notes on derivators - groth

SELECTED TOPICS IN TOPOLOGY: DERIVATORS

MORITZ GROTH

Abstract. These are lecture notes for the course ‘Selected Topics in Topol-

ogy: Derivators’ given by the author during the winter term 2014–2015 at

the University of Bonn, Germany. These notes including the appendix will beexpanded as the course goes on and will be subject to change at least until

the end of the course. I am interested in corrections, feedback, comments, and

suggestions, hence, please, do not hesitate to contact me.

Contents

1. Introduction and overview 22. Abelian categories and classical derived functors 22.1. Review of abelian categories 32.2. Classical derived functors 52.3. Group cohomology as a derived limit functor 93. Derived categories of abelian categories 113.1. Towards derived cokernels 113.2. Derived categories and derived functors 173.3. Cones as derived cokernels 214. Derived categories as triangulated categories 244.1. The homotopy category of an abelian category 244.2. Triangulated categories 274.3. Classical triangulations on derived categories 294.4. Beyond triangulated categories 305. Kan extensions 325.1. Motivation 325.2. Pointwise Kan extensions 335.3. Basic properties and first examples 366. Basics on derivators 396.1. Prederivators 396.2. Derivators 436.3. Examples of derivators 466.4. Limits versus homotopy limits 487. Homotopy exact squares and Kan extensions 507.1. The calculus of mates 507.2. Homotopy exact squares 557.3. First applications to Kan extensions 568. Pointed derivators 618.1. Some basics 618.2. Suspensions, loops, cofibers, and fibers 62

Date: January 27, 2015.

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8.3. Parametrized Kan extensions 668.4. Cartesian and cocartesian squares 698.5. Iterated cofiber constructions 729. Stable derivators 779.1. Basics on stable derivators 779.2. The preadditivity of stable derivators 809.3. The additivity of stable derivators 829.4. Morphisms and natural transformations 849.5. Canonical triangulations in stable derivators 889.6. Exact morphisms of stable derivators 9510. Towards abstract representation theory 9610.1. Tilting theory and strong stable equivalences 9610.2. Abstract tilting theory of An-quivers 10010.3. Universal tilting modules 103Appendix A. Some category theory 104A.1. Adjunctions 104A.2. Limits and colimits 106Appendix B. Examples of derivators 110B.1. Represented derivators 110B.2. Homotopy derivators of model categories 110References 112

1. Introduction and overview

This course offers an introduction to the theory of derivators which is one of themany different approaches to Homological Algebra or Homotopical Algebra. Whilestable derivators are an enhancement of triangulated categories [Nee01], the theoryof derivators is more general as it also captures key features of not necessarily stableor pointed homotopy theories. To put it as a slogan, derivators can be thought ofas a minimal, purely categorical extension of the more classical derived categoriesor homotopy categories to a framework with a well-behaved calculus of homotopylimits and homotopy colimits.

In the remainder of this introduction we try to fill the previous paragraph withmore life; we warn the reader that this section is a rather informal account, whilethe precise mathematics only begins in the next section.

To be continued...Acknowledgements. I thank Thorge Jensen for helpful comments on the notes.

2. Abelian categories and classical derived functors

We assume that the reader is familiar with basic homological algebra includingthe construction of classical derived functors. Nevertheless, for convenience, werecall in this section some basics but refer the reader to the literature for moredetails; see for example [Rot79], the more advanced [Wei94, GM03, KS06], andthe original [CE99, Gro57]. As first examples of derived limit and derived colimitfunctors, we observe that group cohomology and group homology, respectively, arespecial cases of those more general notions.

SELECTED TOPICS IN TOPOLOGY: DERIVATORS 3

2.1. Review of abelian categories. To put it as a slogan, homological algebrais classically the study of abelian categories and derived functors. We begin by re-calling basic definitions concerning abelian categories, emphasizing that all axiomsask for properties as opposed to asking for structure.

Definition 2.1. A preadditive category is a pointed category with finite biprod-ucts.

Thus, denoting the singleton by ∗, a preadditive categoryA satisfies the followingthree defining axioms.

(i) The category A has a zero object, i.e., an object 0 ∈ A which is both finaland initial,

homA(X, 0) ∼= ∗ ∼= homA(0, Y ), X, Y ∈ A.

Such a category is also called a pointed category.(ii) The category A has finite coproducts and finite products.(iii) For any X,Y ∈ A the canonical map X t Y → X × Y from the coproduct to

the product is an isomorphism.

Just to be completely precise, in matrix notation the canonical map of axiom (iii)is given by (

1 0

0 1

): X t Y → X × Y.

Following standard notation, we write X ⊕ Y for any of X t Y ∼= X × Y and referto it as the biproduct of X and Y .

Remark 2.2. (i) All axioms of a preadditive category ask for properties. Theonly structure is the category itself.

(ii) Any preadditive category A can be canonically endowed with an enrich-ment in the category AbMon of abelian monoids. In fact, given morphismsf, g : X → Y , then the sum f + g : X → Y is the composition

f + g : X∆→ X ⊕X f⊕g→ Y ⊕ Y ∇→ Y,

where ∆ is the diagonal X → X×X and∇ is the fold map ∇ : Y tY → Y . Weleave it to the reader to check that this defines an abelian monoid structure onhomA(X,Y ) with neutral element 0: X → 0→ Y and that the compositionis bilinear.

(iii) A closely related perspective on this enrichment is as follows (we will comeback to this later). In a preadditive category, the fold map ∇ : Y ⊕ Y → Yendows any Y ∈ A with the structure of an abelian monoid object. We leaveit as an exercise to verify that such an abelian monoid structure is equivalentto specifying a lift of the represented functor homA(−, Y ) : Aop → Set againstthe forgetful functor AbMon→ Set,

AbMon

Aop

homA(−,Y )//

::

Set.

Similarly, the diagonal map ∆: X → X ⊕ X endows any X ∈ A withthe structure of an abelian comonoid object. And one checks that such an

4 MORITZ GROTH

abelian comonoid structure is equivalent to a lift of the corepresented functorhomA(X,−) : A → Set against the forgetful functor AbMon→ Set.

Asking for an additional exactness property, we obtain additive categories.

Definition 2.3. An additive category is a preadditive category A such that theshear maps (

1 1

0 1

): X ⊕X → X ⊕X, X ∈ A,

are isomorphisms.

Lemma 2.4. The following are equivalent for a preadditive category A.

(i) The category A is additive.(ii) Identity morphisms id : X → X have additive inverses in homA(X,X).

(iii) The abelian monoids homA(X,Y ) are abelian groups.(iv) The fold map ∇ : Y ⊕ Y → Y endows Y ∈ A with the structure of an abelian

group object.(v) The diagonal map ∆: X → X ⊕X endows X ∈ A with the structure of an

abelian cogroup object.

Proof. This proof is left as an exercise.

Thus, to emphasize, from the defining exactness properties of an additive cate-gory one can construct the structure of an enrichment in the category Ab of abeliangroups. The good notion of functors between additive categories are additivefunctors, i.e., functors which preserve finite direct sums.

Finally, abelian categories are additive categories admitting kernels and cokernelsand such that the first Noether isomorphism theorem is true. In more detail, letus consider a morphism f : X → Y in an additive category with finite kernels andfinite cokernels.

(i) The image im(f) of f is the kernel of the canonical map Y → cok(f) to thecokernel, yielding the diagram

im(f)→ Y → cok(f).

(ii) The coimage coim(f) of f is the cokernel of the canonical map ker(f)→ Xfrom the kernel, hence there is the diagram

ker(f)→ X → coim(f).

One easily checks that f factors through a canonical map coim(f) → im(f), andthat in the case of the category of abelian groups A = Ab this map is the usualisomorphism X/ ker(f)→ im(f).

Definition 2.5. An abelian category is an additive category A satisfying thefollowing two properties.

(i) Every morphism has a kernel and a cokernel.(ii) For every morphism f : X → Y the canonical map coim(f) → im(f) is an

isomorphism.

Thus, as a slogan, abelian categories axiomatize categories allowing for the usual‘calculus of short exact sequences’.

Examples 2.6. (i) The category Ab and the category Mod(R) of left R-modulesover a ring R is abelian.


(ii) Given a topological space X, the categories of presheaves or sheaves on Xwith values in an abelian category is again abelian.

(iii) Given an abelian category A, the corresponding category Ch(A) of chaincomplexes is again abelian.

(iv) A category A is abelian if and only if its opposite Aop is abelian.(v) IfAi, i ∈ I, are abelian categories, then the so is the product category

∏i∈I Ai.

Examples 2.6(v) is actually a special case of the following lemma.

Lemma 2.7. Let A be an abelian category and let A be a small category. Thefunctor category AA = Fun(A,A) is again abelian.

Proof. The proof is left as an exercise, the only hint being that all definitions arepointwise.

2.2. Classical derived functors. Let us recall that a functor F : A → B betweenadditive categories is additive if it preserves finite biproducts (including zero ob-jects as a special case). It follows that homA(X,Y ) → homB(FX,FY ) is a grouphomomorphism.

Lemma 2.8. Any additive functor between abelian categories preserves split shortexact sequences.


In general, short exact sequences are not preserved by additive functors. How-ever, many functors showing up in nature preserve exactness ‘on one side’, moti-vating the following definitions.

Definition 2.9. Let F : A → B be an additive functor between abelian categories.

(i) The functor F is left exact if for every exact sequence 0→ X ′ → X → X ′′

in A also the sequence 0→ FX ′ → FX → FX ′′ is exact.(ii) The functor F is right exact if for every exact sequence X ′ → X → X ′′ → 0

in A also the sequence FX ′ → FX → FX ′′ → 0 is exact.(iii) The functor F is exact if it is left exact and right exact.

Examples 2.10. Let R be a commutative ring and let M be an R-module.

(i) The tensor product M ⊗R − : Mod(R) → Mod(R) is right exact, but, ingeneral, not left exact.

(ii) The hom functor homR(M,−) : Mod(R) → Mod(R) is left exact, but, ingeneral, not right exact.

Remark 2.11. (i) By means of Proposition A.15 one observes that any abeliancategory is finitely (co)complete and that it is (co)complete if and only if itadmits (co)products.

(ii) Proposition A.22 implies that an additive functor between abelian categoriesis left exact in the sense of Definition 2.9 if and only if it is left exact in thesense of Definition A.21. Together with Lemma A.23 this yields a formalproof of the positive statements in Examples 2.10.

Let F : A → B be a right exact functor and let A have enough projective objects.Then there is the following classical construction of left derived functors associated

6 MORITZ GROTH

to F . By assumption on A, for every object X ∈ A we can find a projectiveresolution, i.e., an exact sequence

. . .→ P2 → P1 → P0ε→ X

such that all Pi are projective. Since additive functors preserve zero objects, weobtain an induced chain complex FP ∈ Ch(B) which is given by

. . .→ FP2 → FP1 → FP0.

A final application of the homology functors Hn : Ch(B)→ B, n ≥ 0, concludes thedefinition of the (classical) left derived functors LnF : A → B, i.e., we set

(LnF )(X) = Hn(FP ), n ≥ 0.

Since we assume the reader to be familiar with basic homological algebra, wecontent ourselves by claiming that the above construction is well-defined, that it canbe extended to additive functors LnF : A → B, and that, using the right exactnessof F , there is a natural isomorphism L0F ∼= F .

As a first justification of the definition of classical left derived functor, let us notethat they measure systematically the deviation of F from being left exact. Moreprecisely, as a consequence of the horseshoe lemma, every short exact sequence0→ X ′ → X → X ′′ → 0 in A gives rise to a long exact sequence

. . .→ (L1F )(X)→ (L1F )(X ′′)δ→ FX ′ → FX → FX ′′ → 0

in B. In particular, the image im(δ) : (L1F )(X ′′) → FX ′ of the connecting homo-morphism agrees with the kernel of FX ′ → FX.

An even better justification is given by the observation that we constructedan universal example of such approximations. To make this precise we make thefollowing definitions.

Definition 2.12. Let A and B be abelian categories. A homological δ-functorT from A to B consists of

(i) additive functors Tn : A → B, n ≥ 0, and(ii) for every short exact sequence 0 → X ′ → X → X ′′ → 0 in A of morphisms

δn : Tn(X ′′)→ Tn−1(X ′), n ≥ 1,

such that for every such short exact sequence we obtain a natural long exact se-quence

. . .→ T1(X)→ T1(X ′′)δ1→ T0(X ′)→ T0(X)→ T0(X ′′)→ 0.

The naturality means that every morphism of short exact sequences gives riseto a commutative ladder in B. Thus, the connecting homomorphisms δn of (ii)assemble to suitable natural transformations.

Definition 2.13. (i) A morphism of homological δ-functors α : S → Tconsists of natural transformations αn : Sn → Tn, n ≥ 0, such that for everyshort exact sequence 0→ X ′ → X → X ′′ → 0 in A the diagram

Sn(X ′′)δ //

Sn−1(X ′)

Tn(X ′′)δ// Tn−1(X ′)

commutes for all n ≥ 1.


(ii) A homological δ-functor T is universal if for every homological δ-functor Sand every natural transformation α0 : S0 → T0 there is a unique morphismof δ-functors α : S → T extending the given α0.

Here is a justification of the seemingly adhoc construction of classical left derivedfunctors.

Theorem 2.14. Let F : A → B be a right exact functor between abelian categoriesand let A have enough projective objects. The functors LnF together with theconnecting homomorphisms define a universal homological δ-functor.

Proof. We assume that the reader knows that the LnF together with the connectinghomomorphisms assemble to a homological δ-functor. To prove its universality,let T be a homological δ-functor from A to B and let α0 : T0 → F be a naturaltransformation.

For n ≥ 1 we assume by induction that a partial morphism of δ-functors consist-ing of αj , 0 ≤ j < n, has already been constructed and that it is unique. We wantto show that there is a unique way of extending it up to degree n. Given X ∈ Awe choose a short exact sequence

(2.15) 0→ K → P → X → 0

such that P is a projective object. Since the left derived functors LnF, n ≥ 1, vanishon projective objects, by induction assumption we have a commutative (solid arrow)diagram with exact rows

TnXδ //

∃!

Tn−1K //

αn−1

Tn−1P

αn−1

0 // (LnF )Xδ// (Ln−1F )K // (Ln−1F )P.

An easy diagram chase shows that there is a unique dashed morphism αn : TnX →(LnF )X such that the square on the left commutes. Thus, the compatibility withshort exact sequences of the form (2.15) implies that there is at most one way ofdefining the desired αn : Tn → LnF . In particular, the morphism is independent ofthe choice of (2.15).

As for the existence it remains to show that these unique morphisms assembleinto natural transformations and that they are compatible with arbitrary shortexact sequences (and not only the ones of the form (2.15)). To verify the naturalitylet us consider a morphism f : X → Y in A. Such a morphism can be extended toa morphism of short exact sequences

0 // K //

P //

X

// 0

0 // L // Q // Y // 0

in which P,Q are projective objects. Associated to this we obtain the followingdiagram in which all squares commute with possibly the exception of the naturality

8 MORITZ GROTH

square on the left,

TnXδ //

''

Tn−1K

((

TnY //

Tn−1L

(LnF )X

''

// (Ln−1F )K

((

(LnF )Yδ

// (Ln−1F )L.

A diagram chase shows that in that square the two possibly different compositionsTnX → (LnF )Y agree when composed with δ : (LnF )Y → (Ln−1F )L. Since againthis morphism δ is a monomorphism, also the square on the left commutes whichis to say that we constructed a natural transformation αn : Tn → LnF .

It remains to show that αn is compatible with all short exact sequences. Forthis purpose, we note that for any short exact sequence 0 → X1 → X2 → X3 → 0and any short exact sequence 0 → K → P → X3 → 0 such that P is a projectiveobject we can extend the identity morphism id: X3 → X3 to a morphism of shortexact sequences, yielding a commutative diagram

0 // K //

P //

X3

=

// 0

0 // X1// X2

// X3// 0.

Similarly to the previous step, associated to this morphism we obtain a diagram

TnX3δ //

=

((

Tn−1K

))

TnX3//

Tn−1X1

(LnF )X3

= ''

// (Ln−1F )K

((

(LnF )X3δ

// (Ln−1F )X1

in which all squares with possibly the exception of the front square commute (byinduction assumption and by the previous steps). Precomposing the two morphismsTnX3 → (Ln−1F )X1 with id: TnX3 → TnX3, a short diagram chase implies thatalso the front face commutes, concluding the proof.

Dualizing these constructions we are led to right derived functors of left exactfunctors F : A → B. In this case, assuming the existence of enough injective objectsin A, every object X ∈ A has an injective resolution 0 → X → I0 → I1 → . . ..If we denote the truncated cochain complex by I, then the n-th (classical) rightderived functor RnF is defined by

(RnF )(X) = Hn(FI), n ≥ 0.

These classical right derived functors assemble to universal cohomological δ-functorsin the obvious sense.


For variants of these results and a discussion of examples and applications werefer the reader to the literature. Here we only mention that in the case of Exam-ples 2.10 we obtain the torsion products TorRn (M,N) and Ext-groups ExtnR(M,N)as typical examples of classical left derived and right derived functors, respectively.

2.3. Group cohomology as a derived limit functor. In this subsection we ob-serve a close relation between group (co)homology and certain derived limit and de-rived colimit functors. This is meant to be a first illustration that derived (co)limitfunctors encode interesting constructions. The reader who is not familiar withgroup (co)homology can safely skip most of this subsection.

Recall from Lemma 2.7 that for every abelian category A and every small cat-egory A the functor category AA is again abelian. Assuming A to have productsin case A is not finite, the limit functor limA : AA → A exists and is part of anadjunction

(∆A, limA) : A AA,showing that limA is a left exact functor (see Lemma A.23 and Remark 2.11). If wemoreover assume that AA has enough injective objects (which follows for examplefrom A having enough injective objects), then we can apply the techniques of §2.2in order to obtain (classical right) derived limit functors

RnlimA : AA → A, n ≥ 0.

Making dual assumptions on A, we obtain an adjunction (colimA,∆A) : AA Aand associated (classical left) derived colimit functors

LncolimA : AA → A, n ≥ 0.

Here we are only interested in the category A = Ab of abelian groups. The cate-gory of abelian groups is complete and cocomplete. It can be shown that all functorcategories AbA have enough injectives and projectives so that the derived (co)limitfunctors exist. In our special situation we do not need this general statement sincewe can argue more directly. Given a discrete group G, there is an associated cat-egory, again denoted by G, which has one object ∗ only and such that G is thecorresponding endomorphism monoid,

homG(∗, ∗) = G.

