⟨J ,M⟩-arrow (see Denition 1.2.8);(2) pointwise direct universal if each right slice

D

K

⟨J ⟩e // ∗R ⋅ e

X M//

⟨Υ⟩e

A

(see Denition 2.1.8) is a universal conical cell (see Denition 11.2.9), i.e. a direct uni-versal ⟨⟨J ⟩e,M⟩-arrow.

Given a functor K ∶D→X, a direct universal (resp. pointwise direct universal) cell Υ ∶ KR ∶ J M or the pair (R,Υ), or the functor R itself, is called an extension (resp. pointwiseextension) of K direct along ⟨J ,M⟩.

A cell

E

R

J //D

K

X M//

Υ

A

along ⟨J ,M⟩ is called(1) inverse universal if it is an inverse universal ⟨J ,M⟩-arrow (see Denition 1.2.8);

337

11. Extensions

(2) pointwise inverse universal if each left slice

∗ e⟨J ⟩ //

e ⋅ R

D

K

X M//

e⟨Υ⟩

A

(see Denition 2.1.8) is a universal conical cell (see Denition 11.2.9), i.e. an inverseuniversal ⟨e ⟨J ⟩ ,M⟩-arrow.

Given a functor K ∶ D → A, an inverse universal (resp. pointwise inverse universal) cellΥ ∶ R K ∶ J M or the pair (R,Υ), or the functor R itself, is called an extension (resp.pointwise extension) of K inverse along ⟨J ,M⟩.

Remark 11.3.13.(1) A cell Φ ∶ K R ∶ J M is direct universal if and only if to every cell Φ ∶ K L ∶ J M

there is a unique natural transformation Υ/Φ ∶ R→ L such that Φ = Υ Υ/Φ. Dually, a cellΦ ∶ R K ∶ J M is inverse universal if and only if to every cell Φ ∶ L K ∶ J M thereis a unique natural transformation Φ/Υ ∶ L→ R such that Φ = Φ/Υ Υ.

(2) As a special case whereM is given by the hom of a category C, a direct universal (resp. pointwise universal) cell

D

K

J // E

R

C⟨C⟩

//

Υ

C

or the pair (R,Υ), or the functor R itself, is called an extension (resp. pointwise exten-sion) of K direct along J .

an inverse universal (resp. pointwise inverse universal) cell

E

R

J //D

K

C⟨C⟩

//

Υ

C

or the pair (R,Υ), or the functor R itself, is called an extension (resp. pointwise exten-sion) of K inverse along J .

Theorem 11.3.14.

A cell

D

K

J // E

R

X M//

Υ

A

from K to R along ⟨J ,M⟩ is direct universal (resp. pointwise direct universal) if and onlyif the cell

ER

##D

K⟨M⟩//

J;;

Υ

A

338

11. Extensions

from J to R along K ⟨M⟩ is universal (resp. pointwise universal); hence an extension (resp.pointwise extension) of K direct along (J ,M) is the same thing as an extension (resp.pointwise extension) of J direct along K ⟨M⟩. Conversely, a cell

ER

##X M

//

J;;

Υ

A

from J to R along M is universal (resp. pointwise universal) if and only if the cell

X

1

J // E

R

X M//

Υ

A

from the identity X → X to R along ⟨J ,M⟩ is direct universal (resp. pointwise directuniversal); hence an extension (resp. pointwise extension) of J direct alongM is the samething as an extension (resp. pointwise extension) of the identity X→X direct along ⟨J ,M⟩.

A cell

EJ //

R

D

K

X M//

Υ

A

from R to K along ⟨J ,M⟩ is inverse universal (resp. pointwise inverse universal) if andonly if the cell

ER

J##

X⟨M⟩K

//

Υ

D

from R to J along ⟨M⟩K is universal (resp. pointwise universal); hence an extension (resp.pointwise extension) of K inverse along (J ,M) is the same thing as an extension (resp.pointwise extension) of J inverse along ⟨M⟩K. Conversely, a cell

ER

J##

X M//

Υ

A

from R to J along M is universal (resp. pointwise universal) if and only if the cell

E

R

J //A

1

X M//

Υ

A

from R to the identity A → A along ⟨J ,M⟩ is inverse universal (resp. pointwise inverseuniversal); hence an extension (resp. pointwise extension) of J inverse along M is thesame thing as an extension (resp. pointwise extension) of the identity A→A inverse along⟨J ,M⟩.

339

11. Extensions

Proof. The left slice of the module ⟨J ,M⟩ at K and the left slice of the module ⟨K ⟨M⟩⟩Eat J are the same left module over [E,A] given by

R ↦ (J ) ⟨D ∶ E⟩ (K ⟨M⟩R)

. Hence Υ ∶ K R ∶ J M is direct universal i Υ ∶ J R ∶ ⟨K ⟨M⟩⟩E is universal. ByTheorem11.2.11, the slice

D

K

⟨J ⟩e // ∗R ⋅ e

X M//

⟨Υ⟩e

A

of Υ ∶ K R ∶ J M at e ∈ ∥E∥ is universal i the slice

∗R ⋅ e##

DK⟨M⟩

//

⟨J ⟩e;;⟨Υ⟩e

A

of Υ ∶ J R ∶ ⟨K ⟨M⟩⟩E at e ∈ ∥E∥ is universal. Hence Υ ∶ K R ∶ J M is pointwisedirect universal i Υ ∶ J R ∶ ⟨K ⟨M⟩⟩E is pointwise universal.The left slice of the moduleME at J and the left slice of the module ⟨J ,M⟩ at 1X are

the same left module over [E,A] given by

R ↦ (J ) ⟨X ∶ E⟩ (⟨M⟩R)

. Hence Υ ∶ J R ∶ ME is universal i Υ ∶ 1X R ∶ J M is direct universal. ByTheorem11.2.11, the slice

∗R ⋅ e##

X

⟨J ⟩e;;

M//

⟨Υ⟩e

A

of Υ ∶ J R ∶ME at e ∈ ∥E∥ is universal i the slice

X⟨J ⟩e //

1

∗R ⋅ e

X M//

⟨Υ⟩e

A

of Υ ∶ 1X R ∶ J M at e ∈ ∥E∥ is universal. Hence Υ ∶ J R ∶ME is pointwise universali Υ ∶ 1X R ∶ J M is pointwise direct universal.

Note. Theorem11.3.15 and Theorem11.3.6 are reduced to each other by the equivalence ofconditions in Theorem11.3.14 and that in Theorem11.2.11.

Theorem 11.3.15. Let J and M be modules and K be a functor as in Denition 11.3.12. If there is a family of universal conical cells

D

K

⟨J ⟩e // ∗re

X M//

Υe

A

340

11. Extensions


D

K

J // E

R

X M//

Υ

A

such that re = R ⋅ e and ⟨Υ⟩e = Υe, and Υ is pointwise direct universal. If there is a family of universal conical cells

∗re

e⟨J ⟩ //D

K

X M//

Υe

A


E

R

J //D

K

X M//

Υ

A

such that e ⋅ R = re and e ⟨Υ⟩ = Υe, and Υ is pointwise direct universal.

Proof. As noted above, this is reduced to Theorem11.3.6.


Corollary 11.3.16. Let J and M be modules and K be a functor as in Denition 11.3.12.If M is cocomplete (resp. complete) and D is small, then K has a pointwise extension direct(resp. inverse) along ⟨J ,M⟩.

Proof. By Corollary 11.2.14, K has a colimit alongM weighted by ⟨J ⟩e for any e ∈ ∥E∥, andby Theorem11.3.15, a family of universal conical cells Υe ∶ K re ∶ ⟨J ⟩eM, one chosen foreach e ∈ ∥E∥, extends to a pointwise direct universal cell Υ ∶ K R ∶ J M.

Theorem 11.3.17.

Consider a cylinder

D

K

F // E

R

X M//

µ

A

weighted by a fully faithful functor F. If the cell

D

K

F⟨E⟩ // E

R

X M//

µE

A

generated by E inverse along µ (see Example 5.3.5(3)) is pointwise direct universal, so is µ.The converse holds if F is fully faithful and essentially surjective, i.e. if F is an equivalence.

341

11. Extensions

Consider a cylinder

E

R

D

K

Foo

X M//

µ

A


E

R

⟨E⟩F //D

K

X M//

Eµ

A

generated by E direct along µ (see Example 5.3.5(3)) is pointwise inverse universal, so is µ.The converse holds if F is fully faithful and essentially surjective, i.e. if F is an equivalence.

Proof. By the equivalence of conditions in Theorem11.3.14, this follows as a special case ofthe lemma below whereM is given by the composite module K ⟨M⟩. (Conversely, the lemmais a special case of the theorem where K is an identity.)

Lemma. Consider a cylinder

X

1

F // E

R

X M//

µ

A


X

1

F⟨E⟩ // E

R

X M//

µE

A

generated by E inverse along µ is pointwise universal, so is µ. The converse holds if F is fullyfaithful and essentially surjective, i.e. if F is an equivalence.

Proof. The rst assertion follows immediately from the claim below. The second assertionfollows from the claim and Theorem11.3.7.

Claim. Let x ∈ ∥X∥ and e ∈ ∥E∥ be such that x ⋅ F = e. Then the right slice

∗R ⋅ e##

X M//

F⟨E⟩e;;

⟨µE⟩e

A

of µE at e is a universal conical cell i theM-arrow µx ∶ x R ⋅ e is direct universal.

Proof. Depicting µ as

X⟨M⟩R //

F//µ E

342

11. Extensions

, we have a commutative diagram

⟨X⟩x ⟨F⟩x //

Xµx

⟨Xµ⟩x))

F ⟨E⟩e⟨µE⟩e

⟨M⟩ (R ⋅ e) ⟨⟨M⟩R⟩e

by Proposition 5.3.6 and Corollary 5.3.11. Since F is fully faithful, ⟨F⟩x is iso. Hence ⟨µE⟩e ∶F ⟨E⟩e R ⋅ e is a direct universal ⟨M⟩-arrow i so is Xµx ∶ ⟨X⟩x R ⋅ e, and by Theo-rem11.2.4, this is the case i µx ∶ x R ⋅ e is a direct universalM-arrow.

Corollary 11.3.18.

A cylinderE

K

R

##X M

//

µ

A

is direct universal (resp. pointwise direct universal) if and only if so is the cell

E⟨E⟩ //

K

E

R

X M//

⟨µ⟩

A

(see Corollary 5.5.4). A cylinder

ER

K

##X M

//

µ

A

is inverse universal (resp. pointwise inverse universal) if and only if so is the cell

E⟨E⟩ //

R

E

K

X M//

⟨µ⟩

A

(see Corollary 5.5.4).

