Category theory and set theory: algebraic settheory as an example of their interaction

Brice Halimi

May 30, 2014

My talk will be devoted to an example of positive interactionbetween (ZFC-style) set theory and category theory, namelyAlgebraic Set Theory (AST).

I will focus on the first three axioms of the originalaxiomatization of AST by André Joyal and Ieke Moerdijk.

I will argue that AST can be described as a genuine graft ofthe theory of fibered categories onto ZFC, which goesbeyond the unfruitful opposition often established between settheory and category theory.

Championing either Set Theory or Category Theory is justextrapolating a tradition within mathematics (analysis andlogical semantics in the case of Set Theory, algebraic geometryand modern algebraic topology in the case of CategoryTheory).

“Foundations” are provided neither by ST nor by CT.

It is better to try to go beyond the contest.

Presentation of AST

Two different approaches : a logical one (Alex Simpson, SteveAwodey), a geometrical one (Joyal-Moerdijk).

Joyal & Moerdijk: generalization of models of ZF (“freeZF-algebras”).

Awodey & co: general program of completeness resultsbetween different axiomatizations of set theory and differentcollections of “categories of classes.”

Let C be a Heyting pretopos, which means that C is richenough to interpret first-order logic and arithmetic. (Still, C isnot supposed to be a topos, and in particular to have “powerobjects.”)

The basic idea is to characterize a special subcollection ofthe collection of all maps of C: a class S of “small maps”in C.

Intuitively, an arrow is a small map if all its fibers have aset-like size.

Then, an object X of C is said to be small if X → 1 is a smallarrow.

Upshot: Arrows are brought to the fore, instead of sets, andsets themselves are conceived of as “fibers.”

Axioms for the class S:1. Any isomorphism of C belongs to S and S is closed under

composition.This axiom corresponds to the fact that the union of a smallfamily of small sets is a small set.

2. Stability under change base: for any pullback

Y ′ //

g��

Y

f��

X ′ p// X ,

if f belongs to S, so does g.3. “Descent”: for any pullback along an epimorphic arrow p

Y ′ //

g��

Y

f��

X ′ p// // X ,

if g belongs to S, so does f .

4. The arrows 0→ 1 and 1 + 1→ 1 belong to S.This axiom ensures that the empty set (initial object) issmall and that any finite set is small.

5. If two arrows f : Y → X and f ′ : Y ′ → X ′ belong to S, thenso does their sum f + f ′ : Y + Y ′ → X + X ′.This axiom ensures that the disjoint union of two small sets(objets) is a small set.

6. In any commutative diagram

Z // //p //

g ��???????? Y

f����������

B ,

if g belong to S, then so does f .

If L is a partially ordered set (“poset”) in C, then HomC(A,L) isitself a poset.

For g : B → A and λ : B → L, the supremum of λ along g is amap µ : A→ L s.t., for any t : L′ → A and any ν : L′ → L,

L′ ×A Bp2 //___

p1����� B

λ

��========

g��

L′

ν

77t // A

µ //___ L

λ ◦ p2 ≤ ν ◦ p1 in HomC(L′ ×A B,L) iff µ ◦ t ≤ ν in HomC(L′,L)

L is said to be S-complete if any map to L has a supremumalong any map in S.

Definition: A ZF-algebra in C is an S-complete poset L with alljoins, endowed with a map s : L→ L.

The partial order ≤ of L corresponds to inclusion, and the maps corresponds to the singleton operation.

Given a ZF-algebra (L, s), it is possible to define a “membershiprelation” ε // // L× L on L, by setting: x ε y iff s(x) ≤ y forany x and y “in” L.

A notion of homomorphism of ZF-algebras in C is readilyavailable.

Fact: Any object A of C generates a free ZF-algebra in C,written V〈C,S〉(A).

V〈C,S〉(A) is the (generalized) cumulative hierarchy on A.

Fact: Adding two further mild assumptions about S, one gets:

A + V〈C,S〉(A) is a model of ZF set theory with atoms.

The main purpose of my talk is to explain the elliptic phrase“Descent” used to describe the third axiom of AST.

Descent theory is a central tool of abstract algebraic geometry,coming from Grothendieck’s work.

Descent theory makes sense only in the general context of“fibered categories,” a.k.a. “fibrations.”

So I will explain:I what fibrations areI what descent consists inI the rationale of AST’s first three axioms.