We recall that the integral group ring ZG on G is defined as follows. Theunderlying abelian group is the free abelian group on G. Denoting by eg thegenerator associated to g ∈ G, the ring multiplication is the bilinear extensionof the assignment

eg · eg′ = egg′ , g, g′ ∈ G.The reader easily checks that there is an equivalence of categories

(2.16) AbG ' Mod(ZG),

so that AbG clearly has enough injectives and projectives. We will not distinguishnotationally objects corresponding to each other under this equivalence.

The following two constructions apply to modules M ∈ Mod(ZG) and are centralto the theory of (co)homology of groups. Here we simplify notation and writegm = egm, g ∈ G,m ∈M .

(i) The abelian group of invariants is MG = m | gm = m, g ∈ G and thisdefines a functor (−)G : Mod(ZG)→ Ab.

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(ii) The abelian group of coinvariants is MG = M/gm−m | g ∈ G,m ∈ M,defining a functor (−)G : Mod(ZG)→ Ab.

One can check directly that the invariants functor is left exact and that thecoinvariant functor is right exact. A different way of seeing this is as follows. Weobserve that the integral group ring ZG comes with an augmentation map

ε : ZG→ Z :∑

riegi 7→∑

ri,

which is easily seen to be a ring homomorphism. Restriction of scalars on the leftand on the right applied to the regular bimodule ZZZ yields

ZGZZ and ZZZG,

respectively. We refer to these G-actions as trivial G-actions.

Lemma 2.17. Let G be a discrete group.

(i) There is a natural isomorphism (−)G ∼= homZG(Z,−) : Mod(ZG) → Ab,where Z is endowed with the trivial left G-action.

(ii) There is a natural isomorphism (−)G ∼= Z⊗ZG − : Mod(ZG)→ Ab, where Zis endowed with the trivial right G-action.


As a special case of Ext- and Tor-functors we can make the following definitions(but refer the reader to almost any book on homological algebra or, for example,[Bro94] for more details).

Definition 2.18. Let G be a discrete group.

(i) The n-th group cohomology of G with coefficients in M ∈ Mod(ZG) is

Hn(G;M) =(Rn(−)G

)(M) ∼= ExtnZG(Z,M).

(ii) The n-th group homology of G with coefficients in M ∈ Mod(ZG) is

Hn(G;M) =(Ln(−)G

)(M) ∼= TorZGn (Z,M).

Using the equivalences (2.16), a different perspective on group (co)homology isoffered by the following observation.

Lemma 2.19. For every discrete group G the following diagram commutes up tonatural isomorphisms

Mod(ZG)(−)G

xx

(−)G

&&'

Ab Ab.

AbGcolimG

ff

limG

88

Proof. The commutativity of the triangle on the right follows from the generalconstruction of limits in terms of equalizers and products; see (A.16). In fact, inthis particular case, the general construction specializes to the equalizer of abeliangroups

limGM ∼= eq(M ⇒

∏g∈G

M),

where one map is the diagonal map while the other map is m 7→ (gm)g∈G. Clearly,this equalizer can be chosen to be the invariants MG ∈ Ab.


Similarly, using the explicit construction of colimits in (A.17), one observes thatalso the triangle on the left commutes up to natural isomorphisms.

This shows that there is a very close relation between group cohomology functorsHn(G;−) and derived limit functors RnlimG and, similarly, between group homol-ogy functors Hn(G;−) and derived colimit functors LncolimG. Of course, we donot claim that this different perspective simplifies the study of group (co)homology.Instead, the point of these observations was to illustrate that the seemingly very ab-stract derived (co)limit functors specialize to well-known constructions (which areof independent interest). In the remainder of the course we will see that suitablecombinations of derived limit functors (and, more generally, derived Kan extensionfunctors) encode quite some interesting constructions, and that such constructionsare conveniently organized by means of derivators.

3. Derived categories of abelian categories

In this section we recall the construction of the derived category of an abeliancategory, which is a rather refined invariant of the abelian category. We also intro-duce derived functors between derived categories and show that the classical coneconstruction from homological algebra yields a left derived functor of the cokernel.

3.1. Towards derived cokernels. In §2.3 we saw already first examples of abeliancategories A and small categories A such that the functors

limA, colimA : AA → Aexist but fail to be exact. In this subsection we collect an important additionalsuch example.

We already observed that abelian categories are finitely cocomplete and that ar-bitrary finite colimits in abelian categories can be constructed from finite coproductsand cokernels, and dually. Hence we consider these two types of (co)limits inde-pendently. Let us recall that for a discrete category D and an arbitrary category Cthere is a canonical equivalence of categories CD '

∏d∈D C.

Lemma 3.1. Let A be an abelian category and let D be a finite discrete category.The finite biproduct functor ⊕

d∈D

: AD → A

is exact. Hence, finite coproducts and finite products in abelian categories are exact.

Proof. Given an abelian category A, finite coproduct and finite product functorsare part of adjunctions

(∐d∈D

,∆D) : AD A and (∆D,∏d∈D

) : A AD,

showing that∐d∈D is right exact and

∏d∈D is left exact. The preadditivity of

abelian categories yields natural isomorphisms∐d∈D∼=⊕

d∈D∼=∏d∈D, implying

that finite biproducts are exact.

Remark 3.2. Infinite (co)products in abelian categories are, in general, not exact.However, one can impose additional axioms on abelian categories guaranteeingfirst of all the existence of infinite coproducts or infinite products and, second, thatinfinite coproducts or infinite products are exact. In this order, attributing credit

12 MORITZ GROTH

to [Gro57], these assumptions are referred to as axioms (AB3), (AB3*), (AB4), and(AB4*). We refer the reader to the literature for more on these axioms, includinga discussion of examples of abelian categories satisfying these stronger axioms; see[Gro57] or [Fai73, §14].

We now turn to the case of (co)kernels which, in general, happen to fail to beexact. To begin with, as a special case of Lemma 2.7, associated to an abeliancategory A there is the abelian category A[1] of arrows in A. Here, [1] is the poset[1] = (0 < 1), considered as a category. Thus, objects in A[1] are morphismsX0 → X1 in A while morphisms are given by commutative squares

X0//

Y0

X1// Y1.

Since all limits and colimits in A[1] are constructed levelwise (see the proof of (2.7)),a short exact sequence in A[1] is the same thing as a morphism of short exactsequences in A.

Lemma 3.3. Let A be an abelian category.

(i) The functor ker : A[1] → A is left exact.(ii) The functor cok: A[1] → A is right exact.

(iii) In general, the functors ker, cok: A[1] → A are not exact.

Proof. The first two statements are simply a reformulation of the snake lemma. Infact, given a short exact sequence of morphisms in A,

0 // X0

f

// Y0//

g

Z0//

h

0

0 // X1// Y1

// Z1// 0,

the snake lemma yields an exact sequence

0→ ker(f)→ ker(g)→ ker(h)→ cok(f)→ cok(g)→ cok(h)→ 0,

establishing the first two statements.To establish the third statement it suffices to consider the short exact sequence

of morphisms of abelian groups

0 // 0

0

// Z/2Z = //

(1,0)t

Z/2Z //

0

0

0 // Z/2Z(1,0)t

// Z/2Z⊕ Z/2Z(0,1)

// Z/2Z // 0.

In fact, by the snake lemma we obtain an induced exact sequence

0→ 0→ 0→ Z/2Z∼=→ Z/2Z 0→ Z/2Z

∼=→ Z/2Z→ 0,

showing that the kernel is not right exact and that the cokernel is not left ex-act. (To make this example appear less adhoc, we note that this exact sequencecomes from the universal homological δ-functor associated to the right exact ten-sor functor Z/2Z ⊗ − and the short exact sequence 0 → Z → Z → Z/2Z → 0;see Theorem 2.14.)


We now reformulate the failure of exactness using a slightly different perspective.

Definition 3.4. Let A be an abelian category and let f : X → Y be a morphismin Ch(A). The chain map f is a quasi-isomorphism, in notation f : X

∼→ Y, if itinduces isomorphisms in homology,

Hn(f) : Hn(X)∼=→ Hn(Y ), n ∈ Z.

We denote the class of all quasi-isomorphisms by W = WA.

We recall that any additive functor F : A → B induces an additive functorF : Ch(A)→ Ch(B) between associated categories of chain complexes, obtained byapplying F in all degrees.

Lemma 3.5. An additive functor F : A → B between abelian categories is exact ifand only if the induced functor F : Ch(A)→ Ch(B) preserves quasi-isomorphisms.


This lemma is to be seen in contrast to the following result.

Lemma 3.6. Let A,B be additive categories. Any additive functor F : Ch(A) →Ch(B) preserves chain homotopies and hence, in particular, chain homotopy equiv-alences.


For later applications of Lemma 3.5 to limit and colimit functors, it is convenientto have the following refined version of Lemma 2.7.

Lemma 3.7. Let A be a small category and let A be abelian.

(i) The functor category AA = Fun(A,A) is again abelian.(ii) There is an isomorphism of categories Ch(AA) ∼= Ch(A)A which identifies

the quasi-isomorphisms WAA with the morphisms WAA , i.e., the natural trans-

formations of functors A→ Ch(A) which are levelwise quasi-isomorphisms.


Let us recall that kernels and cokernels in Ch(A) are calculated degreewise. Thus,Lemma 3.3, Lemma 3.5, and Lemma 3.7 together imply the following warning.

Warning 3.8. Let A be an abelian category. In general, the functors

cok, ker : Ch(A)[1] → Ch(A)

do not send levelwise quasi-isomorphisms to quasi-isomorphisms.

However, we have partial positive statements along these lines.

Lemma 3.9. Let A be an abelian category and let us consider a levelwise quasi-isomorphism in Ch(A)[1], i.e., a commutative square

(3.10)X0

f//

∼

X1

∼

Y0 g// Y1

in Ch(A) such that the vertical maps are quasi-isomorphisms.

14 MORITZ GROTH

(i) If f and g are monomorphisms, then the induced map cok(f) → cok(g) isalso a quasi-isomorphism.

(ii) If f and g are epimorphisms, then the induced map ker(f) → ker(g) is alsoa quasi-isomorphism.

Proof. We give a proof of (i) and observe that there is a morphism of short exactsequences

0 // X0

f// X1

//

cok(f) //

0

0 // Y0 g// Y1

// cok(g) // 0.

The induced long exact sequences in homology together with the 5-lemma allow usto conclude the proof of (i). The case of (ii) is dual.

Thus, the cokernel and the kernel functors are exact on certain objects. Weconclude this subsection with a seemingly adhoc construction of a ‘corrected version’of the cokernel functor. While a first justification of this construction is providedby Proposition 3.20, a more conceptual one will be given in §3.3. Everything canbe dualized to yield a similar ‘correction’ of the kernel functor.

To motivate these constructions we include the following different description ofthe cokernel. Let f : X0 → X1 be a morphism in an abelian category A. We notethat the cokernel of f can also be constructed by considering pushout squares

(3.11)X0

f//

X1

0 // cok(f).

In fact, the universal property of this pushout square is easily checked to reduce tothe usual universal property of the cokernel. Thus, although the cokernel cok(f) isnot the colimit of the diagram f : X0 → X1 (which would be isomorphic to X1) itis the colimit of the span (0← X0 → X1). We will come back to this intermediatestep later.

Now, as recalled in Warning 3.8, in general the cokernel functor is not exact.Using this different description of the cokernel, a classical way of fixing this problemis by replacing the morphism X0 → 0 quasi-isomorphically by a nice inclusion,namely by the inclusion i : X0 → CX0 of X0 in its cone CX0. We quickly recallthis construction and, since we also need further variants of such constructions, weallow us to include a minor digression.

Digression 3.12. We denote by FZ the category of finitely generated, free abeliangroups and by Chb(FZ) the category of bounded complexes in FZ. One observesthat for every additive category A there is an action

(3.13) ⊗ : Chb(FZ)× Ch(A)→ Ch(A)

which is additive in both variables separately. In fact, this action is obtained froma similar biadditive action ⊗ : FZ ×A → A which is already determined by askingthat there is a natural isomorphism Z⊗X ∼= X. For chain complexes F ∈ Chb(FZ)and X ∈ Ch(A) we define the underlying graded object of F ⊗X as a ‘convolution


type construction’, i.e., we set

(F ⊗X)n =⊕p+q=n

Fp ⊗Xq, n ∈ Z.

The differentials (F ⊗ X)n → (F ⊗ X)n−1 are obtained from those on F and Xusing the usual Koszul sign convention. (Recall that this convention says that, indifferential graded algebra, ‘whenever the order of two homogeneous elements ofdegree p and q is swapped, a sign (−1)pq has to appear’.)

Examples 3.14. (i) Let I = Ccell([0, 1]) ∈ Chb(FZ) be the reduced cellular chaincomplex of the interval endowed with the CW structure consisting of two0-cells and one 1-cell only. More explicitly, I0 = I1 = Z, the differentialI1 → I0 is the identity, and all Ik, k 6= 0, 1, vanish. We call I the intervaland it is easily verified that I is a contractible chain complex. For X ∈ Ch(A)there is a natural isomorphism I ⊗ X ∼= CX where C : Ch(A) → Ch(A) isthe cone functor. (If you did not see a definition of C before, then you cansimply take this construction as a definition and we suggest as an exercise tounravel it.) The notation is of course motivated from similiar constructionsfor pointed topological spaces, where smashing with the interval yields the(reduced) cone construction.

(ii) Let S0 = Ccell(0, 1) ∈ Chb(FZ) be the reduced cellular chain complex of the0-sphere endowed with the CW structure conisting of two 0-cells only. Thus,S0 is the stalk complex Z. For X ∈ Ch(A) there is a natural isomorphismS0 ⊗X ∼= X.

(iii) The continuous map 0, 1 → [0, 1] yields a map S0 → I in Chb(FZ) andhence a natural transformation S0⊗− → I⊗− of functors Ch(A)→ Ch(A).Under the above isomorphisms, for X ∈ Ch(A) this yields the usual naturalinclusion i : X → CX.

(iv) Let S1 ∈ Chb(FZ) be the reduced cellular chain complex of the 1-sphereconsidered as a quotient CW complex [0, 1]/0, 1. For X ∈ Ch(A) there isa natural isomorphism S1⊗X ∼= ΣX where Σ: Ch(A)→ Ch(A) is the usual‘shift against the differential’.

Having recalled the cone construction for chain complexes, we now replace thevertical morphism on the left in (3.11) by X0 → CX0 and hence consider thepushout diagram

(3.15)X0

f//

i

X1

cof(f)

CX0// Cf.

Definition 3.16. (i) The morphism cof(f) : X1 → Cf is the cofiber of f .(ii) The chain complex Cf is the cone of f .

It is immediate that both the cofiber and the cone are functorial in f , i.e., wehave functors

cof : Ch(A)[1] → Ch(A)[1] and C : Ch(A)[1] → Ch(A).

Lemma 3.17. Let A be an abelian category and let X ∈ Ch(A).

(i) The cone CX is contractible.

16 MORITZ GROTH

(ii) The cone of an isomorphism is contractible.

Proof. Since the interval I is contractible and the construction (3.13) is additive inthe first variable it follows from Lemma 3.6 that I ⊗ X ∼= CX is contractible. Iff : X0 → X1 is an isomorphism, then the construction (3.15) yields an isomorphismCX0

∼= Cf . Thus, by the first part, Cf is contractible.

To prove the homotopy invariance of the cone construction we need the followinglemma.

Lemma 3.18. Let A be an abelian category and let us consider a square

X0i //

j

X1

p

X2 q// X

in A. The square is a pushout square if and only if the sequence

X0(i,j)t→ X1 ⊕X2

(p,−q)→ X → 0

is exact.


In the special case of the defining pushout square (3.15) of the cone, we henceobtain a short exact sequence

(3.19) 0→ X0 → X1 ⊕ CX0 → Cf → 0,

since the first map is clearly also a monomorphism. The reader easily checks thatthis short exact sequence is functorial in the morphism f : X0 → X1. This allowsus to give a partial justification of the cone construction (see Lemma 3.5). A betterjustification will be given in §3.3 (see Theorem 3.35).

Proposition 3.20. Let A be an abelian category. The cone C : Ch(A)[1] → Ch(A)sends levelwise quasi-isomorphisms to quasi-isomorphisms.

Proof. We again consider a diagram of chain complexes (3.10) such that the verticalmaps are quasi-isomorphisms, i.e., a levelwise quasi-isomorphism f → g. Associatedto such a diagram we obtain by (3.19) a morphism of short exact sequences

0 // X0//

∼

X1 ⊕ CX0//

Cf //

0

0 // Y0// Y1 ⊕ CY0

// Cg // 0.

Since the objects CX0, CY0 have trivial homology objects, any map between themis a quasi-isomorphism. As finite direct sums of quasi-isomorphisms are againquasi-isomorphisms, in the above diagram also the second vertical morphism is aquasi-isomorphism. The long exact sequence in homology and the 5-lemma implythat the morphism Cf → Cg is a quasi-isomorphism, concluding the proof.


3.2. Derived categories and derived functors. As a preparation for the defi-nition of derived categories, let us reconsider the construction of classical derivedfunctors between abelian categories. Thus, let F : A → B be a right exact functorbetween abelian categories and let A have enough projective objects. The key stepsin the construction of the LiF consisted of choosing projective resolutions

(3.21) . . .→ P2 → P1 → P0ε→ X

for X ∈ A, applying F to the truncated chain complex

P = (. . .→ P2 → P1 → P0),

and then passing to homology. A different way of saying that (3.21) is a resolutionis to say that the commutative diagram

(3.22)

...

...

P2

// 0

P1//

0

P0 ε// X

defines a quasi-isomorphism ε : P → X where X also denotes the associated com-plex concentrated in degree 0, i.e., the stalk complex (. . .→ 0→ 0→ X) ∈ Ch(A).

This simple trick of rewriting the horizontal diagram (3.21) in a vertical fashion asin (3.22) is an important step. In fact, it suggests that, secretely in the constructionof classical derived functors, we already passed from objects in abelian categories(like modules over a ring) to chain complexes. From a more abstract perspectivethis amounts to passing from a good old category to a ‘homotopy theory’, namelyto the ‘homotopy theory’ (Ch(A),WA) consisting of the category Ch(A) togetherwith the class W = WA of quasi-isomorphisms. We plan to say a bit more about‘abstract homotopy theories’ later in the course.