Proof. By the isomorphism in Corollary 5.5.4, the cylinder µ is universal i so is the cell ⟨µ⟩.The pointwise version follows from Theorem11.3.17 by replacing the functor F ∶ D → E withthe identity E→ E.

Remark 11.3.19. By Corollary 11.3.18 and by the bijectiveness of the assignment µ ↦ ⟨µ⟩, wesee that given a moduleM ∶XA, a lift (resp. pointwise lift) of a functor K ∶ E → X direct along M is the same thing as an

extension (resp. pointwise extension) of K direct along ⟨⟨E⟩ ,M⟩. a lift (resp. pointwise lift) of a functor K ∶ E →A inverse alongM is the same thing as an

extension (resp. pointwise extension) of K inverse along ⟨⟨E⟩ ,M⟩.Extensions (resp. pointwise extensions) thus subsume lifts (resp. pointwise lifts), and viceversa as we saw in Remark 11.3.4.

343

11. Extensions

Theorem 11.3.20. Let M ∶XA be a module. The following conditions are equivalent:

(1) M is corepresentable;(2) each right slice ⟨M⟩a ∶ X ∗ preserves colimits and M has a pointwise left extension

(see Denition 11.3.8).When these conditions hold, a cell

AR

##X

⟨X⟩//

M;;

Υ

X

is pointwise universal if and only if the module morphism Υ ∶M → ⟨X⟩R is iso; that is, ifand only if the pair (R,Υ) forms a corepresentation of M.

The following conditions are equivalent:(1) M is representable;(2) each left slice x ⟨M⟩ ∶ ∗ A preserves limits and M has a pointwise right extension

(see Denition 11.3.8).When these conditions hold, a cell

XR

M##

A⟨A⟩

//

Υ

A

is pointwise universal if and only if the module morphism Υ ∶M → R ⟨A⟩ is iso; that is, ifand only if the pair (R,Υ) forms a representation of M.

Proof. By Corollary 6.3.11, M is corepresentable i each right slice ⟨M⟩a ∶ X ∗ is repre-sentable. By Theorem11.3.6,M has a pointwise left extension i each right slice ⟨M⟩a ∶X ∗has a colimit. The rst assertion is thus reduced to Theorem11.2.20. The if part of the sec-ond assertion is already seen in Proposition 11.3.10. Since pointwise universal cell is universal(see Proposition 11.3.5) and a universal cell Υ ∶ M R ∶ XA is the same thing as a di-rect universal ⟨XA⟩-arrow, the only if part follows by applying Theorem6.2.9 to the rightgeneral Yoneda functor XA.

Theorem 11.3.21. Let Φ be a cell from a module M to a module N . Suppose that Φ ∶M N has a right adjoint (see Denition 8.4.1). Then it preserves direct

extensions; that is, each direct universal cell Υ ∶ J M along M yields by compositionwith Φ a direct universal cell Υ Φ ∶ J N along N .

Suppose that Φ ∶M N has a left adjoint (see Denition 8.4.1). Then it preserves inverseextensions; that is, each inverse universal cell Υ ∶ J M along M yields by compositionwith Φ an inverse universal cell Υ Φ ∶ J N along N .

Proof. Since a direct universal cell Υ ∶ J M is the same thing as a direct universal ⟨J ,M⟩-arrow, the assertion is equivalent to saying that if a cell Φ has a right adjoint, then thepostcomposition cell ⟨J ,Φ⟩ (see Denition 1.2.23) preserves direct universals. But since ⟨J ,Φ⟩has a right adjoint as well by Theorem8.4.4, this follows from Theorem8.4.3.

11.4. Kan extensions

Note. Theorem5.5.2 allows the following denition.

344

11. Extensions

Denition 11.4.1. Given a pair of functors

C DKoo F // E

a natural transformation

D

K

F // E

R

C⟨C⟩

//

µ

C

from K to R F or the pair (R, µ), or the functor R itself, is called a left Kan extension (resp.pointwise left Kan extension) of K along F if the cell

D

K

F⟨E⟩ // E

R

C⟨C⟩

//

µE

C

generated by E inverse along µ is direct universal (resp. pointwise direct universal). a natural transformation

E

R

D

K

Foo

C⟨C⟩

//

µ

C

from F R to K or the pair (R, µ), or the functor R itself, is called a right Kan extension(resp. pointwise right Kan extension) of K along F if the cell

E

R

⟨E⟩F //D

K

C⟨C⟩

//

Eµ

C

generated by E direct along µ is inverse universal (resp. pointwise inverse universal).

Remark 11.4.2.(1) A (not necessarily pointwise) left Kan extension of K along F may be dened, as in the

literature, more directly as a universal of K direct along the module ⟨F C⟩ (see Re-mark 4.5.4(3)); by the isomorphism in Theorem5.5.2, a natural transformation µ ∶ K→ RFforms a direct universal ⟨F C⟩-arrow µ ∶ K R if and only if the cell µE is direct univer-sal. Dually, a right Kan extension of K along F may be dened as a universal of K inversealong the module ⟨F C⟩.

(2) A natural transformation µ ∶ K → R F forms a left Kan extension if and only if to everynatural transformation α ∶ K → L F there is a unique natural transformation µ/α ∶ R → Lsuch that α = µ [F µ/α] (where µ [F µ/α] is the pasting composite of µ and µ/α).Dually, a natural transformation µ ∶ F R → K forms a right Kan extension if and onlyif to every natural transformation α ∶ F L → K there is a unique natural transformationα/µ ∶ L→ R such that α = [F α/µ] µ.

Proposition 11.4.3. Given functors as in Denition 11.4.1, the following conditions are equivalent:

(1) R constitutes a left Kan extension (resp. pointwise left Kan extension) of K along F;

345

11. Extensions

(2) R constitutes an extension (resp. pointwise extension)

D

K

F⟨E⟩ // E

R

C⟨C⟩

//

Υ

C

of K direct along the representable module F ⟨E⟩;(3) R constitutes an extension (resp. pointwise extension)

ER

##D

K⟨C⟩//

F⟨E⟩;;

Υ

C

of the representable module F ⟨E⟩ direct along the representable module K ⟨C⟩. the following conditions are equivalent:

(1) R constitutes a right Kan extension (resp. pointwise right Kan extension) of K along F;(2) R constitutes an extension (resp. pointwise extension)

E

R

⟨E⟩F //D

K

C⟨C⟩

//

Υ

C

of K inverse along the corepresentable module ⟨E⟩F;(3) R constitutes an extension (resp. pointwise extension)

ER

⟨E⟩F

##C

⟨C⟩K//

Υ

D

of the corepresentable module ⟨E⟩F inverse along the corepresentable module ⟨C⟩K.

Proof.(1)⇒ (2) By denition.(1)⇐ (2) If R and Υ form an extension (resp. pointwise extension) of K direct along F ⟨E⟩,

then R and ΥE (see Theorem5.5.2) form a left Kan extension (resp. pointwise left Kanextension) of K along F.

(2)⇔ (3) This is Theorem11.3.14.

Remark 11.4.4. A Kan extension is thus regarded as a special instance of an extension asdened in Denition 11.3.12 where J is representable and M is the hom of a category, or asa special instance of an extension as dened in Denition 11.3.1 where both J and M arerepresentable.

Theorem 11.4.5. If D is small and C is cocomplete (resp. complete), then any functorK ∶D→C has a pointwise left (resp. right) Kan extension along any functor F ∶D→ E.

346

11. Extensions

Proof. By the equivalence of (1) and (2) in Proposition 11.4.3, and since the completenessof a category is the same thing as the completeness of its hom (see Remark 7.3.7), this is aspecial case of Corollary 11.3.16 where J is given by the representable module F ⟨E⟩ (resp. thecorepresentable module ⟨E⟩F) andM is given by the hom endomodule ⟨C⟩.

Theorem 11.4.6. If F ∶ D → E is a fully faithful functor, then a pointwise left (resp. right)Kan extension of a functor K ∶ D → C along F is a natural isomorphism. The converse holdsif F is fully faithful and essentially surjective, i.e. if F is an equivalence.

Proof. By the denition of a pointwise Kan extension, this is a special case of Theorem11.3.17whereM is given by the hom ofC, noting that a natural isomorphism K→ RF inC is the samething as a pointwise universal cylinder K R F along the hom ⟨C⟩ (see Proposition 6.4.9).

Theorem 11.4.7.

For a functor F ∶X→A the following conditions are equivalent:(1) F has a right adjoint;(2) F preserves colimits and the identity X→X has a pointwise left Kan extension along F.When these conditions hold, a natural transformation η ∶ 1X → G F forms a pointwise leftKan extension

XF //

1

A

G

X⟨X⟩

//

η

X

of the identity X→X along F if and only if η is the unit of an adjunction F G. For a functor G ∶A→X the following conditions are equivalent:

(1) G has a left adjoint;(2) G preserves limits and the identity A→A has a pointwise right Kan extension along G.When these conditions hold, a natural transformation ε ∶ G F→ 1A forms a pointwise rightKan extension

X

F

AGoo

1

A⟨A⟩

//

ε

A

of the identity A→A along G if and only if ε is the counit of an adjunction G F.

Proof. We see that this faithfully translates into an instance of Theorem11.3.20 where M isgiven by the representable module F ⟨A⟩, noting that(1) a right adjoint of F is the same thing as a corepresentation of the module F ⟨A⟩ (see

Remark 8.2.2(2));(2) F preserves colimits i for each a ∈ ∥A∥ the right module F ⟨A⟩a ∶X ∗ preserves colimits

(see Corollary 7.4.17);(3) a pointwise left Kan extension of the identity X → X along F is the same thing as a

pointwise left extension of the module F ⟨A⟩ (see Proposition 11.4.3);(4) a natural transformation η ∶ 1X → G F is the unit of an adjunction F G i the module

morphism ηA ∶ F ⟨A⟩ → ⟨X⟩G is iso (see Proposition 8.2.5), and by denition, η forms apointwise left Kan extension i the cell

AG

##X

⟨X⟩//

F⟨A⟩ ;;ηA

X

347

11. Extensions

is pointwise universal.

Note. The following is a special case of Theorem11.3.21.

Theorem 11.4.8.

Suppose that a functor H ∶C→ B has a right adjoint. Then it preserves left Kan extensions;that is, each left Kan extension µ ∶ K → R F of a functor K ∶ D → C along F yields bycomposition with H a left Kan extension µ H ∶ K H→ H R F of K H ∶D→ B along F.

