Verdier’s Localization Theorem

John Zhang

June 3, 2012

1 Introduction and Motivation for Category Theory

Category Theory is an area of mathematical study that examines abstractly the proper-ties of mathematical concepts and constructs and the interactions between them, so longas these constructs satisfy certain basic requirements. For a long time, category theoryalso went by the name of “general abstract nonsense.” Today, category theory is takenmore seriously and has evolved into a field of study on its own. Because of its purposeof analyzing relations between concepts (e.g. sets, vector spaces, groups, fields, and soforth), mathematicians studying categories attempted in a variety of ways to be able toinvert morphisms within category. In this paper, we present one of the earlier and morebasic ways of determining this, Verdier’s localization Theorem.

2 Basic Definitions

Notes: Within this paper, letters such as C, etc, will denote categories, normal capi-tal letters F, G will denote functors, greek letters and/or lowercase letters will denotemorphisms, and letters W, X, Y, Z will denote objects. As keeping with the norms ofcategory theory, the composition of f and g will be denoted gf or g ◦ f . Most of thedefinitions given in this section come from [2] and its appendices on basics in categorytheory.

Definition 1: A category C contains the following: a class obj(C) of objects, a setHomC(A,B) for every ordered pair (A,B) of objects, an identity morphism idA ∈HomC(A,A) for each objectA, and a composition functionHomC(A,B)×HomC(B,C)→HomC(A,C) for every ordered triple (A,B,C) of objects. We will use f : A → B todenote a morphism from A to B and we will write gf (note, not fg) for the compositionof two morphisms f : A→ B and g : B → C.

Morphisms and Hom-sets are subject to two axioms.

Associativity Axiom:(hg)f = h(gf) for any morphisms f : A → B, g : B → C, andh : C → D.

Unit Axiom:idBf = fidA for any morphism f : A→ B.

1

Example 1:The fundamental categoy to keep in mind is the category Sets of sets. Theobjects are sets and the morphisms are set functions, that is HomSets(A,B) are thefunctions from A to B. Composition of morphisms is just composition of functions.idA(a) = a, for all a ∈ A.

This example also explains the pedantic insistence of using a class of objects, as theobjects of Sets do not form a set themselves.

Definition 2: A category C is called small if objC forms a set. For example, the categorySets is not small, as explained above.

Definition 3:(Initial and Terminal Objects) Let C be a category. An object I in Cis called an initial object of C if for every object O ∈ C, there is exactly one morphismto from I to O. Similarly, an element T in C is called a terminal object of C if for everyobject O ∈ C there is exactly one morphism from O to T . All initial and terminal objectsare isomorphic. Furthermore, if an object is both initial and terminal, then it is calleda zero object.

Related: The Zero Morphism: Suppose C has a zero object, which we will denote by 0.Then for any two objects B and C in C there is a single morphism in HomC(B,C), themorphism given by B → 0→ C. The morphism is unique because there is precisely onemorphism from B to 0 and one morphism from 0 to C, per the definition of the zeroobjects. The existence of zero morphisms is important because of its uniqueness and thefact that it is relatively easy to work with.

Definition 4:(Products and Coproducts) Suppose {Ci : i ∈ I} is a set of objects ofC, a product

∏i∈I Ci, if it exists, is an object of C, together with maps πj :

∏Ci → Cj ,

for j ∈ I, such that for every object A ∈ C, and every family of morphisms αi : A→ Ci,there is a unique morphism α : A→

∏Ci such that πiα = αi for all i ∈ I.

Dually, we can define a coproduct∐Ci of a set of objects in C as an object of C,

together with maps νj : Cj →∐Cisuch that for every family of morphisms αi : Ci → A

there is a unique morphism α :∐Ci → A such that ανj = αj for all j ∈ I.

With respect to the maps, we can think of one as a projection and the other as an“insertion” of sorts.

Definition 5:(Functors) Let C, D be two categories. By a functor F : C → D, we meana rule that associates an object F (C) ∈ D to every object C ∈ C, and a morphismF (f) : F (C1) → F (C2) in D to objects C1, C2 ∈ C and a morphism f : C1 → C2.A functor is required to preserve identity morphisms i.e. F (idC) = idF (C), as well ascompositions F (gf) = F (g)F (f) for any morphisms g : C1 → C2 and f : C0 → C1.

We may define composition of functors in the obvious way. Suppose E is anothercategory. Then we may define the composite GF : C → E by (GF )(C) = G(F (C)) and(GF )(f) = G(F (f)) for objects C ∈ C and morphisms f .

Example 2: The identity functor idC : C → C is the rule fixing all objects and morphisms,that is, idC(C) = C for any object C ∈ C and idC(f) = f for any morphism f .

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Definition 6: (Subcategories) Suppose C is a category. A subcategory B of C is a col-lection of some of the objects of C and some of the morphisms, such that the morphismsof B are closed under composition and include idB for every B ∈ B. A subcategory isa category in its own right and we can define the inclusion functor F : B → C as theidentity of objects and morphisms of B.