Fibrations

Fibration (fibered category) = category-theoretic generalizationof the notion of surjective map.

A fibration is a functor p : F → C.

For each C, p−1(X ) is a category written FX , and called the“fiber above X .”

Each arrow u : X ′ → X in C gives rise to a “base change”functor u∗ : FX → FX ′ between the corresponding fibers (in thereverse direction).

#"

!•

X ′

AAAAA

�����

FX ′

>u •

X

AAAAA

�����

?

?

?

FX? ??

?� u∗

F = total space

C = base category?

p = fibration?p

A fibration in general

#"

!•

V

AAAAA

�����

ContV

>u •

U

AAAAA

�����

?

?g : V → Y?

u∗(f )

ContU? ??

?f : U → X� u∗ Cont = continuous

functions

Top = topologicalspaces

?

domain functor

?dom

For f : U → X , u∗(f) = f ◦ u : V → X

EXAMPLE of fibration

Given a category C, let C→ be the category of all arrows in C.The codomain functor cod : C→ → C is a fibration.

The fiber above each X ∈ Ob C contains all the arrows in Cwith codomain X .

The class S of small maps is thought of as a sub-fibrationS → C of the codomain fibration cod : C→ → C.

Descent theory

Descent theory = abstract framework geared to describeglueing processes, i.e. the shift from local data to a global item.

TYPICAL EXAMPLE.Let (Ui)i∈I be a covering of a space U, and suppose that foreach i ∈ I a continuous function fi : Ui → V is given, in such away that

∀i , j ∈ I fi |Uij = fj |Uij ,

where Uij := Ui ∩ Uj .

Then there exists a unique function f : U → V s.t.∀i ∈ I f |Ui = fi .

Reformulation in the context of dom : Cont→ Top

Let’s write U ′ :=∐

i∈I Ui . The canonical map α : U ′ → U is bothcontinuous and surjective.And the family (fi)i∈I exactly constitutes an object in ContU′ .

Let’s introduce the following pullback: U ′ ×U U ′p2 //___

p1����� U ′

α

��U ′ α

// UU ′ ×U U ′ '

∐i,j∈I Uij

For x ∈ Uij , p1(x) = x ∈ Ui and p2(x) = x ∈ Uj

For (fi)i∈I ∈ ContU′ :I p∗1((fi)i) = (fi |Uij)i,jI and p∗2((fi)i) = (fi |Uji)i,j = (fj |Uij)i,j .

So the hypothesis ∀i , j ∈ I φij : fi |Uij = fj |Uij becomes:

p∗1((fi)i) = p∗2((fi)i)

U ′ ×U U ′ U ′ U

(Ui ∩ Uj)i,j (Ui)i U

p∗1((fi)i) = p∗2((fi)i) in ContU′×UU′

U ′ ×U U ′ U ′ U

(Ui ∩ Uj)i,j (Ui)i U

More generally, replacing identity with an isomorphism, oneneeds an isomorphism

φ : p∗1((fi)i) ' p∗2((fi)i) in ContU′×U′ ,

i.e. a familyφij : p∗1(fi) ' p∗2(fi)

of isomorphisms in the fibers above the Uij ’s (thoseisomorphisms being compatible when they overlap).

You can glue the local data above the components Ui ofthe covering U ′, so as to get a global item above U =

⋃i Ui .

Grothendieck, “Technique de descente et théorèmesd’existence en géométrie algébrique” (1959).

Let F be a fibration with base category C.Recall that any arrow f : X ′′ → X ′ in C gives rise to a functorf ∗ : FX ′ → FX ′′ .

Given an arrow α : X ′ → X in C and ξ′ ∈ FX ′ , a descent datumon ξ′ ∈ FX ′ w.r.t. α is an isomorphism

p∗1(ξ′) ' p∗2(ξ

′) in FX ′×X X ′

satisfying compatibility conditions:

p∗1(ξ′) ' p∗2(ξ

′) ξ′ ?

X ′ ×X X ′p1 //

p2// X ′ α // X

(Ui ∩ Uj) (Ui) (U)

An arrow α : X ′ → X in C is a strict F -descent morphism if:

(A) αp1 = αp2, and for any two objects ξ, η of FX ,α∗ : FX → FX ′ induces a 1-1 map from HomFX (ξ, η) to thesubset of HomFX ′

(α∗(ξ), α∗(η)) consisting of all the arrowsv : α∗(ξ)→ α∗(η) such that p∗1(v) = p∗2(v)

(B) giving an object of FX exactly amounts to giving an objectof FX ′ equipped with a descent datum w.r.t. α.