Suppose that a functor H ∶C→ B has a left adjoint. Then it preserves right Kan extensions;that is, each Kan extension µ ∶ FR→ K of a functor K ∶D→C along F yields by compositionwith H a right Kan extension µ H ∶ F R H→ H K of H K ∶D→ B along F.

Proof. Since a left Kan extension in C along F is the same thing as a direct universal ⟨F C⟩-arrow (see Remark 11.4.2(1)), the assertion is equivalent to saying that if a functor H has a leftadjoint, then the postcomposition cell ⟨F H⟩ (see Remark 4.5.8(3)) preserves direct universals.Since an adjoint of H is the same thing as an adjoint of its hom ⟨H⟩ (see Remark 8.4.2(2)),⟨F H⟩ = ⟨F ⟨H⟩⟩ has a right adjoint as well by Corollary 8.4.5. Hence the assertion followsfrom Theorem8.4.3.

Problem. In Remark 11.3.4, we saw that pointwise Kan lifts subsume pointwise extensions,hence in particular pointwise Kan extensions. Does the converse hold? Do pointwise Kanextensions subsume pointwise Kan lifts?

11.5. Density

Denition 11.5.1.

(1) A moduleM ∶XA is called dense if its right exponential transpose [M] ∶A→ [X ∶] is fully faithful. codense if its left exponential transpose [M] ∶X→ [∶A]∼ is fully faithful.

(2) A functor F ∶D→ E is called dense if its representable module F ⟨E⟩ ∶D E is dense. codense if its corepresentable module ⟨E⟩F ∶ ED is codense.

(3) A subcategory D of a category E is called dense in E if the inclusion D→ E is dense. codense in E if the inclusion D→ E is codense.

Proposition 11.5.2. Given a module M ∶XA, the following conditions are equivalent:

(1) M is dense;(2) the identity MM forms a pointwise universal cell

A1

##X M

//

M;;1M

A

.(3) for every a ∈ ∥A∥, the identity ⟨M⟩a ⟨M⟩a forms a universal conical cell

∗a

##X

⟨M⟩a;;

M//

1⟨M⟩a

A

.

348

11. Extensions

(4) for every a ∈ ∥A∥, the unit cone

⌈M↓a⌉M↓a

yy

∆a

%%X M

//

1↓⟨M⟩a

A

(see Denition 10.4.1) of ⟨M⟩a forms a universal cone. the following conditions are equivalent:

(1) M is codense;(2) the identity MM forms a pointwise universal cell

X1

M##

X M//

1M

A

.(3) for every x ∈ ∥X∥, the identity x ⟨M⟩ x ⟨M⟩ forms a universal conical cell

∗x

x⟨M⟩

##X M

//

1x⟨M⟩

A

.(4) for every x ∈ ∥X∥, the unit cone

⌈x↓M⌉∆x

yy

x↓M

%%X M

//

1↓x⟨M⟩

A

(see Denition 10.4.1) of x ⟨M⟩ forms a universal cone.

Proof.(1)⇔ (2) By Proposition 11.3.3, the cell

A1

##X M

//

M;;1M

A

is pointwise universal i its right exponential transpose

AMzz

1

""[X ∶]

1M

AMoo

forms a pointwise left Kan lift of M along M. By Theorem6.5.3, this is the case iM is fully faithful.

(2)⇔ (3) Evident by the denition of pointwise universality.(3)⇔ (4) Since the unit cone 1↓⟨M⟩a is the comma transpose of the identity 1⟨M⟩a (see Re-

mark 10.4.9(3)), this is immediate from Proposition 11.2.3.

349

11. Extensions

Theorem 11.5.3. The right (resp. left) Yoneda module is dense (resp. codense).


Corollary 11.5.4. The right (resp. left) Yoneda functor is dense (resp. codense).

Proof. Since the Yoneda module is represented by the Yoneda functor (see Theorem5.2.15),the assertion follows from Theorem11.5.3.

Note. If F in Corollary 6.2.15 is dense (resp. codense), then we have the following.

Theorem 11.5.5. Let F ∶D→ E be functor. If F is fully faithful and dense, then an E-arrow u ∶ r ⋅F→ e is iso if and only if it is universal

from F to e, i.e. the F ⟨E⟩-arrow u ∶ r e is inverse universal. If F is fully faithful and codense, then an E-arrow u ∶ e → F ⋅ r is iso if and only if it is

universal from e to F, i.e. the ⟨E⟩F-arrow u ∶ e r is direct universal.

Proof. By Example 5.2.9(2), the diagram

⟨D⟩ r Du //

⟨F⟩r

F ⟨E⟩e

⟨F ⟨E⟩F⟩ r = F ⟨E⟩ (F ⋅ r)F⟨E⟩u

55

commutes. Since F is fully faithful, ⟨F⟩ r is iso. Hence Du is iso i F ⟨E⟩u is iso. By denition,Du is iso i u is inverse universal. Since ⟨F ⟨E⟩⟩ is fully faithful by the denseness of F, F ⟨E⟩uis iso i u is iso.

Theorem 11.5.6.

If a module M ∶X A is dense and has a representation (R,Υ), then for any a ∈ ∥A∥ thecomposition

⌈M↓a⌉M↓a

∆ // ∗a

X

R

M //

1↓⟨M⟩a

A

1

A⟨A⟩

//

Υ

A

yields a universal cone [M↓a] R a in A. If a moduleM ∶XA is codense and has a corepresentation (R,Υ), then for any x ∈ ∥X∥

the composition

∗x

⌈x↓M⌉∆oo

x↓M

X

1

M //

1↓x⟨M⟩

A

R

X⟨X⟩

//

Υ

X

yields a universal cone x R [x↓M] in X.

350

11. Extensions

Proof. The unit cone 1↓⟨M⟩a is universal by Proposition 11.5.2, and Υ preserves it by Theo-rem7.3.13.

Corollary 11.5.7.

Any right module is a colimit of representables. Specically, given a right moduleM ∶X ∗,the composition

⌈M↓⌉M↓

∆ // ∗M

X

X

⟨Xd⟩ //

1↓M

[X ∶]1

[X ∶]⟨X∶⟩

//

X

[X ∶]

of the unit cone ofM and the Yoneda representation yields a universal coneM↓ [X]Min [X ∶].

Any left module is a colimit of representables. Specically, given a left module M ∶ ∗ A,the composition

∗M

⌈M↓⌉M↓

∆oo

[∶A]∼

1

⟨eA⟩ //

1↓M

A

A

[∶A]∼⟨∶A⟩∼

//

A

[∶A]∼

of the unit cone ofM and the Yoneda corepresentation yields a universal coneM↓ [A]M in [∶A].

Proof. Since the Yoneda module Xd is dense (see Theorem11.5.3), and since ⟨Xd⟩ (M) =M (see Proposition 5.1.3), the assertion follows by applying Theorem11.5.6 to the Yonedarepresentation.

Theorem 11.5.8.

Given functors F ∶X→A and R ∶ [X ∶]→A, the following conditions are equivalent:(1) R constitutes a pointwise left Kan extension

X

F

X // [X ∶]R

A⟨A⟩

//

µ

A

of F along the right Yoneda functor X;(2) R constitutes a pointwise extension

X

F

Xd // [X ∶]R

A⟨A⟩

//

Υ

A

of F direct along the right Yoneda module Xd;

351

11. Extensions

(3) R constitutes a pointwise extension

[X ∶]R

$$X

Xd ::

F⟨A⟩//

Υ

A

of the right Yoneda module Xd direct along the representable module F ⟨A⟩ ∶XA;(4) R constitutes a left adjoint

A⟨F⟨A⟩⟩ //

Υ [X ∶]R

oo

of the right exponential transpose of the representable module F ⟨A⟩ ∶XA. Given functors G ∶A→X and R ∶ [∶A]∼ →X, the following conditions are equivalent:

(1) R constitutes a pointwise right Kan extension

[∶A]∼

R

AAoo

G

X⟨X⟩

//

µ

X

of G along the left Yoneda functor A;(2) R constitutes a pointwise extension

[∶A]∼

R

eA //A

G

X⟨X⟩

//

Υ

X

of G inverse along the left Yoneda module eA;(3) R constitutes a pointwise extension

[∶A]∼R

yy

eA

%%X

⟨X⟩G//

Υ

A

of the left Yoneda module eA inverse along the corepresentable module ⟨X⟩G ∶XA;(4) R constitutes a right adjoint

X⟨⟨X⟩G⟩

//Υ [∶A]∼Roo

of the left exponential transpose of the corepresentable module ⟨X⟩G ∶XA.

Proof.(1)⇔ (2)⇔ (3) Since ⟨Xd⟩ ≅ [X] ⟨X ∶⟩ (see Theorem5.2.15), this follows from Proposi-

tion 11.4.3.(3)⇔ (4) By Remark 11.3.4, and since ⟨Xd⟩ = 1 (see Proposition 5.1.3), R constitutes a

pointwise extension of Xd direct along F ⟨A⟩ i R constitutes a pointwise left Kan lift

[X ∶]1

yyR

$$[X ∶]

η

A⟨F⟨A⟩⟩

oo

352

11. Extensions

of the identity [X ∶] → [X ∶] along ⟨F ⟨A⟩⟩, and by the equivalence of (1) and (5) inProposition 8.2.5, this is the case i η is the unit of an adjunction R [⟨F ⟨A⟩⟩] ∶ A [X ∶].

Corollary 11.5.9.

If a functor F ∶X→A has for each right module M ∶X ∗ an M-weighted colimit

XM //

F

∗RM

A⟨A⟩

//

Υ

A

, then F has a pointwise left Kan extension R ∶ [X ∶] → A along the right Yoneda functor[X] ∶X→ [X ∶], with RM = R ⋅M.

If a functor G ∶A→X has for each left module M ∶ ∗A an M-weighted limit

∗ M //

RM

A

G

X⟨X⟩

//

Υ

X

, then G has a pointwise right Kan extension R ∶ [∶A]∼ → X along the left Yoneda functor[A] ∶A→ [∶A]∼, with RM = R ⋅M.

Proof. By the equivalence of (1) and (2) in Theorem11.5.8, the problem translates into showingthat F has a pointwise extension R ∶ [X ∶] → A direct along the right Yoneda module ⟨Xd⟩ ∶X [X ∶], with RM = R ⋅M. But since ⟨Xd⟩ (M) =M for any right moduleM ∶X ∗ (seeProposition 5.1.3), this follows from Theorem11.3.15.

Theorem 11.5.10.