The passage to stalk complexes defines a fully faithful functor A → Ch(A),suggesting that if we understand Ch(A) we also understand A. For various reasonswe would like to invert the quasi-isomorphisms WA in Ch(A). For example thiswould allow us to identify all projective resolutions of every fixed object X ∈ A.

Definition 3.23. Let A be an abelian category. The derived category D(A) ofA is the localization of Ch(A) at the class WA of quasi-isomorphisms,

D(A) = Ch(A)[W−1A ].

By the very definition, the derived category is actually a pair (D(A), γ) consistingof a category D(A) and a localization functor γ : Ch(A)→ D(A) which sends quasi-isomorphisms to isomorphisms and which is initial with this property. Thus, everyfunctor F : Ch(A) → C which sends quasi-isomorphisms to isomorphisms factors

18 MORITZ GROTH

uniquely through D(A),

Ch(A)∀F //

γ

C, F (WA) ⊆ IsoC ,

D(A).

∃!

<<

Warning 3.24. There is the typical warning in the context of such localizations,namely, that, in general, such localizations do not exist (at least not in a fixeduniverse). A different way of saying this is that such localizations can always beconstructed (see for example [GZ67]), but, in general, the resulting categories arenot locally small, i.e., there might be proper classes of morphisms between certainobjects. There are various ways of dealing with this issue.

(i) One can ignore these issues (as it was done in the more classical literature),reassuring the reader every now and then by claiming that all steps can bejustified.

(ii) One can restrict attention to the large class of Grothendieck abelian categoriessince in that case all these set-theoretic problems happen to disappear. Seefor example [Fai73, §14] for a discussion of Grothendieck abelian categories.

(iii) The language of Quillen model categories [Qui67] is an abstract frameworkwhich, among many other things, allows us to deal with such problems. Assoon as the pair (Ch(A),WA) can be extended to a model category, theset-theoretic problems automatically disappear. And, in fact, in the caseof Grothendieck abelian categories such model structures always exist; see[Hov01].

(iv) If one imposes certain boundedness conditions and assumes the existence ofsufficently many injective or projective objects, then one can show that cor-responding derived categories are equivalent to quotient categories of suitablecategories of chain complexes. In particular, these categories are again locallysmall categories.

For an actual example of such set-theoretic problems we refer to an old example ofFreyd (see the reprint [Fre03]) as discussed in [Kra10, Example 4.15].

For the remainder of this section we assume that the derived categories underconsideration exist. (Thus, we essentially follow approach (i).)

Corollary 3.25. Let F : A → B be an exact functor between abelian categories.There is a unique functor F : D(A)→ D(B) such that

Ch(A)F //

γ

Ch(B)

γ

D(A)F// D(B)

commutes.

Proof. Since the functor γ F : Ch(A) → D(B) sends quasi-isomorphisms to iso-morphisms (by Lemma 3.5) this is immediate from the defining universal propertyof the localization functor γ : Ch(A)→ D(A) (see Definition 3.23).

Here is an important example.


Example 3.26. Let A be an abelian category. The cone C : Ch(A)[1] → Ch(A)induces by Proposition 3.20 and Lemma 3.7 a functor

C : D(A[1])→ D(A).

Thus, to emphasize, there is a functorial cone construction defined on the derivedcategory of the arrow category of an abelian category. We will get back to this later.

As recalled in §2, many additive functors showing up in nature are only left exactor right exact. In such situations one has to work harder in order to obtain certainuniversal induced functors at the level of derived categories.

Definition 3.27. Let A,B be abelian categories and let F : Ch(A) → Ch(B) bea functor. A left derived functor of F is a pair (LF, ε) consisting of a functorLF : D(A) → D(B) and a natural transformation ε : (LF )γ → γF such that forevery other such pair (G : D(A) → D(B), α : Gγ → γF ) there is a unique naturaltransformation α′ : G→ LF such that

α = ε (α′γ) : Gγ → (LF )γ → γF.

Thus, a left derived functor can be depicted by a natural transformation as in

Ch(A)F //

γ

Ch(B)

γ

D(A)LF// D(B).

@H

Remark 3.28. (i) The definition makes precise that a left derived functor is auniversal approximation of the functor F by a functor which sends quasi-isomorphisms to quasi-isomorphisms (assuming that a morphism in Ch(B) isa quasi-isomorphism if and only if it is sent to an isomorphism by γ).

(ii) We warn the reader that this definition is different from the one found intypical books on homological algebra; see [Wei94, GM03]. While in thosereferences in the definition of a left derived functor both F and LF are as-sumed to be exact functors of triangulated categories (see §4), here we definederived functors as it is typically done in the theory of model categories.

(iii) There is the dual notion of a right derived functor.(iv) Since derived functors are defined by a universal property, once one knows

that derived functors exist, then they are unique up to a canonical naturalisomorphism.

In §3.3 we show that C : D(A[1]) → D(A) together with a suitable naturaltransformation is a left derived functor of the cokernel cok: Ch(A[1])→ Ch(A) (seeTheorem 3.35). As a preparation we collect a lemma which is also of independentinterest.

Let us recall that the derived category of an abelian category is defined as alocalization which hence satisfies a certain universal property. We will see nextthat derived categories actually are 2-localizations. Let us consider an abeliancategory A together with the localization functor γ : Ch(A) → D(A). For everycategory C, restriction along γ yields a map hom(D(A), C) → hom(Ch(A), C). If

we denote by homW(Ch(A), C) ⊆ hom(Ch(A), C) those functors which send quasi-isomorphisms to isomorphisms, then the definining universal property of γ is that

20 MORITZ GROTH

for all categories C the restriction along γ induces a bijection

γ∗ : hom(D(A), C)→ homW(Ch(A), C).

It turns out that this can be refined to an isomorphism of categories. For thispurpose, let us denote by

FunW(Ch(A), C) ⊆ Fun(Ch(A), C)

the full subcategory spanned by all functors which send quasi-isomorphisms toisomorphisms.

Lemma 3.29. Let A be an abelian category. The localization functor γ : Ch(A)→D(A) is a 2-localization, i.e., for every category C restriction along γ induces anisomorphism of categories

γ∗ : Fun(D(A), C)→ FunW(Ch(A), C).

Proof. By definition of the localization, γ∗ is bijective on objects and it hencesuffices to show that it is also fully faithful. Let F,G ∈ FunW(Ch(A), C) withcorresponding factorizations F = F ′γ, G = G′γ and let α : F → G be a natu-ral transformation. Note that α : F → G is equivalently specified by the functorα : Ch(A)→ C[1] which is defined by

α(X) = (αX : FX → GX).

If w : X → Y is a quasi-isomorphism, then, since F,G send quasi-isomorphisms toisomorphisms, the vertical morphisms in

FXα //

∼= Fw

GX

∼=Gw

FYα// GY,

are isomorphisms, showing that α ∈ FunW(Ch(A), C[1]). The universal property of

γ implies that there is a unique functor α′ : D(A)→ C[1] such that

Ch(A)α //

γ

C[1]

D(A)∃!α′

II

commutes. This α′ corresponds to a natural transformation α′ : F ′ → G′ definedby

α′(X) = (α′X : F ′X → G′X), X ∈ D(A),

and the reader easily checks that this is the unique natural transformation F ′ → G′

satisfying α′γ = α.


3.3. Cones as derived cokernels. We now take a closer look at the cone con-struction C : Ch(A[1])→ Ch(A) as defined via the pushout square (3.15). Clearly,CX0 → 0 induces a morphism of spans (CX0 ← X0 → X1) → (0 ← X0 → X1),and hence an induced morphism of pushout diagrams

X0f

//

= ##

X1=

%%

X0//

X1

CX0

""

// Cf

$$

0 // cok(f).

The exact sequence of Lemma 3.18 is easily seen to be natural in the pushoutdiagram, hence we obtain a natural morphism of exact sequences

(3.30)

0 // X0//

= ∼

X1 ⊕ CX0

∼

// Cf //

0

X0f

// X1// cok(f) // 0,

and, in particular, a natural transformation

(3.31) φ : Cf → cok(f).

We remark the following concerning this natural morphism of exact sequences.

Remark 3.32. We observe that the first two components of the natural morphismof exact sequences (3.30) define a natural levelwise quasi-isomorphism

(3.33)

X0//

= ∼

X1 ⊕ CX0

∼

X0f

// X1.

Moreover, the domain of this quasi-isomorphism is a monomorphism and we knowby Lemma 3.9 that the cokernel preserves quasi-isomorphisms between monomor-phisms. We refer to this by saying that (3.33) is a functorial resolution onCh(A[1]) which is adapted to the functor cok: Ch(A[1])→ Ch(A). It follows thatthe morphism φ : Cf → cok(f) is a quasi-isomorphism for all monomorphisms f .

As a preparation for the theorem we recall that, by Example 3.26, the coneinduces a functor C : D(A[1])→ D(A). The natural transformation (3.31) yields anatural transformation ε : C γ → γ cok given by

(3.34) ε : C γ = γ C γφ→ γ cok.

Theorem 3.35. Let A be an abelian category.

(i) The cone functor C : D(A[1]) → D(A) together with ε defined in (3.34) is aleft derived functor of the cokernel cok: Ch(A[1])→ Ch(A).

(ii) The fiber functor F : D(A[1]) → D(A) together with a dually defined η is aright derived functor of the kernel ker : Ch(A[1])→ Ch(A).

22 MORITZ GROTH

Proof. We have to show that (C, ε) satisfies the universal property of Definition 3.27.For this purpose let G : D(A[1]) → D(A) be a functor and let α : G γ → γ cokbe a natural transformation,

Ch(A[1])cok //

γ

Ch(A)

γ

D(A[1])G// D(A).

AI

We claim that it is enough to show that there is a unique natural transformationψ : G γ → C γ such that

α = ε ψ : G γ → C γ → γ cok.

In fact, by Lemma 3.29 we then deduce that there is a unique natural transformationα′ : G → C such that ψ = α′γ, which immediately yields the desired universalproperty of (C, ε).

In order to construct such a transformation ψ we consider (X0f→ X1) ∈ Ch(A[1])

and associate to it the following commutative diagram

Gγ(X0 → X1 ⊕ CX0)α //

∼=

γcok(X0 → X1 ⊕ CX0)

Cγ(X0 → X1 ⊕ CX0)ε∼=oo

∼=

Gγ(X0 → X1)α

// γcok(X0 → X1) Cγ(X0 → X1).ε

oo

In this diagram, the vertical morphisms are induced by the resolutions adaptedto the cokernel (see Remark 3.32) and it hence follows that the two outer verticalmorphisms are isomorphisms. Moreover, both squares commute by naturality of αand ε, respectively. Finally, the top horizontal morphism

ε = γφ : Cγ(X0 → X1 ⊕ CX0)→ γcok(X0 → X1 ⊕ CX0)

is an isomorphism since X0 → X1 ⊕ CX0 is a monomorphism; see again Re-mark 3.32. Thus, the above commutative diagram shows us that there is at mostone such natural transformation ψ. These morphisms Gγ(f)→ Cγ(f) actually as-semble to a natural transformation Gγ → Cγ as one can check from the naturalityof the above diagram in (f : X0 → X1).

Remark 3.36. We include a short philosophical remark concerning the constructionof the cone as a derived version of the cokernel. The cokernel is a categoricalconstruction which amounts to ‘collapsing’ the image of a morphism. As observedin Warning 3.8 the cokernel is not an exact or homotopy invariant construction inthat levelwise quasi-isomorphisms are not always sent to quasi-isomorphisms.

One obtains the cone from the cokernel by, instead of collapsing the image,simply ‘adding the potential of collapsing the image’. This is made precise by thedefining pushout square (3.15). In fact, thinking geometrically, by glueing a coneon the image of the morphism we add the potential of collapsing it (since we cannow push it to the apex of the cone).

Similar comments also apply to other derived colimit constructions and the mor-phism (3.31) is a special instance of a general comparison morphism between derivedcolimits and categorical colimits. This picture extends to the topological contextand homotopy colimits.


We conclude this section by one more application of Lemma 3.29, leading to animportant warning. Let us denote by 1 the terminal category consisting of oneobject ∗ and its identity morphism only. For any category A the evaluation at thisunique object yields an isomorphism of categories

A1 ∼= A.

Given an object a ∈ A we also write a : 1 → A for the corresponding functor.Similarly, any morphism f : a→ b in A yields a natural transformation

1

a''

b

88 A.

Lemma 3.37. Let A be a small category, let a ∈ A, and let A be an abeliancategory. The functor a∗ : Ch(AA) → Ch(A) sends quasi-isomorphisms to quasi-isomorphisms and hence induces an evaluation functor a∗ : D(AA)→ D(A) suchthat the following diagram commutes,

Ch(AA)a∗ //

γ

Ch(A)

γ

D(AA)a∗// D(A).

Proof. This is immediate from Lemma 3.7 and Definition 3.23.

Similarly, by Lemma 3.29, every morphism f : a → b induces a unique naturaltransformation

f∗ : a∗ → b∗ : D(AA)→ D(A)

between evaluation functors such that f∗ γ = γ f∗. Given X ∈ D(AA) wesimplify notation by writing a∗(X) = Xa ∈ D(A). Thus, every f : a → b inducesa natural morphism Xf : Xa → Xb in D(A). The reader easily checks that everyX ∈ D(AA) defines a diagram

diaA(X) : A→ D(A), a 7→ Xa, f 7→ Xf .

We refer to diaA(X) as the underlying diagram of X.

Lemma 3.38. Let A be a small category and let A be an abelian category. Thereis an underlying diagram functor

diaA : D(AA)→ D(A)A : X 7→ diaA(X).

Proof. The reader easily defines diaA on morphisms and checks that this defines afunctor.

Warning 3.39. (i) Note the difference between the domain and the target ofthese underlying diagram functors. While the domain is the derived categoryof a diagram category the target is a diagram category of the derived category.In general, the functor diaA is not an equivalence and it is important todistinguish these two categories. Roughly speaking, the functor diaA takes astrict diagram to a diagram which commutes up to chain homotopies only,and applying this functor hence results in a loss of information. We will getback to this later.

24 MORITZ GROTH

(ii) As a special case let us consider A = [1] and the associated underlying di-agram functor dia[1] : D(A[1]) → D(A)[1] from the derived category of thearrow category to the arrow category of the derived category. One sloganconcerning defects of triangulated categories is that cone constructions arenot functorial at that level. In the case of derived categories, this statementtakes the more precise form that the cone functor C : D(A[1]) → D(A) doesnot factor over dia[1] : D(A[1])→ D(A)[1], i.e., in general, there is no dashedarrow

D(A[1])C //

dia[1]

D(A)

D(A)[1]

@

::

making the diagram commute. To re-emphasize, there is a functorial coneconstruction C : D(A[1]) → D(A) (Example 3.26), but only as long as westay in the derived category of the arrow category.

4. Derived categories as triangulated categories

Classically, the derived category is obtained by first passing to the homotopycategory of an abelian category and observing that it can be turned into a trian-gulated category. The derived category is then obtained by a further localizationand, using a theorem of Verdier, this allows one to conclude that derived categoriesare triangulated categories. Here we content ourselves by collecting a few of thekey steps.

Despites being a very sucessful theory, triangulated categories suffer certain de-fects. We conclude this section by a short discussion of such defects. Later in thiscourse we will see that these defects can be fixed by working with stable derivatorsinstead.

4.1. The homotopy category of an abelian category. In this subsection weconsider the homotopy category K(A) of an abelian category A. This categoryoccurs as an intermediate step in the classical construction of the derived cate-gory D(A) = Ch(A)[W−1

A ]. Since the chain homotopy relation ' is a congruencerelation (i.e., we have equivalence relations on all homCh(A)(X,Y ) which are com-patible with compositions), we can make the following definition.

Definition 4.1. Let A be an abelian category. The homotopy category K(A)of A is the quotient category

K(A) = Ch(A)/ ' .

Thus, objects in K(A) are simply chain complexes and morphisms are chainhomotopy classes of morphisms. The composition is defined by choosing represen-tatives and passing to the chain homotopy class of the corresponding composition.

By definition there is a quotient functor γ′ : Ch(A)→ K(A) which is the univer-sal example of a functor identifying chain homotopic maps. Note that a chain mapf : X → Y is a chain homotopy equivalence if and only if γ′(f) is an isomorphism.Similarly, a chain complex X is sent to zero in K(A) if and only if X is contractible.

As a preparation for a different characterization of γ′, we recall the cylinderconstruction at the level of chain complexes. This construction is a further instance


of the biadditive pairing described in Digression 3.12. It might be convenient tobriefly recall Examples 3.14.

Examples 4.2. (i) Let I+ = Ccell([0, 1]+) ∈ Chb(FZ) be the reduced cellularchain complex of [0, 1]+ = [0, 1]t∗ endowed with the CW structure consistingof three 0-cells and one 1-cell only. More explicitly, we have I0 = Z⊕Z, I1 = Z,and Ik = 0, k 6= 0, 1 with non-trivial differential (−id, id) : Z ⊕ Z → Z. Thecontinuous map [0, 1]→ ∗ induces a chain map I+ → S0 which is easily seento be a chain homotopy equivalence. The cylinder cyl(X) of X ∈ Ch(A) isdefined as

cyl(X) = I+ ⊗X.It follows from Digression 3.12, Examples 3.14, and Lemma 3.6 that the chainmap I+ → S0 induces a natural chain homotopy equivalence cyl(X)→ X.

(ii) Let S0+ = Ccell(0, 1+) ∈ Chb(FZ) be the reduced cellular chain complex

of 0, 1+ endowed with the CW structure consisting of three 0-cells only.Thus, S0

+∼= S0 ⊕ S0 and for X ∈ Ch(A) there is a natural isomorphism

S0+ ⊗X ∼= X ⊕X.

(Recall that the space 0, 1+ is the wedge of two copies of 0, 1.)(iii) The continuous map 0, 1 → [0, 1] yields a map S0

+ → I+ in Chb(FZ) andhence a natural transformation S0

+⊗− → I+⊗− of functors Ch(A)→ Ch(A).Under the above isomorphisms, for X ∈ Ch(A) this yields the usual naturalinclusion i : X ⊕ X → cyl(X). We denote the two resulting inclusions byi0, i1 : X → cyl(X).

The point of this cylinder construction is the following.

Lemma 4.3. Let f, g : X → Y be chain maps. There is a bijection between chainhomotopies s : f → g and chain maps H : cyl(X)→ Y making the following diagramcommute

X

i0 $$

f

cyl(X)H // Y.