Definition 7: The Opposite Category Every category C has an opposite category Cop.The objects of Cop are the same as the objects of C, but the morphisms are reversed.

Definition 8: Natural TransformationsSuppose F and G are two functors from C to D. A natural transformation η : F → G

is a rule that associated a morphism ηC : F (C)⇒ G(C) in D to every object C of C insuch a way that for every morphism F : C → C ′ in C, the following diagram commutes,that is to say,

F (C)Ff−−−−→ F (C ′)yη yη

G(C)Gf−−−−→ G(C ′)

We say a diagram commutes if, by proceeding along any path of arrows in thediagram, the result is the same. All of our diagrams will be commmutative.

Types of Subcategories

• Full Subcategories: Suppose B is a subcategory of C, and suppose thatHomB(B,B′) =HomC(B,B

′) for any two objects B and B′ in obj(B). Then we call B a full sub-category of C.

Types of Functors

• Faithful Functors: A functor F : C → D is called faithful if the set mapsHomC(C,C′)→

HomD(F (C), F (C ′)) are injections.

• Full Functors:A functor F : C → D is called full if the set maps HomC(C,C′) →

HomD(F (C), F (C ′)) are surjections.

A functor that is both full and faithful is called fully faithful. For example, theinclusion functor defined for a full subcategory is fully faithful.

Definition 9: Equivalence We call a functor F : C → D an equivalence of categories ifthere is a functor G : D → C and natural isomorphisms idC ∼= GF .

Definition 10: Abelian Category A category C is called a Ab category if every hom-set HomC(C,D) in C is given the structure of an abelian group in such a way thatcomposition distributes over ”addition,” that is to say, given a diagram in C of the form

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Af−−−−→ B

g−−−−→g′

Ch−−−−→ D

we have h(g′+g)f = hg′f+hgf . What does it mean to be able to “add” morphisms?It simply means that we can take a “sum” of morphisms, when h is left invertible i.e.there is a map α with h ◦ α = 1, and when f is right invertible i.e. there is β withβ ◦ f = 1. These restrictions are necessary because only in this case does the additivityproperty g′ + g become well-defined. What this really means is that f and h have quitewell-defined kernels. The property is rather subtle, and we cannot go into full detail. Itsmajor use is in the definition of the additive category, given below.

Definition 11: Additive Category We call C an additive category if it is an Ab-category, with a zero object, and a product A × B, which suffices as both the productand the coproduct, for every pair of objects A and B of C. This product structure makesfinite coproducts and products the same.

Within this section, we have constructed the basic terminology and machinery thatwe will need to tackle category theory in general. Below, we will give a construction ofthe basic study of triangulated categories and thereby prove the main result, Verdier’sLocalization Theorem. Our treatment of triangulated categories is based upon [1], Am-non Neeman’s treatment of triangulated categories.

The modern exposition and construction of triangulated categories is somewhat dif-ferent than Verdier’s original construction. Verdier utilized an axiom known as theOctahedral Axiom, which is marginally weaker than the mapping cone axiom givenhere, and in fact can be proved from the mapping cone axiom. The original expositionof triangulated categories was given by Jean-Louis Verdier, in his 1963 Thesis (French),which was unpublished until 1996. A few years earlier they had been similarly definedby Puppe (German).

3 Triangulated Categories

The notions of triangulated category and derived category (which is a related concept,though we will not discuss it here), were introduced by the French mathematician Jean-Louis Verdier in his 1963 Thesis, based partially on the ideas of Alexander Grothendieck.The concept of a triangulated category is analoguous to certain properties of homotopycategories, as well as the derived category of an Abelian category. Triangulated categoriesform an immensely abstract field of study, and unfortunately we will be unable to provideexamples.

The study of triangulated categories was motivated by certain results in topologyand homotopy theory (Puppe) and algebraic geometry (Verdier).

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We will begin by constructing triangulated categories. To do so, we first construct apre-triangulated category.

Definition 1:(Neeman Defn. 1.1.1) Let C be an additive category, and let Σ : C → C bean additive, invertible automorphism of C. Then with respect to Σ, a candidate trianglein C is a diagram of the form

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

such that the compositions v ◦ u, w ◦ v, and Σu ◦ w are the zero morphisms.

A morphism of candidate triangles is a commutative diagram of the form

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

f

y g

y h

y Σf

yX ′

u′−−−−→ Y ′v′−−−−→ Z ′

w′−−−−→ ΣX ′

where each row is a candidate triangle. We now give the definition of a pre-triangulatedcategory.

Definition 2:(Neeman Defn 1.1.2) A pre-triangulated category T is an additive category,along with an additive automorphism Σ, and a class of candidate triangles with respectto Σ called distinguished triangles, satisfying the following conditions.

TR0: Any candidate triangle which is isomorphic to a distinguished triangle is adistinguished triangle. The candidate triangle

X1−−−−→ X −−−−→ 0 −−−−→ ΣX

is distinguished. 1 is the identity.