If F ∶ X → A is a functor with X small and A cocomplete, then F has a pointwise left Kanextension

X

F

X // [X ∶]R

A⟨A⟩

//

µ

A

along the right Yoneda functor X, and R is characterized by the following properties:(1) R is cocontinuous;(2) F ≅ R [X];that is, if a functor S ∶ [X ∶]→A has the properties above, then S ≅ R.

If G ∶ A → X is a functor with A small and X complete, then G has a pointwise right Kanextension

[∶A]∼

R

AAoo

G

X⟨X⟩

//

µ

X

along the left Yoneda functor A, and R is characterized by the following properties:(1) R is continuous;(2) [A] R ≅ G;

353

11. Extensions

that is, if a functor S ∶ [∶A]∼ →X has the properties above, then S ≅ R.

Proof. By Theorem11.4.5, a pointwise left Kan extension (R, µ) exists. Since R is given bya left adjoint of the functor [⟨F ⟨A⟩⟩] ∶ A → [X ∶] by Theorem11.5.8, R is cocontinuousby Corollary 8.4.8 ([X ∶] is cocomplete by Corollary 7.4.3). Since the Yoneda functor is fullyfaithful, µ is a natural isomorphism by Theorem11.4.6. Now suppose that a functor S ∶[X ∶] → A is cocontinuous and has a natural isomorphism ν ∶ F → S [X]. We will showthat the adjunct of ν along µ, i.e. the unique natural transformation µ/ν ∶ R → S withν = µ [[X] µ/ν] (see Remark 11.4.2(2)), is a natural isomorphism. For this, it sucesto show that the component of µ/ν at each right module M ∶ X ∗ is an isomorphism. ByCorollary 11.5.7, there is a universal cone π ∶M↓ [X] M, and the composite π µ/ν ∶M↓ [X] RM ⋅ S factors as

M↓ [X] R

M↓ [X] µ/ν

π R //M ⋅ RM ⋅ µ/ν

M↓ [X] Sπ S

//M ⋅ S

. Since R and S are cocontinuous, π R and π S are universal cones, and since µ and ν arenatural isomorphisms, so isM↓ [X] µ/ν =M↓ [µ−1 ν]; henceM ⋅µ/ν is an isomorphismby Proposition 6.2.25(2).

Corollary 11.5.11. If F ∶D→ E is a functor between small categories, then the precompositionfunctor [F ∶] ∶ [E ∶]→ [D ∶] (resp. [∶ F] ∶ [∶ E]→ [∶D]) has both left and right adjoints.

Proof. Consider the Yoneda functors as in

DD //

F

[D ∶]

EE

// [E ∶]

. Since D is small and [E ∶] is cocomplete (see Corollary 7.4.3), F [E] has a pointwise leftKan extension

DD //

F

[D ∶]R

EE

//

µ

[E ∶]

along D by Theorem11.5.10, and R is a left adjoint of the functor ⟨[F [E]] ⟨E ∶⟩⟩ byTheorem11.5.8. But

⟨[F [E]] ⟨E ∶⟩⟩ = ⟨F ⟨[E] ⟨E ∶⟩⟩⟩≅ ⟨F ⟨Ed⟩⟩ ∗1

= [F ∶] [⟨Ed⟩] ∗2

= [F ∶] ∗3

(∗1 by Theorem5.2.15; ∗2 by Proposition 2.1.7; ∗3 by Proposition 5.1.3). Hence [F ∶] has a leftadjoint, namely R. Now consider the composition

EE //

[E] [F∶]

[E ∶]

[F∶]xx[D ∶]

354

11. Extensions

. Since [F ∶] is cocontinuous (see Corollary 7.4.4), [F ∶] is a pointwise left Kan extension of[E] [F ∶] along E by Theorem11.5.10, and thus has a right adjoint by Theorem11.5.8.

355

12. Ends

12.1. Ends

Denition 12.1.1. Let E be a category andM ∶XA be a module. A cylinder

∗r

E∼ ×E

K

X M//

π

A

is called universal if it is an inverse universal ⟨E,M⟩-arrow (see Denition 4.4.5). Given abifunctor K ∶ E∼ ×E → A, a universal cylinder π ∶ r K ∶ E M or the pair (r, π), or theobject r itself, is called an end of K along M, with the object r denoted by ∏E K (or justby ∏K).

A cylinderE∼ ×E

K

∗r

X M//

π

A

is called universal if it is a direct universal ⟨E,M⟩-arrow (see Denition 4.4.5). Given abifunctor K ∶ E∼ ×E → X, a universal cylinder π ∶ K r ∶ E M or the pair (r, π), or theobject r itself, is called a coend of K alongM, with the object r denoted by ∐E K (or justby ∐K).

Remark 12.1.2.(1) A cylinder π ∶ r K ∶ EM is universal if and only if to every cylinder α ∶ x K ∶ EM

there is a unique X-arrow α/π ∶ x → r such that α = α/π π. Dually, a cylinder π ∶ K r ∶E M is universal if and only if to every cylinder α ∶ K a ∶ E M there is a uniqueA-arrow π/α ∶ r→ a such that α = π π/α.

(2) As a special case whereM is the hom of a category C, an extraordinary natural transformation π ∶ r K ∶ E → C from an object r ∈ ∥C∥ to a

bifunctor K ∶ E∼×E→C is called universal if it is an inverse universal ⟨E,C⟩-arrow (seeDenition 4.4.7). Given a bifunctor K ∶ E∼ ×E → C, a universal extraordinary naturaltransformation π ∶ r K ∶ E → C or the pair (r, π), or the object r itself, is called anend of K in C.

an extraordinary natural transformation π ∶ K r ∶ E→C from a bifunctor K ∶ E∼×E→C to an object r ∈ ∥C∥ is called universal if it is a direct universal ⟨E,C⟩-arrow (seeDenition 4.4.7). Given a bifunctor K ∶ E∼ ×E → C, a universal extraordinary naturaltransformation π ∶ K r ∶ E → C or the pair (r, π), or the object r itself, is called acoend of K in C.

Theorem 12.1.3. Let E be a category and M ∶XA be a module.

356

12. Ends

A cylinder∗r

E∼ ×E

K

X M//

π

A


∗ ⟨E⟩ //

r

E∼ ×E

K

X M//

⟨π⟩

A

(see Theorem5.5.9) is universal. A cylinder

E∼ ×E

K

∗r

X M//

π

A


E∼ ×E

K

⟨E⟩ // ∗r

X M//

⟨π⟩

A

(see Theorem5.5.9) is universal.

Proof. Immediate by the isomorphism in Theorem5.5.9.

Remark 12.1.4. By Theorem12.1.3 and by the bijectiveness of the assignment π ↦ ⟨π⟩, we seethat given a moduleM ∶XA, an end of a bifunctor K ∶ E∼ ×E → A along M is the same thing as an ⟨E⟩-weighted limit

of K alongM. a coend of a bifunctor K ∶ E∼×E→X alongM is the same thing as an ⟨E⟩-weighted colimit

of K alongM.Weighted limits thus subsume ends.

Theorem 12.1.5. Let E be a category and M ∶XA be a module. A cone π ∶ r K ∶ ∗E M is universal if and only if the cylinder π ∶ r K [∆E∼ ×E] ∶

EM (see Remark 4.6.4(2)) is universal. A cone π ∶ K r ∶ E∗ M is universal if and only if the cylinder π ∶ [∆E∼ ×E] K r ∶

EM (see Remark 4.6.4(2)) is universal.

Proof. Since a cone π ∶ r K ∶ ∗EM is universal i it is an inverse universal ⟨∗E,M⟩-arrow,and the cylinder π ∶ r K [∆E∼ ×E] ∶ EM is universal i it is an inverse universal ⟨E,M⟩-arrow, the assertion follows by applying Theorem6.2.12 to the composite in Theorem4.6.32.

Remark 12.1.6. Theorem12.1.5 says that given a moduleM ∶XA, a limit of a functor K ∶ E → A along M is the same thing as an end of the bifunctor

K [∆E∼ ×E] ∶ E∼ ×E→A alongM.

357

12. Ends

a colimit of a functor K ∶ E → X along M is the same thing as a coend of the bifunctor[∆E∼ ×E] K ∶ E∼ ×E→X alongM.

A limit is thus regarded as an end with a dummy variable; recalling Remark 12.1.4 and Re-mark 11.2.13, we now see that ends, limits, and weighted limits subsume one another.


XM //

P

A

Q

Y N//

Φ

B

is said to preserve (reect, create) ends over a category E if the postcomposition cell ⟨E,Φ⟩ (see

Denition 4.4.17) preserves (reects, creates) inverse universals. preserve (reect, create) coends over a category E if the postcomposition cell ⟨E,Φ⟩ (see



a) preserve ends over E if each universal cylinder π ∶ r K ∶ EM yields by composition with

Φ a universal cylinder π Φ ∶ r ⋅ P Q K ∶ E N . coends over E if each universal cylinder π ∶ K r ∶ E M yields by composition

with Φ a universal cylinder π Φ ∶ K P Q ⋅ r ∶ E N .b) reect

ends over E if a cylinder π ∶ r K ∶ E M is universal whenever the cylinderπ Φ ∶ r ⋅ P Q K ∶ E N is universal.

coends over E if a cylinder π ∶ K r ∶ E M is universal whenever the cylinderπ Φ ∶ K P Q ⋅ r ∶ E N is universal.

c) create ends over E if for every bifunctor K ∶ E∼ ×E → A and for every universal cylinder

κ ∶ s Q K ∶ E N there is exactly one cylinder π ∶ r K ∶ EM with π Φ = κ,and if this π is universal.

coends over E if for every bifunctor K ∶ E∼ ×E→X and for every universal cylinderκ ∶ K P s ∶ E N there is exactly one cylinder π ∶ K r ∶ EM with π Φ = κ,and if this π is universal.

(2) A cell Φ is said to preserve (reect, create) ends, if it preserves (reects, creates) ends over any category. preserve (reect, create) coends, if it preserves (reects, creates) coends over any cate-

gory.

Proposition 12.1.9. A cell Φ preserves (reects, creates) ends (resp. coends) if and only ifΦ preserves (reects, creates) limits (resp. colimits).

Proof.(⇒) By denition, this is equivalent to showing that for any category E, if the cell ⟨E,Φ⟩

preserves (reects, creates) inverse universals, then the cell ⟨∗E,Φ⟩ does the same. But thisfollows by applying Proposition 6.2.22 to the composite in Corollary 4.6.33.