X

i1::

g

@@


Thus, as in topology, the cylinder is the natural domain for homotopies. In thespecial case of the constant chain homotopy of the identity id : X → X one checksthat the resulting homotopy

(4.4)

X

i0 $$

id

cyl(X)HX // X

X

i1::

id

@@

is the chain homotopy equivalence cyl(X) → X from Examples 4.2(i). These ob-servations can be reformulated as follows.

26 MORITZ GROTH

Lemma 4.5. Let A be an abelian category and let C be a category. The followingare equivalent for a functor F : Ch(A)→ C.

(i) For every pair of chain homotopic maps f, g we have Ff = Fg.(ii) The functor F sends chain homotopy equivalences to isomorphisms.

Proof. Clearly, (i) implies (ii). Conversely, (ii) implies that F sends the chainhomotopy equivalences HX in (4.4) to isomorphisms in C. Consequently, the inclu-sions i0, i1 : X → cyl(X) have the same images in C and statement (i) is hence animmediate consequence of Lemma 4.3.

Corollary 4.6. Let A be an abelian category and let HEA be the class of chainhomotopy equivalences in Ch(A). The functor γ′ : Ch(A) → K(A) exhibits K(A)as a localization of Ch(A) at the class of chain homotopy equivalences,

K(A) ∼= Ch(A)[HE−1A ].

Proof. In view of Lemma 4.5 this is simply a reformulation of the universal propertyof the quotient functor γ′ : Ch(A)→ K(A).

This corollary shows us that the homotopy category is indeed an intermediatestep in the construction of the derived category. Since K(A) is a quotient categoryit is simpler to put ones hands on it.

Lemma 4.7. The homotopy category K(A) of an abelian category is additive.

Proof. This is a special case of the more general fact that quotients of additivecategories by additive congruence relations are again additive. We leave the detailsto the reader.

Warning 4.8. It is not true, in general, that the homotopy category of an abeliancategory is abelian.

The additive category K(A) can be endowed with some additional structurewhich can be thought of as shadows of the existence of short existence sequenceson the category Ch(A). First, the suspension functor Σ: Ch(A) → Ch(A) clearlyinduces a suspension functor Σ: K(A) → K(A), which is again an equivalence ofcategories.

A triangle in K(A) is a diagram of the form X → Y → Z → ΣX and amorphism of triangles is a commutative diagram

X //

f

Y //

g

Z //

h

ΣX

Σf

X ′ // Y ′ // Z ′ // ΣX ′.

Note that we insist that the vertical morphism to the very right is the suspension ofthe morphism to the very left. These notions of course make sense in any categorywith an endofunctor.

We now single out a particular class of triangles on K(A). Recall from (3.15)and Definition 3.16 that associated to any chain map f : X → Y there is the coneCf which is endowed with a chain map cof(f) : Y → Cf . There is also a canonicalmap Cf → ΣX. To construct it, we note that by the definition of the cone CX ofa chain complex there is a natural short exact sequence

0→ Xi→ CX

q→ ΣX → 0.


(In fact, by means of Examples 3.14 this is induced by S0⊗− → I ⊗− → S1⊗−.)The universal property of the defining pushout diagram (3.15) applied to q and thezero map 0: Y → ΣX implies that there is a unique map Cf → ΣX making thefollowing diagram

Xf//

i

Ycof(f) 0

CX //

q 00

Cf∃!""

ΣX

commutative. The resulting triangle

X → Y → Cf → ΣX

in K(A) is the standard triangle of f . We say that a triangle in K(A) is distin-guished if it is isomorphic to a standard triangle.

The additional structure on K(A) consisting of the suspension functor and theabove class of distinguished triangles satisfies certain properties as axiomatized bythe notion of a triangulated structure.

4.2. Triangulated categories. In this subsection we take a glimpse at the the-ory of triangulated categories. Here we only include what is strictly necessary inthis motivational section. For more details we refer the reader to the establishedliterature which includes [Nee01, HJR10].

Definition 4.9. Let T be an additive category with a self-equivalence Σ: T → Tand a class of distinguished triangles X → Y → Z → ΣX. The pair consisting of Σand the class of distinguished triangles defines a triangulated structure on T ifthe following four axioms are satisfied.

(T1) For every X ∈ T , the triangle Xid→ X → 0 → ΣX is distinguished. Every

morphism in T occurs as the first morphism in a distinguished triangle and theclass of distinguished triangles is replete, i.e., is closed under isomorphisms.

(T2) If the triangle Xf→ Y

g→ Zh→ ΣX is distinguished then also the rotated

triangle Yg→ Z

h→ ΣX−f→ ΣY is distinguished.

(T3) Given two distinguished triangles and a commutative solid arrow diagram

X //

u

Y //

v

Z //

∃w

ΣXΣu

X ′ // Y ′ // Z ′ // ΣX ′

there exists a dashed arrow w : Z → Z ′ as indicated such that the extendeddiagram commutes.

28 MORITZ GROTH

(T4) For every pair of composable arrows f3 : Xf1→ Y

f2→ Z there is a commutativediagram in which the rows and columns are distinguished triangles:

Xf1 // Y

g1 //

f2

C1h1 //

ΣX

Xf3

// Zg2

g3// C3

h3

//

ΣXΣf1

C2

h2

C2

Σg1h2

h2

// ΣY

ΣYΣg1

// ΣC1

A triangulated category is an additive category together with a triangulatedstructure.

A few comments about these axioms are in order.

Remark 4.10. (i) Axiom (T2) is the rotation axiom, axiom (T3) the (map-ping) cone axiom, and axiom (T4) the octahedron axiom. The nameoctahedron axiom is motivated by a different way of drawing the diagram inaxiom (T4); see for example [Wei94, p. 375].

(ii) Triangulated categories were invented independently by Puppe [Pup67] (mo-tivated by Algebraic Topology) and Verdier [Ver96] (motivated by AlgebraicGeometry) in his 1967 thesis, the difference being that Puppe triangulationsare only asked to satisfy axioms (T1)-(T3).

(iii) One way to think of distinguished triangles is that they are some shadows ofcertain derived cokernel constructions on ‘a model in the background’ (thiswill be justified in §4.3). Motivated by this we refer to the third object in adistinguished triangle as a cone (as already reflected in the name of axiom(T3)). The axioms (T1)-(T4) capture some compatibility properties satisfiedby such derived cokernel constructions.

(iv) Using this interpretation of the third objects in distinguished triangles, wecan think of the octahedron axiom as a triangulated category version of thethird Noether isomorphism theorem. In fact, while the Noether isomorphismtheorem says that ‘the quotient of two quotients is again a quotient’ theoctahedron axiom asks that ‘a cone of two cones is again a cone’.

(v) Note that axiom (T3) asks for a weak functoriality of the cone construction:any ‘partial morphism’ of triangles can be extended to an actual morphism.However, we do not ask for the uniqueness of such an extension which is tosay that the cone is not necessarily a functorial construction T [1] → T . Thislack of functoriality has important implications and we come back to thisin §4.4.

(vi) There is a stronger version of the rotation axiom, asking that a triangle isdistinguished if and only if the rotated diagram is distinguished. It can beshown under the assumption of (T1) and (T3) that this stronger version ofthe rotation axiom is a consequence of the weaker one.

(vii) Also the octahedron axiom is given in a seemingly weaker form than inother references. The observation that the stronger version follows from thisweaker one goes back to [KV87] and this form was recently used by Holmand Jørgenson in their survey article in [HJR10]. For a short discussion of


this fact and related references see [HJR10, Remark 3.2]. The motivation forus to choose these slightly simpler axioms is that this will later simplify theproof that stable derivators give rise to canonical triangulations.

We suggest the reader who just saw the definition of a triangulated category forthe first time to provide the proofs of the following statements.

Proposition 4.11. Let T be a triangulated category and let W ∈ T . For everydistinguished triangle X → Y → Z → ΣX the represented functor homT (W,−)yields an exact sequence of abelian groups

homT (W,X)→ homT (W,Y )→ homT (W,Z).


Using the rotation axiom (TR2) in its symmetric form we see that representedfunctors send distinguished triangles to long exact sequences. This is often referredto by saying that represented functors are homological.

Also the proof of the following 5-lemma is suggested as an exercise.

Proposition 4.12. Let T be a triangulated category and let

X //

f

Y //

g

Z //

h

ΣXΣf

X ′ // Y ′ // Z ′ // ΣX ′

be a morphism of distinguished triangles. If two of the maps f, g, h are isomorphismsthen so is the third one.


Corollary 4.13. Let X → Y be a morphism in a triangulated categories. Any twodistinguished triangles extending the morphism are isomorphic.

Proof. This is immediate from axiom (T3) and the 5-lemma.

Warning 4.14. In general these isomorphisms are not canonical. We will get backto this in §4.4.

Lemma 4.15. A morphism in a triangulated category is an isomorphism if andonly if any of its cones is zero.


Solutions to all these exercises and much more can be found in [Nee01, §1].

4.3. Classical triangulations on derived categories. We again consider anabelian category A. Recall from the end of §4.1 that we defined a suspensionfunctor Σ: K(A)→ K(A) and a class of distinguished triangles on K(A).

Theorem 4.16. The homotopy category K(A) of an abelian category A togetherwith the equivalence Σ: K(A)→ K(A) and the above class of distinguished trianglesis a triangulated category.

Proof. The category K(A) is additive by Lemma 4.7 and Σ: K(A) → K(A) isan equivalence. Axiom (T1) is immediate while the remaining axioms are moreinvolved. We refer the reader to [HJR10] or [Wei94, Proposition 10.2.4] for moredetails.

30 MORITZ GROTH

In the notation of Corollary 4.6 there clearly is an inclusion HEA ⊆WA and, bythat corollary, the localization functor γ : Ch(A) → D(A) hence factors uniquelyover γ′ : Ch(A)→ K(A), yielding the canonical functor

γ′′ : K(A)→ D(A).

Passing through the homotopy category K(A) as an intermediate step, the derivedcategory is obtained as the localization D(A) = K(A)[W−1]. There are generaltechniques which allow us to conclude that such localizations are again triangulatedcategories and that the localization functor is exact in the following sense; see forexample [Wei94, §10] and [Kra10].

Definition 4.17. Let T , T ′ be triangulated categories. An exact functor T → T ′is a pair (F, σ) consisting of

(i) an additive functor F : T → T ′ and(ii) a natural transformation σ : FΣ→ ΣF

such that for every distinguished triangle X → Y → Zh→ ΣX in T the image

triangle FX → FY → FZσFh→ ΣFX is distinguished in T ′.

Remark 4.18. Note that exactness of a functor of triangulated categories is not aproperty but that it amounts to specifying an additional structure. If we want toemphasize this then we refer to σ : ΣF → FΣ as an exact structure on F .

The following theorem is a consequence of the more general Verdier localizationtheorem; again we refer the reader to the literature.

Theorem 4.19. Let A be an abelian category. The derived category D(A) can beturned into a triangulated category and the localization functor γ′′ : K(A)→ D(A)into an exact functor.

4.4. Beyond triangulated categories. The theory of triangulated categories isa large theory and has been successful in many applications in various areas ofpure mathematics for roughly fifty years by now. For a few sample applications invarious areas we refer the reader to the book [HJR10].

Despites all these successes, from the very beginning on (see already the intro-duction to [Hel68]) it was also apparent that the axioms of a triangulated categorycome with certain defects. We include a few comments along these lines.

(i) One crucial observation is that the cone construction at the level of trian-gulated categories is not functorial. For every morphism in T there is adistinguished triangle (axiom (T1)), hence we can associate a cone object toit. There are the following (closely related) drawbacks to this definition of acone.(a) Cone objects are not characterized by a universal property.(b) We already observed that any two cones of a fixed object are isomorphic,

but only by non-canonical isomorphisms.(c) Similarly, every morphism of morphisms in T yields an induced mor-

phism on cones (axiom (T3)), but, again, this morphism is not uniqueand hence it lacks functoriality.

And in fact, it turns out that if there is a cone functor T [1] → T then thetriangulated category is semi-simple ([Ver96, Proposition 1.2.13]). This is tobe seen in contrast to Example 3.26.


(ii) The axioms of a triangulated category ask for the additional structure ofa suspension functor and of distinguished triangles. This datum does notsatisfy any universal property and is hence non-canonical. This is partiallyreflected in the fact that exact morphisms of triangulated categories haveto be defined by means of additional structure as well. But in the typicalexamples the fact that we have an exact morphism of triangulated categoriesreflects the idea that ‘in the background on the models’ there is a morphismof ‘stable homotopy theories’ which has the property of preserving certainfinite homotopy (co)limits.

(iii) As we saw in Theorem 3.35 the cone functor C : D(A[1]) → D(A) is a leftderived cokernel functor, hence satisfies a universal property. There are ofcourse many further derived (co)limit functors which are important in appli-cations. Given a nice abelian category A and a small category A, often onecan define derived (co)limit functors LcolimA, RlimA : D(AA) → D(A). Acrucial observation however is that, say, LcolimA does not factor through theunderlying diagram functors diaA : D(AA)→ D(A)A of Lemma 3.38, i.e., ingeneral, there is not a dashed arrow

D(AA)Lcolim//

diaA

D(A)

D(A)A@

::

making the diagram commutative.A typical slogan referring to this fact is that ‘diagrams in triangulated

categories do not carry enough information (to canonically determine theirhomotopy (co)limits)’. A different but related slogan is that the underly-ing diagram functor amounts to passing from strict diagrams to homotopycommutative diagrams, and that such homotopy commutative diagrams donot carry enough information. (In fact, one would need homotopy coherentdiagrams to construct associated homotopy (co)limits.)

(iv) Still a related observation is as follows. Given a triangulated category Tand a small category A there is no canonical triangulation on the diagramcategory T A.

(v) A further related observation is that the axioms of a triangulated category,while satisfied in many examples, might miss interesting aspects of homolog-ical algebra or homotopy theory. As mentioned above, the axioms encodesome structure which is a shadow of iterated cofiber seqences and certaincompatibilities. The octahedron axiom (T4) goes one step further and con-siders two composable morphisms together with the various associated dis-tinguished triangles. However, the axioms have nothing to say if one wantsto consider longer strings of composable arrows and all the related distin-guished triangles. Pursuing this further leads to higher triangulations; seefor example [BBD82, Remark 1.1.14] and [Mal05].

The aim of the remainder of this course is to show that stable derivators providean alternative theory which successfully adresses the above issues. In fact, one canthink of a stable derivator as a minimal, purely categorical extension of a derivedcategory to a framework which comes with a well-behaved calculus of derived limitsand derived colimits.

32 MORITZ GROTH

5. Kan extensions

In this section we briefly review some basics concerning the theory of Kan exten-sions from ordinary category theory and collect a few first examples. Kan extensionsare certain universal constructions which can be thought of as ‘relative versions oflimits and colimits’. The basic idea behind a derivator is to axiomatize key formalproperties of the calculus of (homotopy) Kan extensions.

5.1. Motivation. To motivate the notion of a Kan extension, we recall from §3(see, in particular, Theorem 3.35) that for every abelian category A the cone functorC : D(A[1]) → D(A) is the derived cokernel. Given a morphism f : X0 → X1 inCh(A), the cone Cf is defined as the pushout object in

X0f//

i

X1

cof(f)

CX0// Cf.

Thus, in all detail, the cone of a chain map is obtained by associating a particularspan to it, passing to the pushout of the span, and then evaluating the pushoutsquare on the final vertex:

(5.1)

X0f// X1

X0f//

i

X1

X0f//

i

X1

cof(f)

CX0 CX0// Cf Cf

Similarly to the case of cokernels, also for cones it is often important to considerthe morphism cof(f) : X1 → Cf instead of the object Cf only. And to obtain adetailed description of the construction of cof(f) it suffices to replace the final stepin (5.1) by the restriction to the vertical morphism on the right.

As an upshot, both constructions pass through intermediate steps which yieldsuitable extensions of the chain map [1]→ Ch(A) to larger diagrams B → Ch(A).The purely categorical notion which is in the background of these intermediatesteps is the notion of a Kan extension.

To abstract from this specific situation, let us consider the following extensionproblem. Let u : A → B be a functor between small categories and let X : A → Cbe a functor taking values in a not necessarily small category C. The aim is to find‘extensions of X along u’,

(5.2)Au

X // C

B.∃?

>>

The notion of ‘extension’ can be made precise in different ways, namely by askingthat the above diagram

(i) commutes on the nose,(ii) commutes up to a natural isomorphism, or(iii) commutes up to a universal natural transformation.

The first condition is ‘against the categorical philosophy that one should not askthat functors are equal but only naturally isomorphic’. While satisfied in manyspecific situations, in general, the second condition turns out to be too strict. The


third condition is the one we would like to axiomatize and, of course, there are twoversions of such universal natural transformations, depending on whether we wantthe datum to be initial or final.

Definition 5.3. Let u : A → B be a functor between small categories, let C be acategory, and let X : A→ C.

(i) A left Kan extension of X along u is a functor LKanu(X) : B → C togetherwith a natural transformation η : X → LKanu(X) u satisfying the followinguniversal property. For every pair (Y : B → C, α : X → Y u) there is aunique β : LKanu(X)→ Y such that α = (βu) · η : X → Y u.

(ii) A right Kan extension of X along u is a functor RKanu(X) : B → Ctogether with a natural transformation ε : RKanu(X) u→ X satisfying thefollowing universal property. For every pair (Y : B → C, α : Y u→ X) thereis a unique β : Y → RKanu(X) such that α = ε · (βu) : Y u→ X.

These notions are dual to each other and in this section we will mostly focus onleft Kan extensions. To develop some intuition for this notion let us consider thefollowing special case. Recall that we denote the terminal category by 1.

Example 5.4. Let A be a small category and let X : A → C be given. A leftKan extension of X along πA : A → 1 consists of a functor LKanπA

(X) : 1 → Cand a universal natural transformation η : X → LKanπA

(X) πA. The functorLKanπA

(X) : 1 → C amounts to picking an object c ∈ C and the natural transfor-mation is of the form η : X → ∆A(c). Thus, the left Kan extension (c, η) is aninitial cocone on X, which is to say that c = colimAX and that η is the colimitingcocone.

Thus, special cases of left Kan extensions always exist as soon as the targetcategory is cocomplete. We will see in the next subsection that this is true moregenerally.

5.2. Pointwise Kan extensions. Let us consider a solid arrow diagram (5.2) suchthat C is cocomplete. The universal property of a left Kan extension suggests thefollowing construction of it.

Let Y : B → C and α : X → Y u be a datum among which the left Kan extensionis supposed to be initial and let us fix an object b ∈ B. For each morphism g : a→ a′

in A we obtain the naturality square

X(a)X(g)

//

αa

X(a′)

αa′

Y u(a)Y u(g)

// Y u(a′).