“What should glue actually glues along α” (see infra).

p∗1(v) = p∗2(v) means that all the components of v haveidentical restrictions to X ′ ×X X ′.

p∗2 (v)

��

p∗1 (v)

��

α∗(ξ)

v��

ξ

u

��= α∗(η) η

X ′ ×X X ′p1 //

p2// X ′ α // X

(Ui ∩ Uj) (Ui) (U)

Condition (A) says that in that case the components of v canactually be glued together along α, so as to come from somearrow u in FX :

v = α∗(u)

A descent datum is a family ξ′ ∈ FX ′ of objects whoserespective restrictions in FX ′×X X ′ all agree (up to isomorphism).

ξ′ij ' ξ′ji ξ′ ξ

X ′ ×X X ′p1 //

p2// X ′ α // X

(Ui ∩ Uj) (Ui) (U)

Condition (B) says that in that case the objects in the family canbe glued together along α, so as to be essentially equivalent toan object in FX :

ξ′ ' α∗(ξ) for some ξ

Reconstruction of the link with AST

An arrow α : X ′ → X is a strict descent morphism if it is a

strict F -descent morphism for F = C→ cod // C .

[Grothendieck 1959, Proposition 2.1]If C has finite products and pullbacks, then the strict descentmorphisms in C are exactly the (universal strict) epimorphismsin C.

[Reformulation of Joyal-Moerdijk’s Axiome 3]Any epimorphism in C is a strict descent morphism for thesub-fibration pS : S → C of the codomain fibrationcod : C→ → C.

Proposition: A subcategory S of C→ satisfies the Axioms 1, 2and 3 of a collection of small maps (in the sense of AST) iff pSsatisfies Grothendieck’s property.

Reinterpretation of AST’s first three axiomsI Axiom 1 says that all isomorphisms in C belong to S,

which implies that, for any object A of C, 1A belongs to S,and thus that there is at least one object (arrow) in S aboveeach object of C. So the category S can really be said tobe “above” C.

I Axiom 2 then says that any pullback of any arrow in Sbelongs itself to S, which ensures that the restriction to Sof the codomain fibration cod : C→ → C is itself a fibration,in other words that pS is a sub-fibration of C→ → C. (Thisis more than just saying that S is a subcategory of C→.)

I Axiom 3 finally says that Grothendieck’s result aboutcod : C→ → C remains true about pS : S → C.(The fact that C is a Heyting pretopos guarantees that anyepimorphism in C is a universal strict epimorphism.)

A class of small maps can be partially characterized as acategory to which descent theory applies.

For further work

In the context of descent theory, the natural extension of afibration is a stack.

Gabriele Vezzosi: If a fibration is a family of categorical objectswhich “pullback like bundles,” a stack is a family of such objectswhich moreover “glue like bundles.”

The notion of stack makes sense only on the condition that thebase category C is endowed with a Grothendieck topology.

A Grothendieck topology J on a category C consists of acollection J(X ) of “covering families” {Xi → X} on X , for eachobject X of C, in such a way that the different J(X )’s areconnected in a systematic way.

Given a base category C endowed with a Grothendiecktopology J, a stack over C is a fibration over C such that anycovering {Xi → X} in J(X ) amounts to a strict descentmorphism.Examples: the fibration dom : Cont→ Top is a stack,cod : C→ → C is a stack for a certain topology on C.

Let 〈C,S〉 a Heyting pretopos C with a class S of small maps.Since C is a pretopos, there is a natural Grothendieck topologyTc on C, called the coherent topology, which is generated by allthe finite jointly epimorphic families of arrows.

Natural question: For which properties of S is pS : S → Censured to be a stack w.r.t. the site 〈C, Tc〉?

Conclusion

From a set-theoretic viewpoint, any map is in fact a set: there isno arrow. The only arrows are the edges of the membershipgraph.

On the contrary, AST reinterprets set theory so as to turn it intoan arrow-based theory (along the codomain fibration), ratherthan an object-based theory (which is typical of thecategory-theoretic point of view).

There is more:

AST = graft of the theory of fibrations onto ZFC

AST = categorical implementation of ZFC with a deepgeometrical twist, and as such an example of combinationbetween “foundational mathematics” and “mathematicalpractice.”

THANK YOU !