(⇐) Because of Proposition 11.2.19, this is equivalent to showing that if Φ preserves (reects,creates) weighted limits, then Φ preserves (reects, creates) ends. But by Theorem12.1.3and since the isomorphism in Theorem5.5.9 is natural in M, for any category E, if a cell

358

12. Ends

Φ ∶M N preserves (reects, creates) limits weighted by ⟨E⟩, then it preserves (reects,creates) ends over E.

12.2. Ends with parameters

Denition 12.2.1. Let E and D be categories andM ∶XA be a module. A cylinder

∗R

D∼ ×D

K

[E,X]⟨E,M⟩

//

ω

[E,A]

along the module ⟨E,M⟩ (see Denition 4.3.6) is called(1) universal if it is an inverse universal ⟨D, ⟨E,M⟩⟩-arrow (see Denition 4.4.5);(2) pointwise universal if each slice

∗R

D∼ ×D

K

[E,X][e,X]

⟨E,M⟩ //

ω

[E,A][e,A]

X M//

⟨e,M⟩

A

(see Remark 4.7.8(2)) is a universal cylinder, i.e. an inverse universal ⟨D,M⟩-arrow.Given a bifunctor K ∶D∼×D→ [E,A], a pointwise universal cylinder ω ∶ R K ∶D ⟨E,M⟩or the pair (R, ω), or the functor R itself, is called a pointwise end of K along ⟨E,M⟩, withthe functor R denoted by ∏E

D K (or just by ∏E K). A cylinder

D∼ ×D

K

∗R

[E,X]⟨E,M⟩

//

ω

[E,A]

along the module ⟨E,M⟩ (see Denition 4.3.6) is called(1) universal if it is a direct universal ⟨D, ⟨E,M⟩⟩-arrow (see Denition 4.4.5);(2) pointwise universal if each slice

D∼ ×D

K

∗R

[E,X][e,X]

⟨E,M⟩ //

ω

[E,A][e,A]

X M//

⟨e,M⟩

A

(see Remark 4.7.8(2)) is a universal cylinder, i.e. a direct universal ⟨D,M⟩-arrow.Given a bifunctor K ∶D∼×D→ [E,X], a pointwise universal cylinder ω ∶ K R ∶D ⟨E,M⟩or the pair (R, ω), or the functor R itself, is called a pointwise coend of K along ⟨E,M⟩,with the functor R denoted by ∐E


Proposition 12.2.2.

359

12. Ends

The following conditions are equivalent:(1) an extraordinary cylinder

∗R

D∼ ×D

K

[E,X]⟨E,M⟩

//

ω

[E,A]


E

R

E

K⊺

Eoo

X⟨D,M⟩

//

ω⊺

[D∼ ×D,A]

(see Denition 4.7.7) is an inverse universal (resp. pointwise inverse universal) ordinarycylinder.

The following conditions are equivalent:(1) an extraordinary cylinder

D∼ ×D

K

∗R

[E,X]⟨E,M⟩

//

ω

[E,A]


E

K⊺

E // E

R

[D∼ ×D,X]⟨D,M⟩

//

ω⊺

A

(see Denition 4.7.7) is a direct universal (resp. pointwise direct universal) ordinarycylinder.

Proof. By the isomorphism in Remark 4.7.8(1), ω is universal i ω⊺ is inverse universal. Sincethe slice of ω at each e ∈ ∥E∥ is given by the component of ω⊺ at e, ω is pointwise universal iω⊺ is pointwise inverse universal.

Remark 12.2.3. By Proposition 12.2.2 and by the bijectiveness of transposition, we see thatgiven a moduleM ∶XA, an end (resp. pointwise end) of a bifunctor K ∶ D∼ ×D → [E,A] along the module ⟨E,M⟩

is the same thing as a lift (resp. pointwise lift) of K⊺ ∶ E → [D∼ ×D,A] inverse along themodule ⟨D,M⟩.

a coend (resp. pointwise coend) of a bifunctor K ∶D∼×D→ [E,X] along the module ⟨E,M⟩is the same thing as a lift (resp. pointwise lift) of K⊺ ∶ E → [D∼ ×D,X] direct along themodule ⟨D,M⟩.

Proposition 12.2.4. A pointwise universal cylinder is universal.

Proof. This is reduced to Proposition 6.4.8 by the equivalence of (1) and (2) in Proposi-tion 12.2.2.

Denition 12.2.5. Let E and D be categories andM ∶XA be a module.

360

12. Ends

A bicylinderE

R

E ×D∼ ×D

K

X M//

ω

A

is called(1) universal if it is an inverse universal ⟨E ×D,M⟩-arrow (see Remark 4.7.2(1));(2) pointwise universal if each left slice

∗R(e)

D∼ ×D

K⊢(e)

X M//

[ω⊢]e

A

(see Denition 4.7.5) is a universal cylinder, i.e. an inverse universal ⟨D,M⟩-arrow.Given a bifunctor K ∶ E×D∼×D→A, a pointwise universal bicylinder ω ∶ R K ∶ E×DMor the pair (R, ω), or the functor R itself, is called an E-parameterized end of K alongM,with the functor R denoted by ∏E

D K (or just by ∏E K). A bicylinder

E ×D∼ ×D

K

E

R

X M//

ω

A

is called(1) universal if it is a direct universal ⟨E ×D,M⟩-arrow (see Remark 4.7.2(1));(2) pointwise universal if each left slice

D∼ ×D

K⊢(e)

∗R(e)

X M//

[ω⊢]e

A

(see Denition 4.7.5) is a universal cylinder, i.e. a direct universal ⟨D,M⟩-arrow.Given a bifunctor K ∶ E×D∼×D→X, a pointwise universal bicylinder ω ∶ K R ∶ E×DMor the pair (R, ω), or the functor R itself, is called an E-parameterized coend of K alongM,with the functor R denoted by ∐E


Proposition 12.2.6.

The following conditions are equivalent:(1) a bicylinder

E

R

E ×D∼ ×D

K

X M//

ω

A


∗R

D∼ ×D

K⊣

[E,X]⟨E,M⟩

//

ω⊣

[E,A]

(see Denition 4.7.5) is a universal (resp. pointwise universal) extraordinary cylinder;

361

12. Ends

(3) its left exponential transpose

E

R

E

K⊢

Eoo

X⟨D,M⟩

//

ω⊢

[D∼ ×D,A]

(see Denition 4.7.5) is an inverse universal (resp. pointwise inverse universal) ordinarycylinder.

The following conditions are equivalent:(1) a bicylinder

E ×D∼ ×D

K

E

R

X M//

ω

A


D∼ ×D

K⊣

∗R

[E,X]⟨E,M⟩

//

ω⊣

[E,A]

(see Denition 4.7.5) is a universal (resp. pointwise universal) extraordinary cylinder;(3) its left exponential transpose

E

K⊢

E // E

R

[D∼ ×D,X]⟨D,M⟩

//

ω⊢

A

(see Denition 4.7.5) is a direct universal (resp. pointwise direct universal) ordinarycylinder.

Proof. By the isomorphisms in Remark 4.7.6, ω is universal i ω⊣ is universal i ω⊢ is inverseuniversal. Since the left slice of ω at each e ∈ ∥E∥ is given by the component of ω⊢ at e, ω ispointwise universal i ω⊢ is pointwise inverse universal. By Proposition 12.2.2, ω⊣ is pointwiseuniversal i [ω⊣]⊺ is pointwise inverse universal. But [ω⊣]⊺ = ω⊢ by the commutative diagramin Remark 4.7.8(1).

Remark 12.2.7. By Proposition 12.2.6 and by the bijectiveness of exponential transposition, wesee that given a moduleM ∶XA, the following are the same thing:

(1) an E-parameterized end of a bifunctor K ∶ E ×D∼ ×D→A alongM;(2) a pointwise end of K⊣ ∶D∼ ×D→ [E,A] along the module ⟨E,M⟩;(3) a pointwise lift of K⊢ ∶ E→ [D∼ ×D,A] inverse along the module ⟨D,M⟩.

the following are the same thing:(1) an E-parameterized coend of a bifunctor K ∶ E ×D∼ ×D→X alongM;(2) a pointwise coend of K⊣ ∶D∼ ×D→ [E,X] along the module ⟨E,M⟩;(3) a pointwise lift of K⊢ ∶ E→ [D∼ ×D,X] direct along the module ⟨D,M⟩.

362

12. Ends

Proposition 12.2.8. A pointwise universal bicylinder is universal.

Proof. This is reduced to Proposition 6.4.8 by the equivalence of (1) and (3) in Proposi-tion 12.2.6.

Theorem 12.2.9. (Parameter Theorem for ends). Let M be a module and K be a bifunctoras in Denition 12.2.5. Suppose that for each object e ∈ ∥E∥, the bifunctor K⊢ (e) ∶ D∼ ×D → A has an end alongM and a universal cylinder ωe ∶ re K⊢ (e) ∶ D M is chosen. Then there is a uniquefunctor R ∶ E → X with R (e) = re such that ω ∶= ([ωe]d)(e,d)∈∥E×D∥ forms a bicylinderR K ∶ E ×DM, and R and ω form an E-parametrized end of K along M.

Suppose that for each object e ∈ ∥E∥, the bifunctor K⊢ (e) ∶ D∼ ×D → X has a coend alongM and a universal cylinder ωe ∶ K⊢ (e) re ∶ D M is chosen. Then there is a uniquefunctor R ∶ E → A with R (e) = re such that ω ∶= ([ωe]d)(e,d)∈∥E×D∥ forms a bicylinderK R ∶ E ×DM, and R and ω form an E-parametrized coend of K along M.

Proof. By the equivalence of (1) and (3) in Proposition 12.2.6, this is reduced to an instanceof Theorem6.4.10 whereM is given by the module ⟨D,M⟩.

Theorem 12.2.10. Let M ∶XA be a module. Suppose that a bifunctor K ∶ E∼ ×E ×D∼ ×D→A has an [E∼ ×E]-parameterized end

E∼ ×E

R

E∼ ×E ×D∼ ×D

K

X M//

ω

A

along M. Then an end of K (regarded as a bifunctor [E ×D]∼ × [E ×D] → A) along Mexists if and only if an end of R in X exists. If this is the case, then for any universalcylinder λ ∶ r K along M, there is a unique universal cylinder π ∶ r R in X such thatthe diagram

r

πe

λe,d

))R (e,e) ωe,e,d

// K (e,e,d,d)

commutes for every (e,d) ∈ ∥E ×D∥. Suppose that a bifunctor K ∶ E∼ ×E ×D∼ ×D→X has an [E∼ ×E]-parameterized coend

E∼ ×E ×D∼ ×D

K

E∼ ×E

R

X M//

ω

A

along M. Then a coend of K (regarded as a bifunctor [E ×D]∼ × [E ×D] → X) along Mexists if and only if a coend of R in A exists. If this is the case, then for any universalcylinder λ ∶ K r along M, there is a unique universal cylinder π ∶ R r in A such thatthe diagram

K (e,e,d,d) ωe,e,d //

λe,d ))

R (e,e)πer

commutes for every (e,d) ∈ ∥E ×D∥.