Given b ∈ B the value Y (b) has to be compatible with these naturality squaresin the following sense. Let us assume we can find morphisms f : u(a) → b andf ′ : u(a′)→ b in B which are compatible with g in the sense that f = f ′ u(g). Foreach such datum the above naturality square extends to the following commutative

34 MORITZ GROTH

diagram

X(a)X(g)

//

αa

X(a′)

αa′

Y u(a)Y u(g)

//

Y f

Y u(a′)

Y f ′zz

Y (b)

in C. Ignoring the line in the middle, we see that Y (b) is part of a cocone on acertain diagram. The universal property of a left Kan extension suggests that weshould form universal such cocones by passing to suitable colimits. This indeedworks and leads to the following result.

Proposition 5.5. Let u : A→ B be a functor between small categories, let C be acategory, and let X : A→ C.

(i) If C is cocomplete, then the left Kan extension of X along u exists.(ii) If C is complete, then the right Kan extension of X along u exists.

The above heuristics suggest that there should be a more precise result alongthese lines coming with explicit formulas which allow us to construct left Kanextensions. Motivated by the above we give the following definition.

Let u : A → B be as above and let b ∈ B. The slice category (u/b) has asobjects pairs consisting of an object a ∈ A together with a morphism f : u(a)→ bin B. A morphism (a, f) → (a′, f ′) is a morphism g : a → a′ in A such thatf ′ u(g) = f ,

ag// a′,

u(a)u(g)//

f

u(a′)

f ′b.

The slice category comes with a projection functor p : (u/b) → A defined byp(a, f) = a and p(g) = g.

Note that the slice category (u/b) depends functorially on b ∈ B. In fact, asso-ciated to h : b → b′ there is the functor h∗ : (u/b) → (u/b′) defined on objects by(a, f) 7→ (a, hf) and which is the identity on morphisms. These functors are com-patible with the projection functors, i.e., we have pb′ h∗ = pb where we wrote pbfor the projection functor (u/b)→ A.

Now, we assume moreover that we are given a functor X : A→ C taking valuesin a cocomplete category C. We define a functor L : B → C on objects by setting

L(b) = colim(u/b)X pb ∈ C.

The universal property of colimiting cocones implies that for every h : b→ b′ thereis a unique morphism L(h) : L(b)→ L(b′) in C,

(5.6) L(h) : colim(u/b)X pb → colim(u/b′)X pb′ ,


which is suitably compatible with the colimiting cocones. In fact, this morphismis obtained by restricting the colimiting cocone of L(b′) along h∗ : (u/b) → (u/b′)in order to obtain a further cocone on X pb′ h∗ = X pb. The universality ofthe colimiting cocone of L(b) implies that there is a canonical L(h) as desired. Asalways, uniqueness implies functoriality, concluding the definition of L : B → C.

In order to obtain a candidate left Kan extension we also have to define a naturaltransformation η : X → L u. For a ∈ A the corresponding component

(5.7) ηa : X(a)→ L(ua) = colim(u/u(a))X pu(a)

is obtained as follows. Note that (a, 1: u(a)→ u(a)) defines an object in (u/u(a))and that (X pu(a))(a, 1u(a)) = X(a). Thus, ηa can be defined as the component ofthe colimiting cocone at the object (a, 1: u(a) → u(a)). We leave it to the readerto check that this defines a left Kan extension (L, η) of X along u.

If one tries to similarly construct a right Kan extension, then one is led to theslice category (b/u). An object in (b/u) is a pair (a, f : b→ u(a)) and morphismsare morphisms in A making the obvious triangles in B commute. This slice categorycomes with a projection functor q = qb : (b/u)→ A.

Proposition 5.8. Let u : A→ B be a functor between small categories, let C be acategory, and let X : A→ C.

(i) If C is cocomplete, then the left Kan extension of X along u exists. Moreover,for b ∈ B we have an isomorphism LKanu(X)(b) ∼= colim(u/b)X pb.

(ii) If C is complete, then the right Kan extension of X along u exists. Moreover,for b ∈ B we have an isomorphism RKanu(X)(b) ∼= lim(b/u)X qb.

Thus, in sufficiently (co)complete categories, Kan extensions always exist andcan be constructed using pointwise formulas.

Warning 5.9. There are examples of Kan extensions (necessarily in contexts lackingcertain limits or colimits) which cannot be calculated using the above formulas.Such Kan extensions will not play a role in this course.

Every functor u : A→ B between small categories induces restriction functorsor precomposition functors u∗ : CB → CA. The following is immediate from thedefinition of Kan extensions and Proposition 5.8.

Corollary 5.10. Let u : A→ B be a functor between small categories.

(i) If C is a cocomplete category, then left Kan extensions define a functorLKanu : CA → CB which is left adjoint to u∗ : CB → CA,

(LKanu, u∗) : CA CB .

(ii) Dually, if C is a complete category, then right Kan extensions define a functorRKanu : CA → CB which is right adjoint to u∗ : CB → CA,

(u∗,RKanu) : CB CA.For left Kan extensions the adjunction unit η : X → u∗LKanuX = LKanu(X)u

has components (5.7), and dually for right Kan extensions.

Notation 5.11. We also denote Kan extension functors by

u! = LKanu : CA → CB and u∗ = RKanu : CA → CB .In the special case that u = b : 1→ B classifies the object b ∈ B we thus write

b! = LKanb : C → CB and b∗ = RKanb : C → CB .

36 MORITZ GROTH

Taking up again Example 5.4 we obtain the following example, leading us tothink of left Kan extensions as ‘relative colimits’, obtained by replacing πA : A→ 1by more general functors u : A→ B.

Example 5.12. Let A be a small category, let C be a cocomplete category, and letus consider the functor πA : A→ 1. Under the isomorphism C1 ∼= C the adjunction(LKanπA

, π∗A) : CA C1 gets identified with (colimA,∆A) : CA C.

5.3. Basic properties and first examples. We begin by considering free andcofree diagrams.

Example 5.13. Let B be a small category, b ∈ B, and C a complete and cocompletecategory (actually, the existence of products and coproducts suffices). We describethe Kan extension functors b!, b∗ : C → CB along b : 1→ B.

(i) The left Kan extension functor b! : C → CB sends an object X ∈ C to the freediagram b!(X) : B → C generated by X. The pointwise formula implies thatthere are isomorphisms

b!(X)b′ ∼= colim(b/b′)X pb′ ∼=∐

homB(b,b′)

X,

where the second isomorphism is a consequence of (b/b′) being the discretecategory on homB(b, b′) so that the colimit reduces to a coproduct.

(ii) The right Kan extension functor b∗ : C → CB sends an object X ∈ C to thecofree diagram b∗(X) : B → C generated by X. In this case there arenatural isomorphisms

b∗(X)b′ ∼= lim(b′/b)

X qb′ ∼=∏

homB(b′,b)

X.

Remark 5.14. Note that this example shows that for X : A → C the diagramsLKanu(X) u,RKanu(X) u : A → C are, in general, not naturally isomorphicto X. Thus, in general, we really change a diagram by passing to a Kan extensionand then restricting back to the original category.

We include a short digression on final functors. For this purpose let u : A→ B bea functor between small categories and letX : B → C be a functor with a cocompletetarget category C. In particular, we hence obtain the two colimits colimA(X u)and colimB X. If we restrict the colimiting cocone of colimB X along u, then forevery f : a→ a′ in A we obtain a commutative diagram

X(ua)

f∗

// colimB X

X(ua′),

99

which is to say that we have a cocone on X u. The universality of colimitingcocones implies that there is a canonical map

(5.15) colimA(X u)→ colimB X,

which is compatible with the cocones. (We already saw an example of these canon-ical maps, namely, when we defined the behavior of pointwise left Kan extensionson morphisms; see (5.6).)


Definition 5.16. A functor u : A → B between small categories is final if thecanonical map (5.15) is an isomorphism for every cocomplete category C and everydiagram X : B → C.

Lemma 5.17. Right adjoint functors between small categories are final.

Proof. Let us consider an adjunction (u, v) : A B between small categories andlet C be a cocomplete category. Passing to the associated precomposition functorswe obtain an induced adjunction

(v∗, u∗) : CA CB .If we combine this with the adjunction (colimB ,∆B) : CB C, then we see thatcolimB v∗ is left adjoint to u∗ ∆B = ∆A and we hence obtain a canonical iso-morphism colimA

∼= colimB v∗. With a bit more care one checks that this is thedesired canonical map (5.15).

Corollary 5.18. Let A be a small category admitting a final object ∗. Every cate-gory C has colimits of shape A and for X : A→ C there is a natural isomorphism

X(∗) ∼= colimAX.

Proof. This is immediate from the proposition since (πA, ∗) : A 1 is an adjunc-tion.

There is a combinatorial criterion which allows us to characterize final functors.Let u : A → B be a functor between small categories and let b ∈ B be an object.We recall that the slice category (b/u) has as objects pairs (a ∈ A, f : b → u(a))and as morphisms (a, f)→ (a′, f ′) morphisms a→ a′ in A such that

bf//

f ′

u(a)

u(a′)

commutes.

Proposition 5.19. A functor u : A → B is final if and only if the categories(b/u), b ∈ B, are non-empty and connected.

For a proof we refer the reader, for example, to [KS06, §2.5]. To be completelyspecific, u is final if and only if for each b ∈ B the following two properties aresatisfied.

(i) There is an object a ∈ A and a morphism b→ u(a) in B.(ii) Any two objects in (b/u) are connected by a finite zigzag of morphisms.

We include a short remark which is intended for the reader who is familiar withsimplicial sets. It can be safely skipped without any loss of continuity.

Remark 5.20. The characterization of final functors given in the above propositionsounds like a geometric statement. To make it precise, let us recall that there isthe nerve functor associating a simplicial set NA to every small category A. Theabove statement can be reformulated by saying that u : A→ B is final if and onlyif all the nerves N(b/u), b ∈ B, are non-empty and have a vanishing π0.

There is a similar theorem for functors u : A → B which tell us that A-shapedand B-shaped homotopy colimits are the same. In that case, the combinatorial

38 MORITZ GROTH

characterization of such functors is that all nerves N(b/u) are weakly contractiblesimplicial sets.

We illustrate this proposition by showing how it can be used to re-establishCorollary 5.18 and Lemma 5.17.

Examples 5.21. (i) Let ∗ ∈ B be a final object and let us consider the functor∗ : 1→ B classifying the final object. For every b ∈ B the slice category (b/∗)is isomorphic to 1 and is hence non-empty and connected.

(ii) Let v : B → A be a right adjoint functor. We want to show that for everya ∈ A the slice category (a/v) is non-empty and connected. The unit of theadjunction (u, v) : A B yields a morphism ηa : a → vu(a), i.e., an object(u(a), ηa) ∈ (a/v), hence showing that (a/v) is non-empty. The explicitdescription of the adjunction isomorphism in terms of the unit (see (A.2))actually shows that (u(a), ηa) is an initial object in (a/v), so that the slicecategory is, in particular, connected.

Proposition 5.22. Let u : A → B be a fully faithful functor between small cate-gories and let C be cocomplete. The functor u! = LKanu : CA → CB is fully faithfuland induces an equivalence onto the full subcategory of CB spanned by all Y suchthat ε : (LKanu u∗)Y → Y is an isomorphism.

Proof. Let us recall from Lemma A.4 that as a left adjoint the functor LKanu is fullyfaithful if and only if the unit η : 1→ u∗LKanu is a natural isomorphism. If we con-sider X : A→ C and a ∈ A then we know from the constructions prior to Proposi-tion 5.8 that the value of LKanu(X) at u(a) can be defined as colim(u/u(a))Xpu(a).Note that the fully faithfulness of u implies that (a, 1: u(a) → u(a)) is a terminalobject in (u/u(a)). The finality of terminal objects (Corollary 5.18) implies that,as a particular instance of (5.15), the component of the colimiting cocone

X(a) = X pu(a)(a, 1)→ colim(u/u(a))X pu(a) = (u∗ LKanu(X))(a)

is an isomorphism. But since this map agrees with the map (5.7) from the con-struction of left Kan etensions, we conclude that the adjunction unit η indeed isa natural isomorphism. The description of the essential image is a special caseof Lemma A.4.

Note that this result justifies the terminology Kan extensions. We conclude thissection by a few more examples.

Examples 5.23. (i) Let A be a small category and let AB be the cocone on A,i.e., the category obtained from A by freely adjoining a new terminal ob-ject ∞ ∈ AB. The cocone AB comes with a fully faithful inclusion functor

i : A→ AB. For every cocomplete category C, the functor LKani : CA → CAB

is fully faithful and one checks that the essential image consists precisely ofthe colimiting cocones.

(ii) Dually, there is the cone AC on a small category A obtained by adjoining anew initial object −∞ and coming with an associated fully faithful inclusion

i : A → AC. For every complete category C, the functor RKani : CA → CAC

is fully faithful with essential image precisely the limiting cones.


(iii) We denote by = [1]× [1] the commutative square, i.e., the category

(0, 0) //

=

(1, 0)

(0, 1) // (1, 1)

and by ip : p→ and iy : y→ the full subcategories obtained by removingthe final object (1, 1) and the initial object (0, 0), respectively. For everyfinitely complete and finitely cocomplete category C the associated Kan ex-tension functors

LKanip : Cp → C and RKaniy : Cy → C

are fully faithful and the essential images consist precisely of the pushoutsquares and pullback squares, respectively.

6. Basics on derivators

In §§6.1-6.2 we define derivators and introduce some basic terminology. To put itas a slogan, derivators can be thought of as a minimal, purely categorical extensionof a derived category to a framework which comes with a powerful calculus ofderived limits and, more generally, derived Kan extensions. Similarly, the derivatorof spaces is such an extension of the homotopy category of spaces to a frameworkwith a calculus of homotopy limits and homotopy Kan extensions.

In §6.3 we mention quite a few examples of derivators, but postpone the proofsthat these actually are examples to a later section. Finally, in §6.4 we emphasizethe difference between categorical limits on the one hand side and derived limits orhomotopy limits on the other side.

6.1. Prederivators. Let A be an abelian category and let D(A) be the derivedcategory. By Theorem 3.35 there is a functorial cone C : D(A[1]) → D(A) which,in general, does not factor through D(A)[1]. More generally, as already emphasizedin §4.4, it is important to distinguish between the following two types of categories.

(i) Derived categories of diagram categories have as objects strict diagrams andthese diagrams allow for many important constructions.

(ii) Diagram categories in derived categories are less well behaved in that theirobjects do not carry enough information and many constructions can not beperformed at that level anymore.

These categories are related by functors diaA : D(AA)→ D(A)A, which, in general,are far from being equivalences; see Lemma 3.38 and Warning 3.39. Since the cat-egories D(AA) can not be reconstructed from D(A), this suggests that the passageto derived categories should be refined by keeping track of the categories D(AA).

The formalization is as follows. Let Cat denote the 2-category of small categories,functors, and natural transformations. Similarly, let CAT be the 2-category of (notnecessarily small) categories, functors, and natural transformations.

Definition 6.1. A prederivator D is a strict 2-functor D : Catop → CAT .

The 2-category Catop is obtained from Cat by reversing the direction of the functorswhile the direction of the natural transformations is unchanged. Thus, given aprederivator D , for every small category A we obtain a category D(A). Associated

40 MORITZ GROTH

to a functor u : A → B in Cat , there is an induced precomposition functor orrestriction functor

u∗ : D(B)→ D(A).

To emphasize the dependence on D , we sometimes also write D(u) or u∗D . Finally,given functors u, v : A → B in Cat and a natural transformation α : u → v, weobtain an induced natural transformation α∗ : u∗ → v∗, as depicted in the diagram

A

u''

v

77 B, D(B)

u∗++

v∗33 D(A).

This datum is compatible with compositions and identities in a strict sense, i.e., wehave equalities of the respective expressions and not only coherent natural isomor-phisms between them.

Recall that 1 denotes the terminal category. Given a prederivator D , we refer toD(1) as the underlying category of D . As we will see in this course, the structureof a derivator D enhances the underlying category D(1) to a more flexible notion.

Examples 6.2. (i) Every category C gives rise to a represented prederivatory(C) defined by

y(C)(A) := CA, A ∈ Cat .

Given a functor u : A→ B in Cat , we define y(C)(u) to be the precompositionfunctor u∗ : CB → CA : X 7→ X u. Finally, given two functors u, v : A → Band a natural transformation α : u→ v the reader easily checks that

α∗X = X α : X u→ X v, X ∈ CB ,

defines a natural transformation α∗ : u∗ → v∗, concluding the definition ofthe 2-functor y(C). The underlying category of y(C) is canonically isomorphicto C itself,

y(C)(1) ∼= C.This example is interesting when it comes to generalizing notions from

ordinary category theory to derivator theory; in fact, the notions should bedefined for derivators such that in the represented case they reduce to theclassical ones.

(ii) Let A be an abelian category, let Ch(A) be the category of unboundedchain complexes, and let W be the class of quasi-isomorphisms in Ch(A).Given a small category A, we again denote by WA the levelwise quasi-isomorphisms, i.e., those natural transformations X → Y : A → Ch(A) suchthat all components are quasi-isomorphisms. The homotopy prederivatorDA : Catop → CAT is defined by the localizations

DA(A) := Ch(A)A[(WA)−1], A ∈ Cat .

Since the restriction functors u∗ : Ch(A)B → Ch(A)A clearly preserve level-wise quasi-isomorphisms, there are canonical functors u∗ : DA(B) → DA(A)compatible with the localization functors (compare to Lemma 3.37). Usingthat these localizations are 2-localizations (see Lemma 3.29), the reader easilyconcludes the definition of the 2-functor DA : Catop → CAT . The underly-ing category of this prederivator is canonically isomorphic to the derived


category D(A). More generally, it follows from Lemma 3.7 that there areisomorphisms

DA(A) ∼= D(AA), A ∈ Cat .

(iii) Let Top denote the category of topological spaces and continuous maps. Werecall that a continuous map f : X → Y is a weak homotopy equivalence iffor all x0 ∈ X and all n ≥ 0 the induced map f∗ : πn(X,x0) → πn(Y, f(x0))is a bijection. For every small category A we denote by WA the levelwiseweak homotopy equivalences in TopA. The (homotopy) prederivator oftopological spaces is defined by

HoTop(A) := TopA[(WA)−1], A ∈ Cat .

As in the previous example, the reader easily checks (using a variant ofLemma 3.29) that this extends to a 2-functor HoTop : Catop → CAT . The un-derlying category of HoTop is canonically isomorphic to the usual homotopycategory of spaces,

HoTop(1) ∼= Ho(Top).