TR1: For any morphism f : X → Y in T there exists a distinguished triangle of theform

Xf−−−−→ Y −−−−→ Z −−−−→ ΣX

TR2: Consider two candidate triangles.

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

and

Y−v−−−−→ Z

−w−−−−→ ΣX−Σu−−−−→ ΣY

If one is a distinguished triangle, so is the other.

TR3: For any diagram of the form

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Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

f

y g

yX ′

u′−−−−→ Y ′v′−−−−→ Z ′

w′−−−−→ ΣX ′

there exists a morphism h : Z → Z ′, not necessarily unique, with

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

f

y g

y h

y Σf

yX ′

u′−−−−→ Y ′v′−−−−→ Z ′

w′−−−−→ ΣX ′

commutative. If a category satisfies the above 4 axioms it is known as a pre-triangulated category. Certain parts of the definition are redundant, for example, Tis not required to be additive. Nevertheless, the above definition will suffice for thepurposes of this paper.

Note on terminology: If T is a pre-triangulated category and we speak of trianglesin T we mean distinguished triangles. Candidate triangles will always have an explicitadjective.

We must know a little more about the automorphism Σ.

Proposition 1(Neeman, Prop 1.1.6) Let T be a pre-triangulated category. Then thefunctor Σ preserves products and coproducts. That is to say, if {Xλ, λ ∈ Λ} is a set ofobjects in T , and the coproduct

∐λ∈ΛXλ exists in T , then the natural map∐

λ∈Λ{ΣXλ} −−−−→ Σ{∐λ∈ΛXλ}

is an isomorphism. In other words, the natural maps induced

ΣXλ −−−−→ Σ{∐λ∈ΛXλ}

give Σ{∐λ∈ΛXλ} the structure of a coproduct in the category T . Similarly if the

product exists then the natural maps

Σ{∏λ∈ΛXλ} −−−−→ ΣXλ

will give Σ{∏λ∈ΛXλ} the structure of a product in T .

ProofSince Σ is invertible, it has a left and right inverse, Σ−1. There are natural isomor-

phisms

Hom(ΣX,Y ) −−−−→ Hom(X,Σ−1, Y )

and

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Hom(X,ΣY ) −−−−→ Hom(Σ−1X,Y )

induced by the inverse. A functor with a left inverse thus respects products and aright inverse respects coproducts. Thus Σ respects both.

Definition 3(Neeman, Defn 1.1.7) Let T be a pre-triangulated category. let H be afunctor from T to an abelian category A. Then the functor H is called homological iffor every distinguished triangle

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

the sequence

H(X)H(u)−−−−→ H(Y )

H(v)−−−−→ H(z)

is exact in the abelian category, that is to say, the composition H(v) ◦H(u) is thezero morphism. Similarly, a functor H is called cohomological if it is homological for theopposite category.

Lemma 1(Neeman, Lemma 1.1.10) Let T be a pre-triangulated category and let Ube an object of T . Then the functor Hom(U,−) is homological. The functor takesX → Hom(U,X).

Proof Suppose we have a triangle

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

We would like to show the exactness of the sequence

Hom(U,X) −−−−→ Hom(U, Y ) −−−−→ Hom(U,Z)

We know that the composite is 0. Let f ∈ Hom(U, Y ) map to 0 in Hom(U,Z). Thatis, let f : U → Y be such that the composite

Uf−−−−→ Y

v−−−−→ Z

is 0. Then we have a commutative diagram

U −−−−→ 0 −−−−→ ΣU−1−−−−→ ΣU

f

y f

y Σf

yY

−v−−−−→ Z−w−−−−→ ΣX

−Σu−−−−→ ΣY

and the top is a triangle by [TR0], [TR2] and the bottom by [TR2]. Thus by [TR3]there is a map h : U → X with Σh : ΣU → ΣX makes the diagram above commute.Thus the square

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ΣU−1−−−−→ ΣU

Σh

y Σf

yΣX

−Σu−−−−→ ΣY

commutes. Thus Σf = Σu ◦ Σh, or f = u ◦ h.

Definition 4 (Neeman, Defn 1.1.12) Let H : T → A be a homological functor. Thefunctor H is called decent if• The abelian category has well-defined products and the product of exact sequences

is exact

• The functor H respects products, that is to say, for any collection {Xλ, λ ∈ Λ} ofobjects Xλ ∈ T the map

H(∏λ∈ΛXλ) −−−−→

∏λ∈ΛH(Xλ)

is an isomorphism.

Definition 5(Neeman, Defn 1.1.14) Let T be a pre-triangulated category. A candidatetriangle

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

is called a pre-triangle, if , for every decent homological functor H : T → A, thesequence

H(Σ−1Z)H(Σ−1w)−−−−−−→ H(X)

H(u)−−−−→ H(Y )H(v)−−−−→ H(Z)

H(w)−−−−→ H(ΣX)

is exact.

Caution There are pre-triangles which are not distinguished.