363

12. Ends

Proof. Given a family of X-arrows αe ∶ x → R (e,e) indexed by the objects e ∈ ∥E∥, letα ω denote the family of M-arrows αe ωe,e,d ∶ x K (e,e,d,d) indexed by the objects(e,d) ∈ ∥E ×D∥. The proof is complete if we show that the assignment α ↦ α ω yields anisomorphism from the right module ⟨E,X⟩ (R) ∶ X ∗ (the right slice at R of the module⟨E,X⟩ ∶ X [E∼ ×E,X]) to the right module ⟨E ×D,M⟩ (K) ∶ X ∗ (the right slice at Kof the module ⟨E ×D,M⟩ ∶ X [[E ×D]∼ × [E ×D] ,A]). But this follows from the claimbelow.

Claim.(1) The composite α ω forms a cylinder x K if and only if α forms a cylinder x R.(2) For any cylinder ξ ∶ x K along M, there is a unique cylinder α ∶ x R in X such that

ξ = α ω.(3) For an X-arrow f ∶ y → x and a cylinder α ∶ x R in X, the associative law

f [α ω] = [f α] ω

holds.

Proof.(1) Clearly αe ωe,e,d is natural in d for each e ∈ ∥E∥. By Theorem4.1.4, it thus suces to

show that for each d ∈ ∥D∥, αe ωe,e,d is natural in e i so is αe. For this, let h ∶ e→ e′ bean E-arrow and consider the diagram

x

αe′

αe // R (e,e)R(e,h)

ωe,e,d // K (e,e,d,d)

K(e,h,d,d)

R (e′,e′)R(h,e′)

//

ωe′,e′,d

R (e,e′)ωe,e′,d

''K (e′,e′,d,d)

K(h,e′,d,d)// K (e,e′,d,d)

; we need to verify that the top left square commutes i the outer square commutes. Bythe naturality of ω, the right and bottom quadrangles commute. Hence if the top leftsquare commutes, then the outer square commutes. Now suppose that the outer squarecommutes. Then a simple diagram chasing shows that

αe R (e,h) ωe,e′,d = αe′ R (h,e′) ωe,e′,d

; henceαe R (e,h) = αe′ R (h,e′)

by the universality of ωe,e′,d.(2) Since ω is pointwise universal, there is a unique family of X-arrows αe ∶ x → R (e,e) such

that αe ωe,e,d = ξe,d for each (e,d) ∈ ∥E ×D∥. By the result above, α ∶= (αe)e∈∥E∥ forms acylinder x R.

(3) Easily veried.

Corollary 12.2.11. Let M ∶XA be a module.

364

12. Ends

Suppose that a bifunctor K ∶ E∼ ×E×D∼ ×D→A has an [E∼ ×E]-parameterized end and a[D∼ ×D]-parameterized end

E∼ ×E

R

E∼ ×E ×D∼ ×D

K

D∼ ×D

R′

E∼ ×E ×D∼ ×D

K

X M//

ω

A X M//

ω′

A

along M. Then an end of R in X exists if and only if an end of R′ in X exists. If this isthe case, then there exist a universal cylinder π ∶ r R and π′ ∶ r R′ in X such that thediagram

r

πe

π′d // R′ (d,d)

ω′e,d,d

R (e,e) ωe,e,d

// K (e,e,d,d)

commutes for every (e,d) ∈ ∥E ×D∥. Suppose that a bifunctor K ∶ E∼ ×E×D∼ ×D→X has an [E∼ ×E]-parameterized coend and

a [D∼ ×D]-parameterized coend

E∼ ×E ×D∼ ×D

K

E∼ ×E

R

E∼ ×E ×D∼ ×D

K

D∼ ×D

R′

X M//

ω

A X M//

ω′

A

along M. Then a coend of R in A exists if and only if a coend of R′ in A exists. If this isthe case, then there exist a universal cylinder π ∶ R r and π′ ∶ R′ r in A such that thediagram

K (e,e,d,d)ω′e,d,d

ωe,e,d // R (e,e)πe

R′ (d,d)

π′d// r

commutes for every (e,d) ∈ ∥E ×D∥.


Remark 12.2.12.(1) The results in Theorem12.2.10 and Corollary 12.2.11 are expressed by

∏E

E∼×E∏D

K ≅ ∏E×D

K ≅∏D

D∼×D∏E

K

or, more informatively, by

∏e∈E∏d∈D

K (e, e, d, d) ≅ ∏(e,d)∈E×D

K (e, e, d, d) ≅∏d∈D∏e∈E

K (e, e, d, d)

, and called the Fubini theorem for ends.(2) Since a limit is regarded as an end with a dummy variable, a similar result holds for limits.

We have the end-limit interchange property

∏e∈E∏d∈D

K(e, d, d) ≅ ∏(e,d)∈E×D

K(e, d, d) ≅∏d∈D∏e∈E

K(e, d, d)

365

12. Ends

by regarding K (e, d, d) as K (e, e, d, d) dummy in the rst variable e, and then the limitsinterchange property

∏e∈E∏d∈D

K(e, d) ≅ ∏(e,d)∈E×D

K(e, d) ≅∏d∈D∏e∈E

K(e, d)

by regarding K (e, d) as K (e, d, d) dummy in the rst variable d.

12.3. Ends of modules

Theorem 12.3.1. Given a small endomodule M ∶ E E, i.e. a bifunctor M ∶ E∼ ×E → Setwith E small, an end of M is given by an equalizer π as in

∏EMπ

π //∏e∈∥E∥ e ⟨M⟩e∏e∈∥E∥(χe,e′)e′∈∥E∥

∏e′∈∥E∥ e′ ⟨M⟩e′

∏e′∈∥E∥(χ′e,e′)e∈∥E∥

//∏e∈∥E∥∏e′∈∥E∥ [e ⟨E⟩e′,e ⟨M⟩e′]

, whereχe,e′ ∶ e ⟨M⟩e→ [e ⟨E⟩e′,e ⟨M⟩e′] ; m↦ [h↦ m h]

andχ′e,e′ ∶ e′ ⟨M⟩e′ → [e ⟨E⟩e′,e ⟨M⟩e′] ; m↦ [h↦ h m]

are the functions given by the exponential transpose of the composition between M-arrowsand E-arrows.

a coend of M is given by a coequalizer π as in

∐e∈∥E∥∐e′∈∥E∥ e′ ⟨M⟩e × e ⟨E⟩e′

∐e∈∥E∥(χe,e′)e′∈∥E∥

∐e′∈∥E∥(χ′e,e′)e∈∥E∥ //∐e′∈∥E∥ e′ ⟨M⟩e′π

∐e∈∥E∥ e ⟨M⟩e π//∐EM

, whereχe,e′ ∶ e′ ⟨M⟩e × e ⟨E⟩e′ → e ⟨M⟩e; (m,h)↦ h m

andχ′e,e′ ∶ e′ ⟨M⟩e × e ⟨E⟩e′ → e′ ⟨M⟩e′; (m,h)↦ m h

are the functions given by the composition between M-arrows and E-arrows.

Proof. By the claim below, an end ofM is given by a universal fork on ∏e∈∥E∥ (χe,e′)e′∈∥E∥ and

∏e′∈∥E∥ (χ′e,e′)e∈∥E∥, i.e. by an equalizer of them.

Claim. Given a small set S, a family of functions αe ∶ S → e ⟨M⟩e, one for each object e ∈ ∥E∥,forms a cylinder S M if and only if the diagram

S(αe′)e′∈E

(αe)e∈E //∏e∈∥E∥ e ⟨M⟩e∏e∈∥E∥(χe,e′)e′∈∥E∥

∏e′∈∥E∥ e′ ⟨M⟩e′

∏e′∈∥E∥(χ′e,e′)e∈∥E∥

//∏e∈∥E∥∏e′∈∥E∥ [e ⟨E⟩e′,e ⟨M⟩e′]

366

12. Ends

commutes, i.e. if and only if the function (αe)e∈E ∶ S → ∏e∈∥E∥ e ⟨M⟩e forms a fork on the

functions ∏e∈∥E∥ (χe,e′)e′∈∥E∥ and ∏e′∈∥E∥ (χ′e,e′)e∈∥E∥.

Proof. The diagram in the claim commutes i the diagram

S(αe′)e′∈E

αe // e ⟨M⟩e(χe,e′)e′∈∥E∥

∏e′∈∥E∥ e′ ⟨M⟩e′

∏e′∈∥E∥ χ′e,e′//∏e′∈∥E∥ [e ⟨E⟩e′,e ⟨M⟩e′]

commutes for each e ∈ ∥E∥, and this diagram commutes i the diagram

Sαe′

αe // e ⟨M⟩eχe,e′

e′ ⟨M⟩e′χ′e,e′

// [e ⟨E⟩e′,e ⟨M⟩e′]

commutes for each e′ ∈ ∥E∥, and this diagram commutes i the diagram

Sαe′

αe // e ⟨M⟩ee⟨M⟩h

e′ ⟨M⟩e′h⟨M⟩e

// e ⟨M⟩e′

commutes for each E-arrow h ∶ e→ e′ since for any s ∈ S,

s ⋅ [αe e ⟨M⟩h] = (s ⋅ αe) h = h ⋅ [(s ⋅ αe) ⋅ χe,e′] = h ⋅ [s ⋅ [αe χe,e′]]

ands ⋅ [αe′ h ⟨M⟩e] = h (s ⋅ αe′) = h ⋅ [(s ⋅ αe′) ⋅ χ′e,e′] = h ⋅ [s ⋅ [αe′ χ′e,e′]]

by the denitions of χe,e′ and χ′e,e′ .

Corollary 12.3.2. For a small endomodule M ∶ E E, the set of frames of M gives an endof M with the universal cylinder

π ∶∏E

MM ∶ E→ Set

dened byπe (α) = αe

such that the component of π at e ∈ ∥E∥ sends each frame α of M to its component at e.

Proof. This follows from the observation that an element α of the product ∏e∈∥E∥ e ⟨M⟩e is aframe ofM i it lies in the equalizer in Theorem12.3.1.

Remark 12.3.3. For a small endomoduleM ∶ E E, the symbol ∏EM thus denotes both theset of frames ofM and an end ofM.