(iv) We conclude by a broad class of examples induced by Quillen model cate-gories; see the original [Qui67], the monographs [Hov99, Hir03] or the in-troductory [DS95]. Let M be a Quillen model category with weak equiv-alences W . For every small category A, the category MA comes with theclass WA of levelwise weak equivalences. Although WA does not necessarilybelong to a Quillen model structure on MA, we can define the homotopyprederivator HoM of M by

HoM(A) := (MA)[(WA)−1],

and it can be shown that these categories are locally small. The underlyingcategory is canonically isomorphic to the usual homotopy category,

HoM(1) ∼= Ho(M) =M[W−1].

These three final examples are the main motivation for the theory of derivators.

Remark 6.3. (i) Our convention for (pre)derivators (which agrees with the oneof Heller [Hel88] and Franke [Fra96]) is based on the idea that we want tomodel diagrams (covariant functors) in a fixed abstract homotopy theory.This results in the use of Catop as domain of definition for prederivators.There is an alternative but isomorphic approach (as used by Cisinski [Cis03,Cis08], Grothendieck [Gro], Maltsiniotis [Mal01, Mal07]) based on presheaves(contravariant functors) in which case Catop is replaced by Catcoop, obtainedby also reversing the orientations of the natural transformations. Althoughthe resulting theories are equivalent, many statements differ slightly since thedirections of various natural transformations in each convention are reversedwith respect to the other.

(ii) Often (for example under certain finiteness conditions) it is convenient to havesome flexibility with respect to the domain of definition of (pre)derivators byrestricting the class of ‘admissible shapes’. There is the notion of a categoryof diagrams Dia, a full sub-2-category of Cat enjoying certain closure proper-ties, and the related definition of a (pre)derivator of type Dia as a 2-functorDiaop → CAT . In this introductory course we will not get into this.

42 MORITZ GROTH

We establish some basic terminology related to prederivators D . Motivated byExamples 6.2 we refer to objects of D(A) as coherent, A-shaped diagrams in D .We will often write X ∈ D if there is a small category A such that X ∈ D(A).

Warning 6.4. We want to warn the reader that this is just terminology; for anabstract prederivator D , an object X ∈ D(A) is not a diagram in whatever sense;see also Warning 6.7. However, every such object induces an ordinary diagram ofshape A as we discuss next.

We keep using implicitly the canonical isomorphisms A ∼= A1 and hence writea : 1 → A for the functor classifying a ∈ A. As a special case of a precompositionfunctor we obtain an evaluation functor a∗ : D(A) → D(1). Given a morphismg : X → Y in D(A) we will write ga : Xa → Ya for the induced map in the underlyingcategory D(1).

Similarly, every morphism f : a→ b in A yields a natural transformation of thecorresponding functors 1→ A, and for every prederivator we hence obtain

1

a''

b

77 A, D(A)

a∗**

b∗44 D(1) .

Evaluated at X ∈ D(A) this defines a map f∗ : Xa → Xb. We summarize thefunctoriality of this construction in the following lemma.

Lemma 6.5. Let D be a prederivator and let g : X → Y be a morphism in D(A).

(i) The assignment diaA(X) : A → D(1) : a 7→ Xa, f 7→ f∗, defines a functordiaA(X) : A→ D(1), the underlying (incoherent) diagram of X ∈ D(A).

(ii) There is a natural transformation diaA(g) : diaA(X) → diaA(Y ) : A → D(1)with components diaA(g)a = ga : Xa → Ya, a ∈ A.

(iii) The assignments X 7→ diaA(X) and g 7→ diaA(g) define a functor, the un-derlying diagram functor

diaA : D(A)→ D(1)A.

Proof. The proof is left as an exercise. We suggest the reader to really do thisexercise in order to get used to the concept of 2-functoriality.

This construction is an abstract version of Lemma 3.38.

Examples 6.6. We take up again Examples 6.2 and describe the associated under-lying diagram functors.

(i) For represented prederivators y(C) the functors diaA : CA → (C1)A reduce tothe canonical isomorphisms CA ∼= (C1)A.

(ii) In the case of the homotopy prederivator DA of an abelian category, thefunctor diaA : DA(A)→ DA(1)A corresponds under DA(A) ∼= D(AA) to thefunctor diaA : D(AA) → D(A)A of Lemma 3.38. In general, this functor isnot an equivalence of categories.

(iii) For the (homotopy) prederivator of spaces we obtain the forgetful functor

diaA : Ho(TopA)→ Ho(Top)A.(iv) Similarly, if (M,W ) comes from a Quillen model category, then the functor

diaA : HoM(A)→HoM(1)A gets identified withMA[(WA)−1]→ Ho(M)A.If we use the description of Ho(M) as the category of cofibrant and fibrantobjects and homotopy classes of morphisms, then this example makes precisethat diaA sends a strict diagram to a homotopy commutative diagram.


Warning 6.7. Given a coherent diagram X ∈ D(A) we will often draw it as anordinary diagram which is to say that we draw diaA(X). However, it is important tonote that, in general, X is not determined by diaA(X), even not up to isomorphism.More specifically, the functor diaA : D(A) → D(1)A is usually not an equivalence,and one has hence to be careful and always distinguish between coherent diagramsand incoherent ones. The main point of the theory of derivators is that manyimportant constructions which are available at the level of coherent diagrams cannot be performed anymore at the level of underlying incoherent diagrams. To putit more frankly, this suggests that the functors diaA should not be applied.

Prederivators allow us to describe systems of categories of diagrams and associ-ated restriction functors and similarly for derived categories or homotopy categoriesof such diagrams. However, to be able to perform more interesting constructionswe have to impose additional axioms, leading to the notion of a derivator.

6.2. Derivators. The basic idea behind derivators is to encode collections of de-rived categories or homotopy categories together with a well-behaved calculus ofhomotopy Kan extensions. For (pre)derivators we follow the established terminol-ogy for ∞-categories and simply speak of Kan extensions as opposed to homotopyKan extensions (and similarly for related notions). This does not result in a riskof ambiguity since in the context of an abstract (pre)derivator the concept ‘Kanextension’ is meaningless.

Definition 6.8. Let D be a prederivator and let u : A→ B be a functor.

(i) The prederivator D admits left Kan extensions along u if the restrictionfunctor u∗ : D(B)→ D(A) has a left adjoint u! : D(A)→ D(B),

(u!, u∗) : D(A) D(B).

(ii) The prederivator D admits right Kan extensions along u if the restrictionfunctor u∗ : D(B)→ D(A) has a right adjoint u∗ : D(A)→ D(B),

(u∗, u∗) : D(B) D(A).

Motivated by Example 5.12, in the case of left Kan extensions along functorsπA : A → 1 to the terminal category we also speak of colimits of shape A andwrite (πA)! = colimA. Similary, right Kan extensions along πA will be referredto as limits of shape A and will occasionally be denoted by (πA)∗ = limA .Sometimes we want to emphasize notationally with respect to which prederivatorcertain functors are considered. In that case, we will also write

u∗D , uD! , colimD

A , uD∗ , and limD

A .

In classical category theory, a key formal property of Kan extensions is that insufficiently (co)complete categories they can be calculated pointwise; see §5.2. LetC be a complete and cocomplete category and let X : A → C be a diagram. Forevery u : A → B in Cat the left Kan extension u!(X) : B → C and the right Kanextension u∗(X) : B → C exist (Proposition 5.5). Moreover, by Proposition 5.8they can be calculated pointwise in that for every b ∈ B certain canonical naturaltransformations

colim(u/b)X p→ u!(X)b and u∗(X)b → lim(b/u)X qare isomorphisms. Here, (u/b) and (b/u) are slice categories and p : (u/b)→ A andq : (b/u)→ A are the corresponding canonical projections functors.

44 MORITZ GROTH

In the represented case one can use these pointwise (co)limit expressions toconstruct a diagram of shape B which will be a Kan extension. This is impossiblefor an abstract prederivator D since objects in D(B) are just abstract objects,in particular, not ordinary diagrams. For a derivator we nevertheless ask thatKan extensions exist and can be calculated pointwise. To make this precise, let usconsider the slice squares

(6.9)

(u/b)p

//

π(u/b)

A

u

1b

// B

and

(b/u)q

//

π(b/u)

A

u

1b

// B.

AI

Let us focus on the square on the left, and consider an object (a, f : u(a) → b) in(u/b), and trace it through the diagram. Passing through the upper right cornerwe obtain u(a) while the other path yields b. Thus, in general the square does notcommute but we claim that a canonical natural transformation lives in that square,and dually.

Lemma 6.10. Let u : A→ B be a functor and let b ∈ B.

(i) The assignment (a, f : u(a)→ b) 7→ f defines a transformation u p→ b π.(ii) The assignment (a, f : b→ u(a)) 7→ f defines a transformation b π → u q.


Now, let us assume that the prederivator D admits the necessary Kan exten-sions. Associated to the slice squares (6.9) we obtain the following canonical mate-transformations

colim(u/b) p∗ = π!p

∗ η→ π!p∗u∗u! → π!π

∗b∗u!ε→ b∗u! and(6.11)

b∗u∗η→ π∗π

∗b∗u∗ → π∗q∗u∗u∗

ε→ π∗q∗ = lim(b/u)q

∗,(6.12)

where the transformations denoted η and ε are suitable adjunction units and counits,respectively. The transformation (6.11) is hence defined as the following pasting

D(1)

ε

D(u/b)π!oo

D(A)p∗oo

η

D(1)

π∗

OO

id

YY

D(B)b∗

oo

u∗

OO

D(A),u!

oo

idjj

and similarly for (6.12). (We will say a bit more about the passage to canonicalmates in §7.1.) It turns out that the good way to express that Kan extensions in Dare pointwise is by asking these canonical mates to be isomorphisms.

Definition 6.13. A prederivator D is a derivator if it has the following properties.

(Der1) D : Catop → CAT takes coproducts to products, i.e., the canonical func-tor D(

∐i∈I Ai) →

∏i∈I D(Ai) is an equivalence. In particular, D(∅) is

equivalent to the terminal category.(Der2) For any A ∈ Cat , a morphism f : X → Y in D(A) is an isomorphism if

and only if each fa : Xa → Ya is an isomorphism in D(1).(Der3) Each restriction functor u∗ : D(B)→ D(A) has both a left adjoint u! and

a right adjoint u∗.


(Der4) For any functor u : A → B and any object b ∈ B, the canonical mate-transformations (6.11) and (6.12) associated to (6.9),

colim(u/b) p∗ ∼−→ b∗u! and b∗u∗

∼−→ lim(b/u)q∗,

are isomorphisms.

A few comments about these axioms are in order.

Remark 6.14. (i) Axiom (Der1) simply says that a coherent diagram on a dis-joint union is canonically determined by its restrictions to the summands.Axiom (Der2) encodes the idea that a transformation of two coherent dia-grams of the same shape is an isomorphism if and only if all componentsare. This is an abstraction of the fact that a natural transformation is invert-ible if and only if each component is. The remaining two axioms encode a‘(homotopical) completeness and cocompleteness property’ together with thepointwise formulas for Kan extensions.

(ii) The formalism behind axiom (Der4) (based on the calculus of canonicalmates) is arguably the key tool in the theory of derivators and will be studiedin more detail in §7. Although a bit heavy on a first view, this formalismturns out to be a very convenient tool allowing for rather mechanical proofs.Often it allows us to show that certain ‘obvious maps which clearly are iso-morphisms’ first of all exist and second turn out to be isomorphisms.

(iii) Note that axioms (Der1)-(Der4) are asking for properties while the only ac-tual structure is the underlying prederivator. This is in contrast to moreclassical approaches including triangulated categories in which case some non-canonical structure is part of the notion. We will get back to this later.

(iv) The axioms of a derivator seem to simply capture ‘rather obvious’ formalproperties of the calculus of Kan extensions. In a way, it is surprising thatthese axioms encode anything essential about homotopy theory. Nevertheless,it turns out that this is the case.

(v) Although already mentioned in Remark 6.3 we want to reemphasize thatthere is some flexibility with respect to the allowable shapes in the definitionof a derivator. By the very definition a derivator encodes the idea of having acomplete and cocomplete homotopy theory (enjoying some nice properties).Under certain finiteness assumptions (like in K-theoretic contexts) one wantsto restrict to finite shapes only and this can be done by means of categoriesof diagrams. Related to this, there are also variants of derivators satisfyingonly half of (Der3) and half of (Der4), leading to left derivators and rightderivators. We will not get into these variants in this introductory course.

Related to Warning 6.7 we remark the following.

Warning 6.15. Note that we defined a derivator to be a prederivator which ishomotopically cocomplete and homotopically complete in that all restriction functorsu∗ : D(B) → D(A) have left adjoints and right adjoints. It is important to beaware of the fact that, in general, this does not imply that the categories D(A)are complete or cocomplete in the categorical sense. Thus, in the context of anabstract derivator, in general, ordinary categorical Kan extensions and categorical(co)limits do not exist.

We will expand on this warning in §6.4.

46 MORITZ GROTH

6.3. Examples of derivators. In this subsection we collect a few examples ofderivators (see again Examples 6.2). Detailed proofs that we actually obtain deriva-tors are a bit more involved and we suggest the reader to treat them as black boxeson a first reading. For convenience, in the appendix we nevertheless include detailedproofs at least in the case of represented derivators and in the case of homotopyderivators of combinatorial model categories.

We begin with the examples originating from ordinary category theory.

Proposition 6.16. Let C be a category. The represented prederivator y(C) is aderivator if and only if C is complete and cocomplete.

As already mentioned, this example is convenient when it comes to generalizingnotions from ordinary category theory to derivator theory.

We next turn to examples from homological algebra and homotopical algebra.It turns out that various different approaches to homotopical algebra forget toderivators, i.e., that the typical passages to homotopy categories factor throughderivators. Here we will focus on Quillen model categories. To begin with, let usrecall that a Quillen model category is combinatorial if the model structure is cofi-brantly generated and if the underlying category is locally presentable. We do notinclude a discussion of locally presentable categories here and instead only mentionthat chain complexes over a ring (more generally, chain complexes in Grothendieckabelian categories) and simplicial sets yield examples of locally presentable cate-gories. For an introduction to locally presentable categories we refer to [Gro10, §3.1]while detailed accounts can be found in [GU71, MP89, AR94] and [Bor94a, Bor94b].

Theorem 6.17. Let (M,W ) be a combinatorial model category. The homotopyprederivator

HoM : Catop → CAT : A 7→ MA[(WA)−1]

is a derivator, the homotopy derivator of M.

A proof of this result [Gro13] is included in §B.2. Before we collect more specificexamples, let us recall the following definition.

Definition 6.18. Let A be an abelian category.

(i) A generator is an object G ∈ A such that f : X → Y in A is an isomorphismif and only if f∗ : homA(G,X)→ homA(G, Y ) is a bijection.

(ii) The category A is Grothendieck abelian if it is cocomplete, if filteredcolimits in A are exact, and if it has a generator.

It can be shown that Grothendieck abelian categories are also complete, see forexample [KS06, §8.3]. To illustrate this notion we include the following examples.

Examples 6.19. (i) The category of R-modules, R a ring, is a Grothendieckabelian category.

(ii) The category of quasi-coherent OX -modules on an arbitrary scheme X is aGrothendieck abelian category.

(iii) For every Grothendieck abelian category A the category Ch(A) of chain com-plexes in A is again a Grothendieck abelian category.

A proof of the following result can be found in [Hov01].

Theorem 6.20. Let A be a Grothendieck abelian category. The category Ch(A)admits a combinatorial model structure with the class of quasi-isomorphisms asclass of weak equivalences.


The results stated so far already allow us to obtain interesting examples of deriva-tors arising in homological algebra, algebraic geometry, and topology (see Exam-ples 6.23). To indicate that there are many additional examples of derivators, westate the following two theorems. The first result is due to Cisinski [Cis03] and itis a rather deep result.

Theorem 6.21. Let (M,W ) be a model category. The homotopy prederivator

HoM : Catop → CAT : A 7→ MA[(WA)−1]

is a derivator, the homotopy derivator of M.

Similarly, ∞-categories give rise to derivators. A sketch proof of the followingresult can be found in [GPS14]. Recall that the nerve NA of a small category Ais the simplicial set which in simplicial degree n is given by the set of strings of ncomposable arrows a0 → a1 → . . .→ an.

Theorem 6.22. Let C be a complete and cocomplete∞-category. There is a deriva-tor

HoC : Catop → CAT : A 7→ Ho(CNA),

the homotopy derivator of the ∞-category C.

Thus, derivators encode aspects of the calculus of derived Kan extensions in niceabelian categories and, more generally, of the calculus of homotopy Kan extensionsin model categories and∞-categories. We now include a few more specific examples.

Examples 6.23. (i) For every ring R the category of R-modules is Grothendieckabelian and

DR : Catop → CAT : A 7→ Ch(R)A[(WA)−1]

hence defines a derivator, the derivator of the ring R. In particular, wehave the derivator DZ of the integers and the derivator Dk of a field k. Thesederivators enhance the more classical derived categories.

Note that DR encodes a lot of information concerning the ring R as weillustrate next.(a) Given a discrete group G there is an isomorphism DR(G) ∼= D(RG),

i.e., DR knows about derived categories of all group algebras RG. More-over, as special cases of (derived) Kan extension functors, DR encodesgroup cohomology and group homology as well as their versions for chaincomplexes (see §2.3).

(b) Given a (finite) quiver Q there is an isomorphism DR(Q) ∼= D(RQ),which is to say that the derived categories of path algebras RQ of anyquiver Q are encoded by DR.

(c) As a further variant, DR also encodes derived categories of incidencealgebras RP of (finite) posets P as well as derived categories of categoryalgebras RA (of finite categories A).

(ii) Given a scheme X, let Ch(X) be the category of chain complexes of quasi-coherent OX -modules on X. For every small category A we denote by WA

the levelwise quasi-isomorphisms in Ch(X)A. The prederivator

DX : Catop → CAT : A 7→ Ch(X)A[(WA)−1]

is a derivator, the derivator of the scheme X. The underlying categoryDX(1) is isomorphic to the more classical derived category D(X).