Lemma 2(Neeman, Lemma 1.1.17) Let Λ be a set, and suppose that for every λ ∈ Λ weare given a pre-triangle

Xλ −−−−→ Yλ −−−−→ Zλ −−−−→ ΣXλ

and that the products∏λ∈ΛXλ,

∏λ∈Λ Yλ, and

∏λ∈Λ Zλ exist in T . The sequence∏

λ∈ΛXλ −−−−→∏λ∈Λ Yλ −−−−→

∏λ∈Λ Zλ −−−−→ {

∏λ∈Λ ΣXλ}

is identified as∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ −−−−→

∏λ∈Λ Zλ −−−−→ Σ{

∏λ∈ΛXλ}

and this candidate triangle is a pre-triangle. Thus the product of pre-triangles is apre-triangle.

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Proof Let H : T → A be a decent homological functor. Because for each λ ∈ Λ thesequence

Xλ −−−−→ Yλ −−−−→ Zλ −−−−→ ΣXλ

is a pre-triangle, applying H we obtain an exact sequence in A,

H(Σ−1Zλ) −−−−→ H(Xλ) −−−−→ H(Yλ) −−−−→ H(Zλ) −−−−→ H(ΣXλ)

the product of such sequences are exact by the definition of a decent homologicalfunctor. Thus the maps

H(∏λ∈ΛXλ) −−−−→

∏λ∈ΛH(Xλ)

H(∏λ∈Λ Yλ) −−−−→

∏λ∈ΛH(Yλ)

H(∏λ∈Λ Zλ) −−−−→

∏λ∈ΛH(Zλ)

are all isomorphisms. We may apply the functor H to the sequence

H(Σ−1Zλ) −−−−→ H(Xλ) −−−−→ H(Yλ) −−−−→ H(Zλ) −−−−→ H(ΣXλ)

in order to obtain a long exact sequence. Thus the sequence is a pre-triangle.

Proposition 2(Neeman, Prop 1.2.1) Let T be a pre-triangulated category and let Λ beany set. Suppose that for any λ ∈ Λ we are given a distinguished triangle

Xλ −−−−→ Yλ −−−−→ Zλ −−−−→ ΣXλ

Suppose moreover that the products∏λ∈ΛXλ,

∏λ∈Λ Yλ, and

∏λ∈Λ Zλ exist in T .

Then we already know that the triangle formed by the product∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ −−−−→

∏λ∈Λ Zλ −−−−→ Σ{

∏λ∈ΛXλ}

is a pre-triangle. We assert that it is a distinguished triangle.

Proof By [TR1], the map ∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ

can be completed to a triangle.∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ −−−−→ Q −−−−→ Σ{

∏λ∈ΛXλ}

Now for each λ ∈ Λ there is a corresponding triangle, so we have a diagram withrows as triangles.

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∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ −−−−→ Q −−−−→ Σ{

∏λ∈ΛXλ}y y y

Xλ −−−−→ Yλ −−−−→ Zλ −−−−→ ΣXλ

which by [TR3] may be completed as a morphism of triangles∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ −−−−→ Q −−−−→ Σ{

∏λ∈ΛXλ}y y y y

Xλ −−−−→ Yλ −−−−→ Zλ −−−−→ ΣXλ

Take the product over Λ to obtain∏λ∈ΛXλ −−−−→

∏λ∈Λ Yλ −−−−→ Q −−−−→ Σ{

∏λ∈ΛXλ}y y y y∏

λ∈ΛXλ −−−−→∏λ∈Λ Yλ −−−−→

∏λ∈Λ Zλ −−−−→ Σ{

∏λ∈ΛXλ}

as a map of pre-triangles. Thus the map is actually an isomorphism, so since the toprow is a distinguished triangle, the bottom row is a triangle.

We will give a definition of a triangulated category. The new axiom based on thoseof pre-triangulated categories is a stronger version of [TR3].Definition 6 (Neeman Defn 1.3.13) Let T be a pre-triangulated category. T is calledtriangulated if for any morphism of candidate triangles

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

f

y g

y h

y Σf

yX ′

u′−−−−→ Y ′v′−−−−→ Z ′

w′−−−−→ ΣX ′

There is a way to form a new candidate triangle out of this data. It is the diagram

Y⊕X ′

Γ−−−−→ Z⊕Y ′

Λ−−−−→ ΣX⊕Z ′

Ξ−−−−→ ΣY⊕

ΣX ′

for Γ, Λ, and Ξ related to the original morphisms in some way. Specifically, we maywrite them in matrix notation as (

−v 0g u′

)for Γ, (

−w 0h v′

)for Λ,

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and (−Σu 0Σf w′

)for Ξ

The matrix notation is to take the direct sums intuitively as vectors and to multiplyas you would vectors. Mathematicians do not know of a category that is pre-triangulatedbut not triangulated.

Now that we have defined triangulated categories, we proceed to define homotopyCartesian squares and a couple results related to them.