367

12. Ends

Denition 12.3.4. Given a pair of left modulesM,N ∶ ∗ E, the endomodule

⟨M,N ⟩ ∶ E E


EM // Set

⟨Set⟩ // Set ENoo

, where ⟨Set⟩ denotes the hom of the category Set; that is,

e ⟨M,N ⟩d ∶= [⟨M⟩ e, ⟨N ⟩d]

for e, d ∈ E (see Remark 1.1.16(3)).

Remark 12.3.5. A frame Φ of the endomodule ⟨M,N ⟩ ∶ E E is the same thing as a naturaltransformation Φ ∶M→ N ∶ E→ Set, i,e, a left module morphism Φ ∶M→ N ∶ ∗ E.

Proposition 12.3.6. Let M,N ∶ ∗ E be a pair of left modules. Given a small set S, thereis a canonical bijection between the set of cylinders

∗S

E∼ ×E

⟨M,N ⟩

Set⟨Set⟩

//

[α]

Set

and the set of conical cells

∗ M //

S

E

N

Set⟨Set⟩

//

⟨α⟩

Set

, with each component⟨α⟩e ∶ ⟨M⟩e→ [S, ⟨N ⟩e]

of a cell ⟨α⟩ given by the simple transpose of the component

[α]e ∶ S → [⟨M⟩e, ⟨N ⟩e]

of the corresponding frame [α]; moreover, the bijection is natural in S.

Proof. Immediate since the transposition

[A, [S,B]] ⊺Ð→ [S, [A,B]]

is bijective and natural in all three variables.

Remark 12.3.7. Hence anM-weighted limit of N in Set is the same thing as an end of ⟨M,N ⟩in Set:

M∏N ≅∏ ⟨M,N ⟩

Proposition 12.3.8. If M,N ∶ ∗ E are small left modules, then an end of the endomodule⟨M,N ⟩ ∶ E E exists and is given by the set of module morphisms M→ N ∶ ∗ E:

∏E

⟨M,N ⟩ ≅M ⟨∶ E⟩N

368

12. Ends

Proof. SinceM and N are small, so is the the endomodule ⟨M,N ⟩. Hence the set of framesof ⟨M,N ⟩ gives an end of ⟨M,N ⟩ by Corollary 12.3.2. But the set of frames of ⟨M,N ⟩ is thesame thing as the set of left module morphismsM→ N (see Remark 12.3.5).

Theorem 12.3.9. For any left moduleM ∶ ∗ E and any object r ∈ ∥E∥, there is an isomor-phism

⟨M⟩ r ≅∏E

⟨r ⟨E⟩ ,M⟩.

Proof. By Theorem5.2.10 and Proposition 12.3.8,

⟨M⟩ r ≅ (r ⟨E⟩) ⟨∶ E⟩ (M) ≅∏E

⟨r ⟨E⟩ ,M⟩

.

Remark 12.3.10.(1) To apply Proposition 12.3.8, the universe may have to be enlarged temporarily so that E

becomes small. However, the result holds in the original universe, in which M is locallysmall.

(2) Some authors call the isomorphism in Theorem12.3.9 the end form of the Yoneda lemma.

Denition 12.3.11. Given a right module M ∶ E ∗ and a left module N ∶ ∗ E, theirproduct is the endomodule

⟨M ×N ⟩ ∶ E E

dened by the composition

E∼ ×EM×N // Set × Set × // Set

, where Set × Set ×Ð→ Set is the set theoretic product; that is

e ⟨M ×N ⟩d ∶= e ⟨M⟩ × ⟨N ⟩d

for e, d ∈ E.Proposition 12.3.12. Let M ∶ E ∗ be a right module and N ∶ ∗ E be a left module.Given a small set S, there is a canonical bijection between the set of cylinders

E∼ ×E

⟨M×N ⟩

∗S

Set⟨Set⟩

//

[α]

Set

and the set of conical cells

EM //

N

∗S

Set⟨Set⟩

//

⟨α⟩

Set

, with each componente ⟨α⟩ ∶ e ⟨M⟩→ [⟨N ⟩e,S]

of a cell ⟨α⟩ given by the exponential transpose of the component

[α]e ∶ e ⟨M⟩ × ⟨N ⟩e→ S

of the corresponding frame [α]; moreover, the bijection is natural in S.

369

12. Ends

Proof. Immediate since the exponential transposition

[A × B,S] ⊢Ð→ [A, [B,S]]

is bijective and natural in all three variables.

Remark 12.3.13. Hence an M-weighted colimit of N in Set is the same thing as a coend of⟨M ×N ⟩ in Set:

M∐N ≅∐ ⟨M ×N ⟩

Theorem 12.3.14.

Given a right module M ∶ E ∗ and an object r ∈ ∥E∥, there is a canonical universalcylinder

χ ∶ ⟨M × r ⟨E⟩⟩ r ⟨M⟩with each component χe ∶ e ⟨M⟩ × r ⟨E⟩e→ r ⟨M⟩ given by the composition

(m,h)↦ h m

. Given a set S and a cylinder α ∶ ⟨M × r ⟨E⟩⟩ S, the adjunct χ/α ∶ r ⟨M⟩ → S is givenby the partial evaluation of αr ∶ r ⟨M⟩ × r ⟨E⟩ r→ S at 1r, i.e. by the assignment

m↦ αr (m,1r)

. Given a left moduleM ∶ ∗ E and an object r ∈ ∥E∥, there is a canonical universal cylinder

χ ∶ ⟨⟨E⟩ r ×M⟩ ⟨M⟩ r

with each component χe ∶ e ⟨E⟩ r × ⟨M⟩e→ ⟨M⟩ r given by the composition

(h,m)↦ m h

. Given a set S and a cylinder α ∶ ⟨⟨E⟩ r ×M⟩ S, the adjunct χ/α ∶ ⟨M⟩ r → S is givenby the partial evaluation of αr ∶ r ⟨E⟩ r × ⟨M⟩ r→ S at 1r, i.e. by the assignment

m↦ αr (1r,m)

.

Proof. The naturality of χ follows immediately from the functoriality of M. To prove theuniversality of χ, it suces to show that χ/α dened as above gives a unique factorizationα = χ χ/α. Let h ∶ r→ e be an E-arrow and consider the diagram

e ⟨M⟩ × r ⟨E⟩ rh⟨M⟩×1

1×r⟨E⟩h // e ⟨M⟩ × r ⟨E⟩eχe

αe

r ⟨M⟩ × r ⟨E⟩ r χr//

αr 00

r ⟨M⟩χ/α

(( S

370

12. Ends

. The inner commutative square maps (m,1r) ∈ e ⟨M⟩ × r ⟨E⟩ r as follows:

(m,1r)_h⟨M⟩×1

1×r⟨E⟩h // (m,h)_χe

(h m,1r)

χr// h m

; hence, by the commutativity of the outer diagram and the denition of χ/α,

[χ/α] (h m) = αr (h m,1r) = αe (m,h)

. The diagrame ⟨M⟩ × r ⟨E⟩e

χe

αe

''r ⟨M⟩

χ/α// S

thus commutes for arbitrary e ∈ ∥E∥; hence α = χ χ/α. Since χr maps (m,1r) to 1r m = m,in order for the diagram

r ⟨M⟩ × r ⟨E⟩ r χr //

αr))

r ⟨M⟩χ/αS

to commute, χ/α must map m to αr (m,1r); the uniqueness of χ/α follows.

Corollary 12.3.15.

For any right module M ∶ E ∗ and any object e ∈ ∥E∥, there is an isomorphism

r ⟨M⟩ ≅∐E

⟨M × r ⟨E⟩⟩

. For any left module M ∶ ∗ E and any object r ∈ ∥E∥, there is an isomorphism

⟨M⟩ r ≅∐E

⟨⟨E⟩ r ×M⟩

.

Proof. We have shown in Theorem12.3.14 that there is a canonical universal cylinder ⟨M × r ⟨E⟩⟩r ⟨M⟩.

Remark 12.3.16. Some authors call the isomorphisms in Corollary 12.3.15 the coend form ofthe Yoneda lemma (cf. Remark 12.3.10(2)). [Lor15] contains a slick proof for the isomorphismsusing the Yoneda lemma without constructing a universal cylinder.

Denition 12.3.17. Given a small right moduleM ∶ E ∗ and a small left module N ∶ ∗ E,their tensor product, writtenM⊗EN or justM⊗N , is the quotient set dened by

M⊗EN = ∐e∈∥E∥ e ⟨M ×N ⟩e≈

, where ≈ is the equivalence relation generated by all (m,n) ≈ (m′,n′) such that the diagram

e

h

m

))∗n

55

n′ ((

∗

e′ m′

66

commutes for some E-arrow h.

371

12. Ends

Remark 12.3.18. An amalgamation of a right moduleM and a left module N is a pushout asshow in

E

M

N // ⌊N ⌋

⌊M⌋ // ⌊M⌋ +E ⌊N ⌋

. For smallM and N , the amalgamation ⌊M⌋ +E ⌊N ⌋ is constructed by pasting the collagecategories ⌊M⌋ and ⌊N ⌋ together in the obvious way and dening the hom-set between thetwo endpoints by the tensor productM⊗EN .

Proposition 12.3.19. IfM and N are small, then a coend of the endomodule ⟨M ×N ⟩ ∶ EE exists and is given by the tensor product of M and N :

∐E

⟨M ×N ⟩ ≅M⊗EN

Proof. The equivalence relation ≈ in Denition 12.3.17 is given by the coequalizer π as in

∐e∈∥E∥∐e′∈∥E∥ e′ ⟨M ×N ⟩e × e ⟨E⟩e′

∐e∈∥E∥(χe,e′)e′∈∥E∥

∐e′∈∥E∥(χ′e,e′)e∈∥E∥ //∐e′∈∥E∥ e′ ⟨M ×N ⟩e′

π

∐e∈∥E∥ e ⟨M ×N ⟩e π//M⊗EN

, where χe,e′ denotes the function e′ ⟨M ×N ⟩e× e ⟨E⟩e′ → e ⟨M ×N ⟩e given by the composi-tion ((m′,n) ,h)↦ (h m′,n), and χ′

e,e′ denotes the function e′ ⟨M ×N ⟩e×e ⟨E⟩e′ → e′ ⟨M⟩e′given by the composition ((m′,n) ,h) ↦ (m′,n h). The assertion thus follows from Theo-rem12.3.1.