48 MORITZ GROTH

(iii) More generally, for every Grothendieck abelian category A there is the deriva-tor

DA : Catop → CAT : A 7→ Ch(A)A[(WA)−1],

the homotopy derivator of the Grothendieck abelian category A.(iv) The category Top of topological spaces can be endowed with the Serre model

structure such that the weak equivalences are the weak homotopy equiva-lences. Denoting by WA the levelwise weak homotopy equivalences in TopA,there is a derivator

HoTop : Catop → CAT : A 7→ TopA[(WA)−1],

the (homotopy) derivator of topological spaces.(v) Let us recall the notion of a spectrum in the sense of topology. A spec-

trum X consists of pointed topological spaces Xn, n ≥ 0, and pointed mapsΣXn → Xn+1. A morphism f : X → Y of spectra is a collection of pointedmaps fn : Xn → Yn compatible with the structure maps. The category Spof spectra can be endowed with the stable model structure, having the prop-erty that the homotopy category Ho(Sp) is the classical stable homotopycategory SHC (see [Vog70] and [Ada74, Part III]). We denote the associatedhomotopy derivator HoSp by

(6.24) Sp : Catop → CAT : A 7→ SpA[(WA)−1]

and refer to it as the derivator of spectra.

There are many additional examples and we will mention some of them in ourshort outlook on abstract representation theory.

Recall the duality principle from ordinary category theory which allows one often,say, to formally deduce results concerning colimits from similar results concerninglimits. This duality principle is based upon the passage to opposite categories. Inorder to obtain a similar principle we include the following example.

Example 6.25. Given a prederivator D , the opposite prederivator Dop is definedby setting

Dop(A) = D(Aop)op.

The following result formalizes the self-duality of the axioms of a derivator. Itleads to the important duality principle. In more detail, many statements in thetheory of derivators have dual versions and by the proposition it will be sufficientto state and prove only one of them.

Proposition 6.26. A prederivator D is a derivator if and only if the opposite Dop

is a derivator.

Proof. The proof is left as an exercise. As a hint we mention that for u : A → Bthe left Kan extension functor uDop

! with respect to Dop is closely related to theright Kan extension functor (uop)D

∗ with respect to D .

6.4. Limits versus homotopy limits. We expand a bit on Warning 6.15 andconsider a derivator D and a small category A, asking ourselves whether the cat-egory D(A) admits categorical (co)limits of shape B. In this subsection only, weuse the expressions categorical (co)limits and categorical Kan extensions in orderto refer to the usual constructions from category theory and the terms homotopy


(co)limits and homotopy Kan extensions to refer to adjoints to restriction functorsu∗ : D(D)→ D(C) in derivators D .

By definition of categorical (co)limits these would be adjoint functors to thediagonal functor ∆B : D(A) → D(A)B which sends X ∈ D(A) to the constant B-shaped diagram with value X. The point of this subsection is that, in general, theaxioms of a derivator do not imply the existence of such adjoints.

However, there are adjoints to a closely related functor. Let πB : B ×A→ A bethe projection away from B with restriction functor π∗B : D(A) → D(B × A). Bydefinition of a derivator, π∗B admits a left adjoint (πB)! : D(B ×A)→ D(A) and aright adjoint (πB)∗ : D(B ×A)→ D(A). Note that the two functors

(6.27) ∆B : D(A)→ D(A)B and π∗B : D(A)→ D(B ×A)

have the same domain but that the targets, in general, are essentially differentcategories. In fact, objects in D(B ×A) are diagrams which are coherent in the A-and the B-direction, while objects in D(A)B are coherent in the A-direction only.

The two functors in (6.27) are related by the following partial underlying diagramfunctors. Given categories A and B, evaluating on objects and morphisms in Bonly we obtain a partial underlying diagram functor

diaB,A : D(B ×A)→ D(A)B .

In formulas, associated to b ∈ B there is b × 1A : A ∼= 1 × A → B × A and givenX ∈ D(B ×A) we set

(6.28) diaB,A(X)(b) = (b× 1A)∗X ∈ D(A).

Lemma 6.29. Let D be a prederivator and let A,B ∈ Cat.

(i) There is a partial underlying diagram functor diaB,A : D(B × A) → D(A)B

defined by (6.28).(ii) The functor diaB,A satisfies ∆B = diaB,A π∗B : D(A)→ D(B ×A).


The underlying diagram functor diaB : D(B) → D(1)B of Lemma 6.5 is canon-ically isomorphic to diaB,1 : D(B × 1) → D(1)B . We emphasize that, in general,also the functors diaB,A, which make diagrams partially incoherent, are far frombeing equivalences.

Thus, given a derivator D and small categories A,B there is the diagram

(6.30)

D(B ×A)diaB,A

//

(πB)!

(πB)∗

D(A)B

D(A)

π∗B

OO

id// D(A).

∆B

OO

If the partial underlying diagram functor diaB,A : D(B×A)→ D(A)B is an equiv-alence, then also ∆B : D(A) → D(A)B has adjoints on both sides, which is to saythat D(A) has categorical (co)limits of shape B.

Lemma 6.31. Let D be a derivator and let A ∈ Cat.

(i) The category D(A) admits a terminal object ∗ and an initial object ∅.(ii) The category D(A) admits products and coproducts.

50 MORITZ GROTH

Proof. Let S be a set which we consider as a discrete category, i.e., a categorywith identity morphisms only. As a special case of axiom (Der1), the canonicalfunctor D(

∐s∈S A) →

∏s∈S D(A) induced from the inclusions is an equivalence.

One easily checks that the diagram

D(∐s∈S A)

' //

'

∏s∈S D(A)

'

D(S ×A)diaS,A

// D(A)S

commutes where the vertical maps are the canonical identifications. Thus, fordiscrete categories S the functors diaS,A are equivalences as well. It is immediatefrom the above discussion (see (6.30) in the case of B = S) that D(A) hence hasS-fold products and coproducts. Specializing to S = ∅ we deduce the existence ofterminal and initial objects.

Remark 6.32. (i) One way to refer to this result is by saying that in derivatorscategorical (co)products and homotopy (co)products both exist and that theyagree. This is particular to (co)products in that, in general, D(A) does notadmit further categorical (co)limits.

(ii) Using categories of diagrams one can also include examples of derivators suchthat only finite (co)products exist (this comment is related to Remark 6.3(ii)).

7. Homotopy exact squares and Kan extensions

In this section we discuss in more detail the formalism behind axiom (Der4),making precise that Kan extensions in derivators are pointwise. This formalismrelies on the calculus of mates, a certain calculus which applies to natural trans-formations. We begin in §7.1 by reviewing this calculus and illustrate it by someexamples.

In §7.2 we introduce the related notion of homotopy exact squares, arguably themain technical tool in the theory of derivators. While working with derivators, oneoften runs into the situation that outputs of certain constructions ‘obviously areisomorphic’. The formalism of homotopy exact squares often allows one to, first,construct canonical maps between such gadgets and, second, to deduce that thesecanonical maps actually are isomorphisms. To conclude this section, in §7.3 wecollect a few first applications of homotopy exact squares to the calculus of Kanextensions in derivators.

7.1. The calculus of mates. Before we define canonical mates, let us quicklyrecall the notion of pasting. This is a composition operation which applies to naturaltransformations living in ‘larger diagrams’. Here we refrain from giving a precisedefinition of such diagrams and instead only mention a few relevant examples. Letus consider the diagram

(7.1)

C1

C2

v∗1oo C3v∗2oo

D1

u∗1

OO

D2

u∗2

OO

w∗1

oo D3,w∗2

oo

u∗3

OO


which consists of categories, functors, and transformations α1 : v∗1u∗2 → u∗1w

∗1 and

α2 : v∗2u∗3 → u∗2w

∗2 . We define the natural transformation α1 α2 as

α1 α2 = (α1w∗2) (v∗1α2) : v∗1v

∗2u∗3 → v∗1u

∗2w∗2 → u∗1w

∗1w∗2 ,

and refer to it as the horizontal pasting of the squares in (7.1). Similarly, wedefine the vertical pasting of squares. The pasting operation also applies to morecomplicated such diagrams as long as all natural transformations ‘point in the samedirection’. As an additional simple example we include the following.

Example 7.2. Let (L,R) : C D be an adjunction with unit η : 1→ RL and counitε : LR → 1. We recall from §A.1 that the unit and the counit are subject to thetriangular identities,

LLη//

=++

LRL

εL

RηR//

=++

RLR

Rε

L, R.

Using the pasting operation we can rewrite the triangular identities concerning theleft adjoint L as

C L //

id ++

D

R

id

=

C

L

CL//

;Cη

D

;Cε

D,

and there is a similar picture for R.

As a special case of this pasting operation we now define mates. Let us considera natural transformation α : p∗u∗ → v∗q∗ living in a square of not necessarily smallcategories

(7.3)

C1 oop∗

OO

v∗

C2OOu∗

D1ooq∗D2.

Having our applications in mind, in this section functors are often denoted likeprecomposition functors u∗ and left adjoints and right adjoints to such functors aredenoted by u! and u∗, respectively. If the necessary adjoint functors exist, then anynatural transformation α as in (7.3) has two associated canonical mates, namely

α! : v!p∗ η→ v!p

∗u∗u!α→ v!v

∗q∗u!ε→ q∗u! and(7.4)

α∗ : u∗q∗η→ p∗p

∗u∗q∗α→ p∗v

∗q∗q∗ε→ p∗v

∗.(7.5)

Here, η denotes the units of the adjunctions (u!, u∗) and (p∗, p∗), and ε the counits

of the adjunctions (v!, v∗) and (q∗, q∗), respectively. Thus, α! is defined as the

following pasting on the left, while α∗ is defined as the following pasting on the

52 MORITZ GROTH

right,

C2

D1

C1v!oo

C2p∗oo

C1

p∗

OO

C2p∗oo

=kk

D1

v∗

OO

=

UU

D2q∗oo

u∗

OO

C2,u!oo

=kk

D1

v∗

OO

D2q∗oo

u∗

OO

D1.

q∗

OO

=

UU

Remark 7.6. (i) Note that the construction of the canonical mates depends onchoosing certain adjoint functors and adjunction (co)units. By Lemma A.11adjoint functors are unique up to unique isomorphism in a way that theisomorphism is compatible with adjunction units and counits. This impliesthat the canonical mates are well-defined up to some pasting with certaincanonical natural isomorphisms. In particular, the question if a canonicalmate is an isomorphism does not depend on the choices. This motivates usto refer to the transformations (7.4) and (7.5) as the canonical mates.

(ii) Let us again consider the square (7.3) and let us assume that the four functorshave adjoints on both sides. Using ‘conjugations by units and counits fromopposite sides’ the two natural transformations (7.4) and (7.5) are the onlyones which make sense for arbitrary α. Thus, the notation α 7→ α! andα 7→ α∗ is unambiguous.

Examples 7.7. (i) Associated to a functor u∗ : D → C there is the identity trans-formation u∗ → u∗, which populates the two squares

C

| id

Du∗oo C

id

C=oo

D

u∗

OO

D,

=

OO

=oo C

=

OO

D.

u∗

OO

u∗oo

If u∗ admits a left adjoint u! : C → D then the canonical mates (7.4) ofthe two squares yield the counit ε : u!u

∗ → id and the unit η : id → u∗u!,respectively. Similarly, if u∗ admits a right adjoint u∗ : C → D, then thecanonical mates (7.5) of these two squares are the unit η : id→ u∗u

∗ and thecounit ε : u∗u∗ → id, respectively.

(ii) Let us consider functors u∗, v∗ : D → C and a natural transformation u∗ → v∗

which we want to rewrite as two different squares

C

| id

C=oo C

id

Du∗oo

D

v∗

OO

D,

u∗

OO

=oo C

=

OO

D.

=

OO

v∗oo

populated by this transformation. If u∗ and v∗ admit left adjoints u! andv!, respectively, then the canonical mate (7.4) of the square on the left is


the conjugate transformation or total mate v! → u! in the sense of (A.8).Similarly, if u∗ and v∗ admit right adjoints u∗ and v∗, respectively, then thecanonical mate (7.5) of the square on the right is the total mate v∗ → u∗ asdefined in (A.7).

(iii) Let A be a small category and let F : C → D be a functor. We abuse notationand simply write F : CA → DA for the induced functor. By the very definitionwe have the two commutative squares

DA F // CA DA F // CAYa

D

∆A

OO

F//

!

C,

∆A

OO

D

∆A

OO

F// C.

∆A

OO

Assuming the existence of the necessary (co)limits, the canonical mate (7.4)of the square on the left and the canonical mate (7.5) of the square on theright, respectively, are the canonical maps

colimA F → F colimA and F limA → limA F,

measuring whether F : C → D preserves (co)limits of shape A; see (A.19) and(A.20).

(iv) Let C be a complete and cocomplete category, let u : A → B be a functorbetween small categories, and let b ∈ B. In the background of Proposition 5.8there were the slice squares

(u/b)p

//

π(u/b)

A

u

1b

// B

and

(b/u)q

//

π(b/u)

A

u

1b

// B.

AI

Here, the component of the natural transformation in the square on the leftat the object (a, f : u(a) → b) is given by f , and similarly in the square onthe right. Passing to diagram categories we obtain induced natural transfor-mations

C(u/b)

CAp∗

oo

C

π∗(u/b)

OO

CBb∗

oo

u∗

OO

and

C(b/u) CAq∗

oo

C

π∗(b/u)

OO

CB .b∗

oo

AIu∗

OO

The canonical mate (7.4) of the square on the left and the canonical mate(7.5) of the square on the right are the natural isomorphisms

colim(u/b) p∗ → b∗ LKanu and b∗ RKanu → lim(b/u) q∗

of Proposition 5.8, expressing that Kan extension in sufficiently (co)completecategories are pointwise.

Lemma 7.8. The passages to canonical mates are compatible with respect to hori-zontal and vertical pasting as expressed by the formulas

(α1 α2)! = (α2)! (α1)! and (α1 α2)∗ = (α2)∗ (α1)∗.

54 MORITZ GROTH

Proof. Let us consider the horizontal pasting α1α2 of two natural transformationsα1 : v∗1u

∗2 → u∗1w

∗1 and α2 : v∗2u

∗3 → u∗2w

∗2 ,

C1

C2

v∗1oo C3v∗2oo

D1

u∗1

OO

D2

u∗2

OO

w∗1

oo D3.w∗2

oo

u∗3

OO

Assuming that the functors u∗1, u∗2, and u∗3 have left adjoints, the pasting (α2)!(α1)!

is by definition given by

D1

C1

(u1)!oo C2

v∗1oo

D1

=

UU

u∗1

OO

D2

w∗1

oo

u∗2

OO

C2

=kk

(u2)!

oo C3

v∗2

oo

D2

=

UU

u∗2

OO

D3w∗2

oo

u∗3

OO

D3.

=kk

(u3)!

oo

Note that the two triangles in the middle cancel out by a triangular identity (seeExample 7.2) and we are left with (α1 α2)! as intended.

Let us consider a natural transformation (7.3) such that the canonical mate α!

is defined. This mate can then also be written as

(7.9)

D1oov!

OO

q∗

C1OOp∗

D2oou!C2,

and obviously the horizontal functors now have right adjoints. The canonical mate(α!)∗ is hence defined and can be chosen to be α.

Lemma 7.10. The two different formations of mates α 7→ α! and α 7→ α∗ areinverse to each other, i.e., we have α = (α!)∗ and α = (α∗)!.


Let us now consider a natural transformation (7.3) such that all four functorshave left adjoints. The canonical mate α! can again be written as in (7.9), and byour assumption it has a further canonical mate (α!)! : q!v! → u!p!.

Lemma 7.11. Let α be a natural transformation (7.3) such that all functors haveleft adjoints. The canonical mate (α!)! : q!v! → u!p! is conjugate to α : p∗u∗ → v∗q∗.


Lemma 7.12. Given a natural transformation α as in (7.3). The canonical matesα! : v!p

∗ → q∗u! and α∗ : u∗q∗ → p∗v∗ are conjugate. In particular, α! is an iso-

morphism if and only if α∗ is an isomorphism.


Proof. The canonical mate α! can again be written as in (7.9). Note that for thatnatural transformation all four functors have right adjoints. By Lemma 7.10 wehence conclude that the canonical mate α∗ : u∗q∗ → p∗v

∗ is given by ((α!)∗)∗ andthe dual version of Lemma 7.11 then implies that α! and α∗ are conjugate. Finally,the second claim is guaranteed by Corollary A.9.

7.2. Homotopy exact squares. We are mostly interested in applications of thecalculus of mates to derivators. Let D be a derivator and let us consider a naturaltransformation in Cat which lives in a square

(7.13)

Cp//

v

| α

A

u

Dq// B.

Since a derivator is a 2-functor Catop → CAT we have an induced natural transfor-mation

D(C) oop∗

OO

v∗ α∗

D(A)OO

u∗

D(D) ooq∗

D(B).

As summarized in §7.1, such a transformation has canonical mates

α! : v!p∗ η→ v!p

∗u∗u!α∗→ v!v

∗q∗u!ε→ q∗u! and(7.14)

α∗ : u∗q∗η→ p∗p

∗u∗q∗α∗→ p∗v

∗q∗q∗ε→ p∗v

∗.(7.15)

(Again we generically write η for adjunction units and ε for adjunction counits.)Thus, α! is a natural transformation of functors D(A) → D(D) and α∗ a naturaltransformation of functors D(D) → D(A). It follows from Lemma 7.12 that α! isan isomorphism if and only if α∗ is an isomorphism.

Definition 7.16. Let α : up→ qv be a natural transformation in Cat as in (7.13).The square (7.13) is homotopy exact if the canonical mate (7.14) or, equivalently,the canonical mate (7.15) is an isomorphism for every derivator D .

Example 7.17. (i) By definition of a derivator, Kan extensions can be calculatedpointwise and this is made precise by axiom (Der4). Note that the tworelevant natural transformations (6.11) and (6.12) in that axiom are instancesof canonical mates, namely the ones associated to the two types of slicesquares (6.9). Thus, (Der4) is precisely saying that slice squares are homotopyexact.

(ii) As a special case of the slice squares (6.9) we reconsider the following situationfrom Examples 5.23. Let A be a small category and let AB be the coconeon A, i.e., the category obtained from A by adjoining a new terminal object∞ ∈ AB. The maps a → ∞, a ∈ A, yield a natural transformation living in

56 MORITZ GROTH

the square

A1 //

πA

A

i

1 ∞// AB.

In fact, it is easily seen that this square is isomorphic to the slice squareassociated to i : A → AB and ∞ ∈ AB, and the square is hence homotopyexact.

(iii) The class of homotopy exact squares is closed under horizontal and verticalpasting. In fact, this is immediate from Lemma 7.8.