Definition 7 (Neeman Defn 1.4.1) Let T be a triangulated category. Then a commutativesquare

Yf−−−−→ Z

g

y g′y

Y ′f ′−−−−→ Z ′

is called homotopy Cartesian if there is a distinguished triangle

YΛ−−−−→ Y ′

⊕Z

Ξ−−−−→ Z ′δ−−−−→ ΣY

where Λ = [g,−f ]t and Ξ = [f ′, g′] for some δ : Z → ΣY .

Notation and Names If we have a homotopy square

Yf−−−−→ Z

g

y g′y

Y ′f ′−−−−→ Z ′

then we call Y the homotopy pullback of

Z

g′y

Y ′f ′−−−−→ Z ′

and Z ′ the homotopy pushout

Yf−−−−→ Z

g

yY ′

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It follows from [TR1] that any diagram

Yf−−−−→ Z

g

yY ′

has a homotopy pushout, since the morphism Y → Y ′⊕Z can be completed to a

triangle, which in turn defines the homotopy cartesian square. Similarly by an oppositecategory construction homotopy pullbacks also exist. Using more advanced results theyare pushouts and pullbacks are unique up to ismorphism.

We now proceed to give a definition of a triangulated subcategory.Definition 8(Neeman Defn 1.5.1) Let T be a triangulated category. A full additivesubcategory S in T is called a triangulated category if every object isomorphic to anobject of S is in S, if ΣS = S, and if for any distinguished triangle

X −−−−→ Y −−−−→ Z −−−−→ ΣX

with the objects X and Y lying in S, we also have Z lying in S.Thus from [TR2] we deduced that if S is a triangulated subcategory of T and

X −−−−→ Y −−−−→ Z −−−−→ ΣX

is a triangle in T , then if any two objects of X, Y , and Z are in S, so is the third.We proceed to define the subcategory MorC .

Definition 9 (Neeman Defn 1.5.3) Suppose that T is a triangulated category and that Sis a triangulated subcategory. We define a collection of morphisms MS by the followingrule. A morphism f : X → Y belongs in MS iff in some triangle

Xf−−−−→ Y −−−−→ Z −−−−→ ΣX

the object Z lies in S. The category MorS is then take to be the category with theset of objects obj(T ) and the set of morphisms MS .

Our next task is to define a triangulated functor.Definition 10 (Neeman Defn 2.1.1) Let C and D be triangulated categories. A trian-gulated functor F : C → D is an additive functor F : C → D together with naturalisomorphisms

φX : F (Σ(X))→ Σ(F (X))

such that for any distinguished triangle

Xu−−−−→ Y

v−−−−→ Zw−−−−→ ΣX

in C the candidate triangle

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F (X)F (u)−−−−→ F (Y )

F (v)−−−−→ F (Z)F (w)−−−−→ Σ(F (X))

is a distinguished triangle in D.

Definition 11(Neeman Defn 2.1.3) Let F : D → T be a triangulated functor. The kernelof F is definied to be the full subcategory C of D whose objects map to objects of Tisomorphic to 0. This is similar to most any definition of kernel.

Definition 1(Neeman Defn 2.1.6) A subcategory C of a triangulated category D is calledthick if it is triangulated and it contains all direct summands of its objects.

Some basic examples of triangualted categories in higher mathematics are homotopycategories of complexes, derived categories, and the stable homotopy category of spectraof CW-complexes in topology.

4 The Verdier Localization Theorem

History: This theorem, due indeed to Verdier, was slightly more general than is givenhere and constructs the quotient D/C if S is an arbitrary thick subcategory. The proofis also somewhat different from Verdier’s proof, whose characterization of thick subcat-egories was unnecessarily opaque, although his definition is equivalent as the one givenabove. One can see Verdier’s Proof in [5].

Verdier Localization Theorem:(Neeman, Theorem 2.1.8) Let D be a triangulated cate-gory, and let C ⊂ D be a triangulated subcategory (not necessarily thick). Then there isa universal functor F : D → T with C ∈ ker(F ). In other words, there exists a triangu-lated category D/C, and a triangulated functor Funiv : D → D/C, so that C is the kernelof Funiv, and that Funiv is universal with this property. If F : D → T is a triangulatedfunctor whose kernel contains C, then it factors uniquely as D → D/C → T , where thefunctor from D to D/C is Funiv.

The category D/C is called the Verdier quotient of D by C and functor Funiv iscalled the Verdier localisation map. What localization essentially does is it takes someof the morphisms in D and makes them invertible. The construction of this inversion isanalogous to the construction of a quotient group in group theory, where if one has agroup G, and a normal subgroup H, then one can construct a homomorphism with Has the kernel.

5 Proving the Theorem

We have a couple things to prove: D/C is triangulated, Funiv has the requisite universalproperty and is triangulated. WE will begin by constructing the category D/C.