Remark 12.3.20. Given a small right moduleM ∶ E ∗, there are bijections

(⟨E⟩ r) ⟨E ∶⟩ (M) ≅ r ⟨M⟩ ≅M⊗E r ⟨E⟩

, dually, for a small left moduleM ∶ ∗ E,

(r ⟨E⟩) ⟨∶ E⟩ (M) ≅ ⟨M⟩ r ≅ ⟨E⟩ r⊗EM

; the bijections on the left are the Yoneda lemma, and the bijections on the right follow fromCorollary 12.3.15 and Proposition 12.3.19. Some authors call the latter bijections the coYonedalemma (cf. Remark 11.1.6(4)).

Theorem 12.3.21. A left small module N ∶ ∗ E has a pointwise left Kan extension alongthe right Yoneda functor E, given by the tensor product as shown:

E

N

E // [E ∶]−⊗N

Set⟨Set⟩

//

µ

Set

; moreover, µ is a natural isomorphism.

Proof. By Corollary 11.5.9, it suces to show that for any right module M ∶ E ∗, M ⊗Ngives an M-weighted colimit of N . But since a M-weighted colimit of N is the same thingas a coend of ⟨M ×N ⟩ (see Remark 12.3.13), this follows from Proposition 12.3.19. Since theYoneda functor E is fully faithful, the second assertion follows from Theorem11.4.6.

372

A. List of Symbols

Symbol Usage Meaning Reference[∶] [X ∶] category of right modules over X 1.1.3

[∶A] category of left modules over A 1.1.3[K ∶] precomposition with K 1.1.3[∶ K] precomposition with K 1.1.3[X ∶A] category of modules XA 1.1.9[S ∶ T] precomposition with S ×T 1.1.9[M ∶ N ] category of cellsM N 1.3.2

⟨⟩ ⟨C⟩ hom of a category C 1.1.15⟨H⟩ hom of a functor H 1.2.27⟨τ⟩ hom of a natural transformation τ 1.3.5⟨X⟩x representable right module of x 2.3.2a ⟨A⟩ representable left module of a 2.3.2⟨X⟩G corepresentable module of G ∶A→X 2.3.6F ⟨A⟩ representable module of F ∶X→A 2.3.6

⟨, ⟩ ⟨E,C⟩ hom of category [E,C] 1.1.15⟨J ,M⟩ module of cells J M 1.2.8⟨J ,Φ⟩ postcomposition with Φ 1.2.23⟨E,M⟩ module of cylinders EM 4.3.6, 4.4.5⟨E,Φ⟩ postcomposition with Φ 4.3.16, 4.4.17⟨H,M⟩ precomposition with H 4.3.28

⟨∶⟩ ⟨X ∶⟩ hom of category [X ∶] 1.2.27⟨∶A⟩ hom of category [∶A] 1.2.27⟨X ∶A⟩ hom of category [X ∶A] 1.2.27

∥∥ ∥C∥ set of objects of C 0.3⌊⌋ ⌊M⌋ category of a collageM 3.1.1⌈⌉ ⌈K⌉ category of a comma K 3.2.7∗ ⟨∗E,M⟩ module of cones ∗EM 4.6.5

⟨E∗,M⟩ module of cones E∗M 4.6.5⟨∗E,Φ⟩ postcomposition with Φ 4.6.15⟨E∗,Φ⟩ postcomposition with Φ 4.6.15⟨∗H,M⟩ precomposition with H 4.6.26⟨H∗,M⟩ precomposition with H 4.6.26⟨E × ∗D,M⟩ module of wedges E × ∗DM 4.8.3⟨E ×D∗,M⟩ module of wedges E ×D∗M 4.8.3⟨∶ ∗E⟩ module of cones ∗ ∗E 4.10.1⟨E∗ ∶⟩ module of cones E∗ ∗ 4.10.1⟨X ∶ ∗E⟩ module of wedges X ∗E 4.10.1⟨E∗ ∶A⟩ module of wedges E∗A 4.10.1

∼ C∼ opposite module of C 0.6M∼ opposite module ofM 1.1.27

373

A. List of Symbols

Symbol Usage Meaning Reference↑ [X ↑A] category of collages XA 3.1.4

[M ↑ N ] category of collage cellsM N 3.1.8K↑ module of K 3.2.211↑K unit cylinder of K 10.1.3

unit cone of K 10.4.1Φ↑ collage transpose of Φ 10.4.9

↓ [X ↓] category of right commas over X 3.2.4[↓A] category of left commas over A 3.2.4[X ↓A] category of commas XA 3.2.10M↓ comma ofM 3.2.17M↓a right comma ofM at a 3.2.28x↓M left comma ofM at x 3.2.28X↓x right comma of X at x 3.2.30a↓A left comma of A at a 3.2.281↓M unit cylinder ofM 10.1.3

unit cone ofM 10.4.1Φ↓ comma transpose of Φ 10.4.9

⟨X A⟩ cylinder module [X ↓A] [X ↑A] 10.3.1⟨X ⟩ cone module [X ↓] [X ∶] 10.4.2⟨A⟩ cone module [↓A] [∶A] 10.4.2

M right exponential transpose ofM 2.1.1ME right action ofM ∶XA on [E,A] 2.2.1X right Yoneda functor for X 2.3.1XA right general Yoneda functor for [A,X] 2.3.5

M left exponential transpose ofM 2.1.1EM left action ofM ∶XA on [E,X] 2.2.1A left Yoneda functor for A 2.3.1XA left general Yoneda functor for [X,A] 2.3.5

M right comma exponential transpose ofM 3.2.27X right comma exponential transpose of X 3.2.29

M left comma exponential transpose ofM 3.2.27A left comma exponential transpose of A 3.2.29

d Xd right Yoneda module for X 5.1.1XdA right general Yoneda module for [A,X] 5.1.6

e eA left Yoneda module for A 5.1.1XeA left general Yoneda module for [X,A] 5.1.6

M corepresentable module ofM 11.1.1X corepresentable module of X 11.1.3ME corepresentable module ofME 11.1.7XE corepresentable module of XE 11.1.9

M representable module of M 11.1.1A representable module of A 11.1.3EM representable module of EM 11.1.7EA representable module of EA 11.1.9

M corepresentable module ofM 11.1.5 M representable module of M 11.1.5⊢ K⊣ right exponential transpose of K 0.11

374

A. List of Symbols

Symbol Usage Meaning Referenceα⊣ right exponential transpose of α 4.7.5, 4.8.5

⊣ K⊢ left exponential transpose of K 0.11α⊢ left exponential transpose of α 4.7.5, 4.8.5

⊺ K⊺ simple transpose of K 0.11α⊺ simple transpose of α 4.7.7, 4.8.7

⟨F M⟩ module of cylinders left weighted by F 4.5.3⟨F Φ⟩ postcomposition with Φ 4.5.7

⟨F M⟩ module of cylinders right weighted by F 4.5.3⟨F Φ⟩ postcomposition with Φ 4.5.7

⟨XM⟩ right hom ofM ∶XA 5.2.1⟨XM⟩ right Yoneda morphism forM 5.2.5⟨XM⟩E right general Yoneda morphism for ⟨E,M⟩ 5.3.3Xm module morphism generated by X direct along m 5.2.4Xα module morphism generated by X direct along α 5.3.2X Yoneda representation 5.2.17XA general Yoneda representation 5.3.18

⟨MA⟩ left hom ofM ∶XA 5.2.1⟨MA⟩ left Yoneda morphism forM 5.2.5E⟨MA⟩ left general Yoneda morphism for ⟨E,M⟩ 5.3.3mA module morphism generated by A inverse along m 5.2.4αA module morphism generated by A inverse along α 5.3.2A Yoneda corepresentation 5.2.17XA general Yoneda corepresentation 5.3.18

X inverse of X 5.3.12 A inverse of A 5.3.12U ⟨MUN ⟩ module of right slant cellsM N 8.1.5

⟨XUA⟩ module of right slant cells XA 8.3.1Q ⟨N QM⟩ module of left slant cells N M 8.1.5

⟨AQX⟩ module of left slant cells AX 8.3.1/ m/u adjunct of m inverse along u 6.1.2/ u/m adjunct of m direct along u 6.1.2

∏ ∏EM set of frames ofM ∶ E E 4.1.1

∏E Φ postcomposition with Φ 4.1.5

∏H precomposition with H 4.1.8

∏∗EM set of frames ofM ∶ ∗ E 4.2.1

∏E∗M set of frames ofM ∶ E ∗ 4.2.1

∏∗E Φ postcomposition with Φ 4.2.7

∏E∗ Φ postcomposition with Φ 4.2.7

∏∗H precomposition with H 4.2.11

∏H∗ precomposition with H 4.2.11

∏K limit of K 7.1.1end of K 12.1.1

∏E K E-parameterized limit of K 7.2.7E-parameterized end of K 12.2.5

∏J K J -weighted limit of K 11.2.9

∐ ∐K colimit of K 7.1.1coend of K 12.1.1

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A. List of Symbols

Symbol Usage Meaning Reference

∐E K E-parameterized colimit of K 7.2.7E-parameterized coend of K 12.2.5

∐J K J -weighted colimit of K 11.2.9∆ ∆E functor E→ ∗ 0.17

∆c constant functor on c 0.18∆E∗ representable module of E→ ∗ 1.1.23∆∗E corepresentable module of E→ ∗ 1.1.23

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Bibliography

[AHS09] J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories: TheJoy of Cats. Dover Publications, 2009.

[Awo06] Steve Awodey, Category Theory (Oxford Logic Guides). Oxford University Press,2006.

[Bor94] Francis Borceux, Handbook of Categorical Algebra: Volume 1, Basic Category Theory,Cambridge University Press, 1994.

[Ell06] David Ellerman, A Theory of Adjoint Functors - with Some Thoughts on Their Philo-sophical Signicance, in What Is Category Theory?, edited by Giandomenico Sica, 127-83.Polimetrica, 2006.

[Ell07] David Ellerman, Adjoint Functors and Heteromorphisms, arXiv:0704.2207, 2007.

[Kel05] Max Kelly, Basic Concepts of Enriched Category Theory, Reprints in Theory andApplications of Categories, No. 10, 2005.

[Lor15] Fosco Loregian, This is the (co)end, my only (co)friend, arXiv:1501.02503.

[ML98] Saunders MacLane, Categories for the Working Mathematician, Springer, 2nd ed. 1998.

[nL] nLab. http://ncatlab.org/nlab/show/HomePage.

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Documents

Takahiro Kato May 24, 2017 - viXravixra.org/pdf/1503.0085v6.pdf · The term profunctor is quite appropriate in this regard. However, the notion has another, probably more important,