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The objects of D/C are simple; they are just the objects of D. We now turn tothe morphisms. We previously defined a category MorC ∈ D. Recall that a morphismf : X → Y lies in MorC iff the triangle

Xf−−−−→ Y −−−−→ Z −−−−→ ΣX

has the object Z lying in C. Now a triangulated functor F takes an object Z to 0 iffit takes

Xf−−−−→ Y −−−−→ Z −−−−→ ΣX

to a triangle isomorphic to the image of

X1−−−−→ X −−−−→ 0 −−−−→ ΣX

so F (f) : F (X) → F (Y ) is an isomorphism. Thus in D/C the morphisms in MorCwill be isomorphisms, thus invertible. We construct the morphisms in D/C as diagramsof the form

Xf−−−−→ Y

g

yX

where f is a morphism in MorC . If a morphism f : X → Y lies in MorC then this istaken to

Xf−−−−→ Y

1

yX

Its inverse is

X1−−−−→ X

f

yY

We will explain this strange setup later. This is a direct consequence of the fact thatmorphisms in MorC are invertible. Now let us proceed to prove the theorem.

Why should this setup work? The identity morphism for a given X is just

X1−−−−→ X

1

yX

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and composition is defined as follows. Suppose we have two morphisms f : X → Yand g : Y → Z. Then their composition in D/C is given as follows. Suppose the twomorphisms become

Xf−−−−→ Yy

X

and

Yg−−−−→ Zy

Y

Then their composition is,

Xf−−−−→ Y

g−−−−→ Zy yX

f−−−−→ YyX

Suppress the Y in the second row to get

Xg◦f−−−−→ Zy

X

Thus D/C is clearly a category.

Lemma 1(Neeman, Lemma 2.1.22) For any map in D/C of the form

Wg−−−−→ Y

f

yX

it can be written as the composition of the two maps

W1−−−−→ W

f

yX

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and

Wg−−−−→ Y

1

yW

Proof: The proof is trivial, by the diagram

W1−−−−→ W

g−−−−→ Y

1

y 1

yW

1−−−−→ W

f

yX

we compute the composite.

Lemma 2 (Neeman,Lemma 2.1.21) Earlier, we asserted that the inverse of

Xf−−−−→ Y

1

yX

is

X1−−−−→ X

f

yY

if f is a morphism in MorC .We see this by direct computation.

Proof : On the one hand

X1−−−−→ X

1−−−−→ X

1

y f

yX

f−−−−→ Y

1

yX

shows that the composite X → Y → X is the identity in D/C. Meanwhile thecomposite Y → X → Y is

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X1−−−−→ X

f−−−−→ Y

1

y 1

yX

1−−−−→ X

1

yX

which gives

Xf−−−−→ Y

f

yY

and since f lies in MorC this is equivalent to

Y1−−−−→ Y

1

yY

We have to show that D/C is a triangulated category, Funiv : D → D/C is a triangu-lated functor and that Funiv is universal. The proof is taken from [1].

Note that we only defined triangulated categories in term of pre-triangulated cate-gories, and pre-triangulated categories in term of additive categories. Thus we need toprove that D/C is additive. This requires a couple results.

We first prove several important equivalences.Proposition 1 (Neeman, Prop 2.1.26) Suppose f and g are two morphisms X → Y in D.Then the following are equivalent.•Funiv(f) = Funiv(g)

• There exists a map α : W → X in MorC with fα = gα

• The map f − g : X → Y factors as X → C → Y with C ∈ C.Proof We first prove that the first two are equivalent. The morphisms Funiv(f) andFuniv(g) will agree in D/C iff the diagrams

Xf−−−−→ Y

1

yX

and

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Xg−−−−→ Y

1

yX

are equivalent. This will happen iff there is an object W ∈ D and maps α1 : W → Xand α2 : W → X in MorC such that the squares

Wα1−−−−→ X

α2

y 1

yX

1−−−−→ X

and

Wα1−−−−→ X

α2

y f

yX

g−−−−→ Y

By commutativity of the first square we obtain that α1 = α2 = α and the commuta-tivity of the second square gives us fα = gα.

Now let us prove the equivalence of the second and third equivalences. The secondstatement is equivalent to saying that there is some α : W → X in MorC with (f−g)α =0. Consider the triangle

Wα−−−−→ X −−−−→ C −−−−→ ΣW

Since the functor Hom(−, Y ) is cohomological, (f −g)α = 0 iff f −g factors throughC. But α ∈MorC iff C ∈MorC . Consequently there exists an α ∈MorC with (f−g)α =0 iff f − g factors through C ∈MorC .

Lemma 3 (Neeman, Lemma 2.1.27) Any commutative square in D/C is isomorphic tothe image of a commutative square in D. More precisely, if

W −−−−→ Xy yY −−−−→ Z

is a commutative square in D/C, there is a commutative square in D

W ′ −−−−→ X ′y yY ′ −−−−→ Z ′

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and maps in MorC , W′ → W , X ′ → X, Y ′ → Y and Z ′ → Z, which, being

isomorphisms in D/C, represent the isomorphism of the two diagrams.

Proof We know that we can lift the composites W → X → Z and W → Y → Z toW1 → X ′ → Z ′ and W2 → Y ′ → Z ′. Let us now replace W1 and W2 by the homotopypullback

W3 −−−−→ W1y yW2 −−−−→ W

and thus we may assume that W1 = W2. By the commutativity in D/C of thediagram

W3 −−−−→ X ′y yY ′ −−−−→ Z

By the previous result, there is a map W ′ → W3 such that the two composites areequal in D. Thus there is a commutative diagram in D

W ′ −−−−→ X ′y yY ′ −−−−→ Z ′

and we may take Z ′ → Z as the identity. This will allow us to prove that D/C is anadditive category, which is what we need to establish that it is triangulated.

Lemma 4 (Neeman, Lemma 2.1.28) The object 0 ∈ D is a terminal and initial object inD/C.Proof The two statements are equivalent. We prove that 0 is terminal. Let X be anobject of D, also an object of D/C. Then the diagram

X −−−−→ 0

1

yX

exhibits a morphism X → 0 inD/C. meanwhile, given any other

P −−−−→ 0

f

yX

then the map f : P → X shows that the maps

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P −−−−→ 0

f

yX

and

X −−−−→ 0

1

yX

are equivalent. Thus there is only one map X → 0 in D/C. Similarly, if we have adiagram

X1−−−−→ X

0

y0

then the map g : X → P shows that

Pg−−−−→ X

0

y0

and

X1−−−−→ X

0

y0

are equivalent. Thus the object 0 in D is also a zero object in D/C.

Lemma 5 (Neeman, Lemma 2.1.29): Let X and Y be two objects of D and thus twoobjects of D/C. Then the direct sum in D, X

⊕Y , is a biproduct in D/C, i.e. it satisfies

the universal properties of coproduct and product.

Proof There are maps in D detailing X⊕Y → X, X

⊕Y → Y for the product and

X → X⊕Y , Y → X

⊕Y for the coproduct. The two statements are dual, so we satisfy

ourselves by describing the maps for the coproduct. Given two morphisms X → Q,Y → Q in D/C is the same as giving the equivalent classes of diagrams

Pf−−−−→ Q

α

yX

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and

P ′g−−−−→ Q

α′yY

with α and α′ lying in MorC . Then α and α′ fit into triangles

Pα−−−−→ X −−−−→ Z −−−−→ ΣP

and

P ′α′−−−−→ Y −−−−→ Z −−−−→ ΣP ′

where Z and Z’ lie in C. Thus the direct sum of these is a triangle, i.e. we have atriangle

P⊕P ′

α⊕α′−−−−→ X

⊕Y −−−−→ Z

⊕Z ′ −−−−→ ΣP

⊕ΣP ′

But Z⊕Z ′ lies in C, so α

⊕α′ is in MorC .

Thus the diagram

P⊕P ′

f⊕g−−−−→ Qy

X⊕Y

is a well-defined representative for a morphism in D/C.We may compose it withX → X

⊕Y , the map into the coproduct. This is computed by the commutative

diagram to be

P −−−−→ P⊕P ′

f⊕g−−−−→ Q

α

y yC

i1−−−−→ X⊕Y

and since P → P⊕P ′ is just the projection, this is just

Pf−−−−→ Q

α

yX

and we may proceed similarly with the map from Y → X⊕Y . Hence a pair of maps

X → Q and Y → Q in D/C do factor through the object X⊕Y . Thus a factorization

exists. We would like to show the uniqueness of the factorization.

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Suppose the are given two morphisms in D/C, using the opposite category construc-tion,

QyX⊕Y

f−−−−→ P

and

QyX⊕Y

g−−−−→ P ′

such that their composites with the maps X → X⊕Y and Y → X

⊕Y exists and

agree. Let us assume that P = P ′ and that the vertical maps are the same, which wedo by replacing P and P’ by their homotopy pushout.

Thus we may assume that we have two diagrams

QyX⊕Y

f−−−−→ P

and

QyX⊕Y

g−−−−→ P

so that their composites with X → X⊕Y and Y → X

⊕Y agree in D/C. Let

i : X → X⊕Y and j : Y → X

⊕Y . Then Funiv(α)−1Funiv(fi) = Funiv(α)−1Funiv(gi),

Funiv(α)−1Funiv(fj) = Funiv(α)−1Funiv(gj). If we now multiply through by Funiv(α) weobtain that Funiv(fi) = Funiv(gi) and Funiv(fj) = Funiv(gj).

The uniqueness of the factorization now follows from the above reasoning, since(f − g)i factors through some C ∈ C and (f − g)j factors through C ′ ∈ C. Thus (f − g)factors through C

⊕C ′.

Let us define the suspension functor on D/C to be the suspension functor of D onobjects, and the suspension functor on diagrams defining morphisms for morphisms.Furthermore, let Φ : ΣFuniv → FunivΣ be the identity. Define candidate triangles inD/C as all candidate triangles isomorphic to

FunivX −−−−→ FunivY −−−−→ FunivZ −−−−→ FunivΣX

where

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X −−−−→ Y −−−−→ Z −−−−→ ΣX

is a distinguished triangle in D, which is aided by our definition of Φ. Then [TR0]and [TR2] are obvious for D/C. To prove [TR1] note that any morphism in D/C canbe written in the form Funiv(u)Funiv(f)−1 for f : P → X in MorC and u : P → Y anymorphism in D. By [TR1] on D we may complete u to a triangle

Pu−−−−→ Y

v−−−−→ Zw−−−−→ ΣP

Then

Funiv(X)Funiv(u)Funiv(f)−1

−−−−−−−−−−−−→ Funiv(Y )Funiv(v)−−−−−→ Funiv(Z)

Funiv(Σf)Funiv(w)−−−−−−−−−−−−→ ΣFuniv(X)

is isomorphic in D/C to

Funiv(P )Funiv(u)−−−−−→ Funiv(Y )

Funiv(v)−−−−−→ Funiv(Z)Funiv(w)−−−−−−→ ΣFuniv(X)

and hence is a distinguished triangle. We are left with proving [TR3] and [TR4],which is just a stronger version of [TR3]. We need to show that given a diagram in D/Cwhere the rows are triangles,

X −−−−→ Y −−−−→ Z −−−−→ ΣXy y yX ′ −−−−→ Y ′ −−−−→ Z ′ −−−−→ ΣX ′

there is a way to choose Z → Z ′ such that the mapping cone is a triangle([TR4]),whereas [TR3] asserts only that the such a map Z → Z ′ exists and makes the diagramcommutative. In that square, which we will denote

X −−−−→ Yy yX ′ −−−−→ Y ′

Observe, by a previous lemma, the commutative square

X −−−−→ Yy yX ′ −−−−→ Y ′

can be lifted from D/C, the rows can be extended to triangles in D, and the diagram

X −−−−→ Y −−−−→ Z −−−−→ ΣXy y y yX ′ −−−−→ Y ′ −−−−→ Z ′ −−−−→ ΣX ′

23

can be extended to a good morphism of triangles in D, and thus in D/C. Note byanother lemma, that the commutative square with vertical isomorphisms

X −−−−→ Yy yX −−−−→ Y

will extend to an isomorphism of candidate triangles

X −−−−→ Y −−−−→ Z −−−−→ ΣXy y y yX −−−−→ Y −−−−→ Z −−−−→ ΣX

and similarly the commutative square

X ′ −−−−→ Y ′y yX ′ −−−−→ Y ′

will extend similarly to an isomorphism of candidate triangles in D/C.

X ′ −−−−→ Y ′ −−−−→ Z ′ −−−−→ ΣX ′y y y yX ′ −−−−→ Y ′ −−−−→ Z ′ −−−−→ ΣX ′

Composing, we have a good morphism of triangles, as required, given by

X −−−−→ Y −−−−→ Z −−−−→ ΣXy y y yX ′ −−−−→ Y ′ −−−−→ Z ′ −−−−→ ΣX ′

Thus the category D/C is triangulated.

Now we wish to prove that Funiv is a triangulated functor. Given how we definedtriangles in D/C, Funiv takes triangles to triangles. By previous results, C is containedin the kernel of Funiv, and universality follows since a triangulated functor taking C to 0we find that universality is obvious. By a previous result, Funiv is the universal functorwith this property. This concludes the proof of the Theorem.

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6 Odds and Ends

We can take the study of triangulated categories several ways. The essential propertyof triangulated categories is the fact that maps can be composed to obtain other mapsthat are essentially “0,” and through this basic principle, authors have created similarconstructs, known as n-angulated categories, see for example [3] or [4]. In addition,there are different ways to construct localizations, either with Thomason localization orBousfield localization. These are more complicated than Verdier localization, and we willnot discuss them. As discussed previously, localization provides a means to constructinga quotient modulo some equivalency class, or the like.

Verdier’s Localization Theorem is a powerful result. In homological algebra, thefollowing result is a direct consequence of Verdier’s Localization Theorem.

Theorem The derived category of an abelian category is triangulated. A chain mapbecomes invertible in the derived category iff it is a quasi-isomorphism. A triangulatedfunctor F : K(A) → K′ factors over a triangulated functor D(A) → K′ iff F sendsquasi-isomorphisms to isomorphisms in K′.

This result was also proved by Verdier and is a crucial result in homological algebra.

References

[1] Amnon Neeman, Triangulated Categories. Princeton: Princeton UniversityPress, 2001.

[2] Charles Weibel, An Introduction to Homological Algebra. Cambridge Studiesin Advanced Mathematics; New York: Cambridge University Press, 1994.

[3] Christof Geiss, Bernhard Keller, and Seffen Oppermann, n-Angulated CategoriesarXiv:1006.4592v3

[4] Petter Andreas Bergh and Marius Thaule, The Axioms for n-Angulated categoriesarXiV:1112.2533v2

[5] Jean-Louis Verdier Categories derivees, etat 0 SGA 4.5 Lecture Notes in Math,Volume 569, Springer-Verlag 1977, pp.262-308. (French)

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