The point of this definition is that there is a battery of further examples ofhomotopy exact squares. These squares allow us to show that in the context ofabstract derivators (like homotopy derivators of model categories or complete andcocomplete∞-categories) certain canonical maps exist and are isomorphisms. A bitmore sloppy, these squares hence establish ‘rules which allow us to simultaneouslymanipulate categorical Kan extensions, derived Kan extensions, and homotopy Kanextensions’; see §6.3. For example, one ‘of course’ expects that given a categoryA ∈ Cat admitting a terminal object t ∈ A then for every derivator D and diagramX ∈ D(A) there should be a canonical isomorphism Xt

∼= colimAX in D(1). Thisand further first results along these lines are established in §7.3.

7.3. First applications to Kan extensions. In this section we illustrate the useof homotopy exact squares by extending a few results about the calculus of Kan ex-tensions from ordinary category theory to the framework of abstract derivators. Bythe duality principle, it suffices to make statements for left or right Kan extensions.

Lemma 7.18. For v : B → A a right adjoint between small categories the followingsquare is homotopy exact

Bv //

πB

| 1

A

πA

1id// 1,

i.e., for every derivator D and X ∈ D(A) the canonical mate

(πB)!v∗(X)→ (πA)!(X)

is a natural isomorphism.

Proof. For every adjunction (u, v, η, ε) : A B we obtain an induced adjunction

(v∗, u∗, η∗, ε∗) : D(B) D(A).

Thus, u∗ is a model for the right Kan extension functor v∗. By definition of ahomotopy exact square we hence have to show that

η∗π∗A : π∗A → u∗v∗π∗A

is an isomorphism for every derivator D . But this natural transformation is givenby η∗π∗A = (πAη)∗ = 1∗ = 1, and hence clearly is an isomorphism.


In more standard notation, we thus have canonical isomorphisms

colimB v∗X → colimAX, X ∈ D(A),

saying that colimits of coherent diagrams are not affected by restrictions along rightadjoint functors. This is a derivator version of the ‘finality of right adjoints’.

Definition 7.19. A functor u : A → B in Cat is final if the canonical mate(πA)!u

∗ → (πB)! is an isomorphism.

Thus, u : A→ B is final if and only if the square

Au //

πA

| 1

B

πB

1 =// 1

is homotopy exact.

Examples 7.20. (i) Identity functors are final as are compositions of final func-tors.

(ii) Right adjoint functors and, in particular, equivalences are final.

Remark 7.21. If u : A→ B is final then the canonical mate colimA u∗ → colimB is

an isomorphism in every derivator D . Since this applies to represented derivators,such a functor is final in the usual sense of category theory, which, by Proposi-tion 5.19 can be characterized by the fact that for every b ∈ B the slice categories(b/u) are non-empty and connected. It can be shown that finality in the sense ofDefinition 7.19 is much more restrictive. In fact, it turns out that a functor u isfinal if and only if all slice categories (b/u) have weakly contractible nerves.

Corollary 7.22. Let A ∈ Cat admit a terminal object t. For X ∈ D(A) there is acanonical isomorphism

Xt

∼=→ colimAX.

Proof. This is immediate from Lemma 7.18 since t : 1 → A is right adjoint toπA : A→ 1.

Corollary 7.23. Let A ∈ Cat admit a terminal object t. There is a natural iso-morphism π∗A

∼= t∗ : D(1) → D(A). Moreover, t∗ is fully faithful and the essentialimage consists precisely of those X ∈ D(A) such that all morphisms in diaA(X)are isomorphisms.


Example 7.24. The inclusion of the terminal object 1 : 1 → [1] induces a fullyfaithful functor i∗ : D(1)→ D([1]). An object X ∈ D([1]) lies in the essential imageof i∗ if and only if the morphism dia[1](X) : X0 → X1 in D(1) is an isomorphism.We abuse terminology and refer to such a diagram X ∈ D([1]) as an isomorphism.

As in ordinary category theory, the following result justifies that we speak ofKan extensions. Since this is the first proof of this kind in this course, we give afairly detailed proof.

58 MORITZ GROTH

Proposition 7.25. If u : A→ B is fully faithful, then the square

(7.26)

Aid //

id

| 1

A

u

Au// B

is homotopy exact. This is equivalent to saying that for every derivator D thefunctors u!, u∗ : D(A)→ D(B) are fully faithful.

Proof. The canonical mate (7.14) associated to this square is the adjunction unitη : 1 → u∗u! while the canonical mate (7.15) is the counit ε : u∗u∗ → 1. Thus, thesquare (7.26) is homotopy exact if and only if the Kan extension functors u!, u∗ arefully faithful.

Let us now show that this square is homotopy exact by showing that the unitη : 1→ u∗u! is an isomorphism. Since isomorphisms can be detected pointwise (byaxiom (Der2)) it is enough to show that all components ηa, a ∈ A, are isomorphisms.For this purpose, let us consider the following pasting

(7.27)

(A/a)p//

π(A/a)

~

A

1

id //

id

A

u

1a// A

u// B

where the square on the left is a slice square (6.9). The functoriality of mates withrespect to pasting implies that the canonical mate of this pasting is

(π(A/a))!p∗ ∼=→ a∗

ηa→ a∗u∗u!,

where the first map is an isomorphism by (Der4). Thus, ηa is an isomorphism ifand only if the canonical mate of the pasting (7.27) is an isomorphism.

Since u : A→ B is fully faithful, the reader easily verifies that the functor

(A/a)→ (A/u(a)) : (a′, f : a′ → a) 7→ (a′, u(f) : u(a′)→ u(a))

is an isomorphism of categories. Using this isomorphism we can rewrite the pasting(7.27) as the pasting

(7.28)

(A/a)∼= //

π(A/a)

(A/u(a))p//

π(A/u(a))

A

u

1 // 1u(a)

// B

in which the square on the right is a slice square (6.9). Since isomorphisms are finalfunctors (see Examples 7.20) the square on the left is homotopy exact. Thus, by(Der4) and Example 7.17(ii), the pasting (7.28)=(7.27) is homotopy exact, whichconcludes the proof.

Thus, Kan extensions along fully faithful functors u : A → B are fully faithful.In particular, an object X ∈ D(B) lies in the essential image of u! or u∗ if and onlyif the adjunction counit ε : u!u

∗(X) → X or the adjunction unit η : X → u∗u∗(X)

is an isomorphism, respectively. By (Der2) it is enough to check this for every


object b ∈ B. The point of the following lemma is that it suffices to take care ofthe objects which do not lie in the image of u.

Lemma 7.29. Let u : A→ B be fully faithful and let X ∈ D(B).

(i) The diagram X lies in the essential image of u! if and only if the adjunctioncounit εb : u!u

∗(X)b → Xb is an isomorphism for all b ∈ B − u(A).(ii) The diagram X lies in the essential image of u∗ if and only if the adjunction

unit ηb : Xb → u∗u∗(X)b is an isomorphism for all b ∈ B − u(A).

Proof. By duality it suffices to take care of the case of left Kan extensions. ByProposition 7.25, u! is fully faithful and X lies in the essential image of u! if andonly if the adjunction counit ε : u!u

∗(X)→ X is an isomorphism. By (Der2) this isthe case if and only if we all components εb, b ∈ B, are isomorphisms, establishingone direction. The converse direction follows from the triangular identity (see (A.3))

1 = u∗ε · ηu∗ : u∗ → u∗u!u∗ → u∗.

In fact, since 1 and ηu∗ are isomorphisms this is also the case for u∗ε, and it ishence enough to check the objects which do not lie in the image of u.

As a further application of homotopy exact squares we show that derivators arestable under shifting.

Example 7.30. Associated to B ∈ Cat there is the 2-functor B × − : Cat → Catwhich sends A to the product B × A. If we are also given a prederivator D , thenwe can define the shifted prederivator DB by setting

DB(A) = D(B ×A).

More precisely, the shifted prederivator is defined as

DB : Catop B×−→ Catop D→ CAT .

In particular, given u : A → A′ in Cat , the induced functor u∗ : DB(A′) → DB(A)is (1B×u)∗ : D(B×A′)→ D(B×A). The underlying category of DB is canonicallyisomorphic to D(B).

Let us illustrate the shifting operation by the following examples. In thoseexamples we already use the concept of an isomorphism of (pre)derivators – anotion which will be formally introduced in a later section.

Examples 7.31. (i) Let DA be the homotopy derivator of a Grothendieck abeliancategory A and let B,A ∈ Cat . A combination of Lemma 3.7 with thecategorical exponential law yields canonical isomorphisms

DBA (A) = DA(B ×A)

= Ch(A)B×A[(WB×AA )−1]

∼= Ch(AB)A[(WAAB )−1]

= DAB (A).

Thus, there is an isomorphism of derivators DBA∼= DAB .

(ii) We now specialize to the derivator DR of a ring R (see also Examples 6.23(i)).(a) For every discrete group G there is an isomorphism DG

R∼= DRG, i.e., the

shifting operation encodes the passage to group algebras.

60 MORITZ GROTH

(b) For every (finite) quiver Q there is an isomorphism DQR∼= DRQ, i.e., the

shifting operation encodes the passage to path algebras.(c) For every (finite) poset P there is an isomorphism DP

R∼= DRP , i.e., the

shifting operation encodes the passage to incidence algebras.

Proposition 7.32. For every derivator D and every B ∈ Cat the prederivatorDB is again a derivator, the shifted derivator or the derivator of (coherent)B-shaped diagrams in D . The Kan extension functors of DB are calculated as

uDB

! = (1B × u)D! and uDB

∗ = (1B × u)D∗ .

Proof. The verification of the axioms (Der1)-(Der3) for DB is left to the reader. Itremains to check the pointwise formulas encoded by axiom (Der4). Thus, given afunctor u : A→ A′ and a′ ∈ A′ let us consider the square

(7.33)

B × (u/a′)1×p//

1×π

B ×A

1×u

B × 11×a′

// B ×A′

which is obtained from the slice square (6.9) by forming the product with B. Wehave to show that the canonical mate (1 × π)!(1 × p)∗ → (1 × a′)∗(1 × u)! is anisomorphism of functors D(B ×A)→ D(B × 1). Since isomorphisms are detectedpointwise by (Der2), it is enough to show that all components of the mate areisomorphisms. For b ∈ B we consider the following pasting diagram

(7.34)

((1× u)/(b, a′))

1

φ

∼=//

π

((1× π)/b)p′//

π

B × (u/a′)1×p//

1×π

B ×A

1×u

1 =// 1

b// B × 1

1×a′// B ×A′

in which the square in the middle is the slice square (6.9) associated to the functor1× π and b ∈ B ∼= B× 1. The square on the left is induced from the isomorphismsof categories

(7.35) ((1× π)/b) ∼= (B/b)× (u/a′) ∼= ((1× u)/(b, a′))

with (b, a′) : 1 → B × A′. The functoriality of mates with pasting (Lemma 7.8)implies that the canonical mate associated to (7.34) factors as

π!φ∗(p′)∗(1× p)∗

∼=→ π!(p′)∗(1× p)∗

∼=→ b∗(1× π)!(1× p)∗ → b∗(1× a′)∗(1× u)!.

Here, the second arrow is an isomorphism by (Der4) applied to D and the corre-sponding slice square. The first arrow is also an isomorphism because the isomor-phism (8.32) is final; see Examples 7.20. In order to conclude the proof it sufficesto show that (7.34) is homotopy exact for every b ∈ B. Since the pasting (7.34)agrees with the slice square (6.9) associated to 1 × u : B × A → B × A′ and theobject (b, a′) ∈ B ×A′, this follows by a further application of (Der4).


8. Pointed derivators

In this section we introduce pointed derivators as derivators admitting zero ob-jects. Typical examples are homotopy derivators of abelian categories or homotopyderivators of pointed model categories, like the homotopy derivator of pointed topo-logical spaces. We show in §8.1 that certain Kan extension functors along (co)sievesamount to extending diagrams by zero objects. This yields a convenient tool al-lowing us to ‘add zero objects where desired’. In §8.2 we illustrate this tool andconstruct suspension, loop, cofiber, and fiber functors in pointed derivators, therebygeneralizing these classical constructions from homological algebra and homotopytheory. In §8.3 we include a short discussion of parametrized Kan extensions orKan extensions in shifted derivators, while in §8.4 we extend a few well-known re-sults concerning pushout squares in ordinary categories to cocartesian squares inderivators. These techniques are illustrated in our discussion of iterated cofiberconstructions in §8.5.

8.1. Some basics. Let us recall from Lemma 6.31 that, given a derivator D and asmall category A, the category D(A) admits an initial object ∅ and a final object ∗.

Definition 8.1. A derivator D is pointed if the underlying category D(1) has azero object.

Thus, we ask axiomatically that the unique map ∅ → ∗ in D(1) is an iso-morphism. Following standard conventions, any zero object will be denoted by0 ∈ D(1). We again take up the examples from §6.3.

Examples 8.2. (i) A represented derivator y(C) is pointed if and only if the rep-resenting category C is pointed.

(ii) Homotopy derivators of Grothendieck abelian categories are pointed, hence,in particular, derivators of fields, rings, and schemes are pointed.

(iii) Homotopy derivators of pointed model categories are pointed. As specialcases, the homotopy derivator HoTop∗ of pointed topological spaces and thederivator Sp of spectra are pointed.

Lemma 8.3. (i) A derivator D is pointed if and only if Dop is pointed.(ii) If D is a pointed derivator, then D(A), A ∈ Cat , are pointed categories, i.e.,

the shifted derivators DA are again pointed.(iii) Let D be a pointed derivator and u : A→ B. The functors u∗ : D(B)→ D(A)

and u!, u∗ : D(A)→ D(B) preserve zero objects.


We know from Proposition 7.25 that Kan extensions along fully faithful functorsu : A → B are fully faithful and, by Lemma 7.29, that the essential images canbe characterized by the components of the adjunction (co)units lying in B − u(A).For special classes of fully faithful functors these characterizations of the essentialimages admit a particularly simple form.

Definition 8.4. Let u : A→ B be a fully faithful functor.

(i) The functor u is a cosieve if for every morphism u(a) → b in B it followsthat b lies in the image of u.

(ii) Dually, u is a sieve if for every morphism b→ u(a) in B it follows that b liesin the image of u.

62 MORITZ GROTH

The following proposition will be of constant use in later sections, for example inthe construction of suspensions, cofibers, and cofiber sequences in pointed deriva-tors; see §8.2 and §8.5. To indicate its natural generality, we formulate the resultfor arbitrary derivators.

Proposition 8.5. Let D be a derivator and let u : A→ B be a functor.

(i) If u is a cosieve, then u! : D(A)→ D(B) is fully faithful and X ∈ D(B) liesin the essential image of u! if and only if Xb

∼= ∅ for all b ∈ B − u(A).(ii) If u is a sieve, then u∗ : D(A) → D(B) is fully faithful and X ∈ D(B) lies

in the essential image of u∗ if and only if Xb∼= ∗ for all b ∈ B − u(A).

Proof. We give a proof of the first statement. Proposition 7.25 guarantees thatu! : D(A)→ D(B) is fully faithful and by Lemma 7.29 a diagram X ∈ D(B) lies inthe essential image of u! if and only if the counit εb : u!u

∗(X)b → Xb, b ∈ B−u(A),is an isomorphism. By (Der4) there are canonical isomorphisms

u!u∗(X)b ∼= colim(u/b) p

∗u∗(X), b ∈ B.We note that if b ∈ B−u(A) then the slice category (u/b) is empty, since u : A→ Bis a cosieve. By (Der1) the category D(∅) is equivalent to the terminal categoryand any object therein is hence a zero object. Since left adjoint functors preserveinitial objects, we deduce that for b ∈ B − u(A) there are isomorphisms

u!u∗(X)b ∼= colim(u/b) p

∗u∗(X) = colim∅ p∗u∗(X) ∼= ∅.

Thus, for b ∈ B − u(A) the map εb : u!u∗(X)b → Xb is an isomorphism if and only

if Xb∼= ∅, concluding the proof.

In the case of pointed derivators the two characterizations of the essential imagesagree. We say that X ∈ D(B) vanishes at an object b ∈ B if Xb

∼= 0, and wesimilarly speak of diagrams which vanish on a subcategory of B.

Corollary 8.6. Let D be a pointed derivator and let u : A→ B be a functor.

(i) If u is a cosieve, then u! : D(A) → D(B) is fully faithful and induces anequivalence onto the full subcategory of D(B) spanned by all diagrams whichvanish on B − u(A).

(ii) If u is a sieve, then u∗ : D(A)→ D(B) is fully faithful and induces an equiva-lence onto the full subcategory of D(B) spanned by all diagrams which vanishon B − u(A).

In the situation of the corollary, we refer to u! as left extension by zero andto u∗ as right extension by zero. These results already allow us to carry outsome interesting constructions in arbitrary pointed derivators.

8.2. Suspensions, loops, cofibers, and fibers. In this section we define sus-pensions, loops, cofibers, and fibers for arbitrary pointed derivators, thereby gener-alizing the classical constructions from homological algebra and homotopy theory.The following definition is motivated by Examples 5.23(iii). Let us recall that wedenote by = [1]× [1] the commutative square, i.e., the category

(0, 0) //

=

(1, 0)

(0, 1) // (1, 1)


and by ip : p→ and iy : y → the full subcategories obtained by removing thefinal object (1, 1) and the initial object (0, 0), respectively.

Definition 8.7. A square in a derivator D is an object in D(). A square in Dis cocartesian or cartesian if it lies in the essential image of (ip)! : D(p)→ D()or (iy)∗ : D(y)→ D(), respectively.

Remark 8.8. For every derivator D , the category of (coherent) spans D(p) and thecategory of cocartesian squares are equivalent categories, and dually for cospansand cartesian squares.

In §§8.3-8.4 we collect a few general results concerning (co)cartesian squares.Here, instead, we immediately want to apply this notion.

Example 8.9. A square in a represented derivator is cocartesian if and only if it is apushout square (Examples 5.23(iii)). In homotopy derivators of abelian categoriesor model categories the notion of cocartesian squares reduces to derived pushoutsquares and homotopy pushout squares.

We now reconsider the rather detailed description of the cone construction ofa morphism of chain complexes as given at the beginning of §5.1; see, in particu-lar, (5.1). Starting from a morphism of chain complexes f : X0 → X1, we first con-struct the span which is obtained by adding the inclusion in the cone i : X0 → CX0

and then pass to the pushout square. It is obvious how to mimic this second stepin the context of abstract pointed derivators. As for the first step, let us recall thatcones of chain complexes are contractible and are hence sent to zero objects in de-rived categories. As observed in Corollary 8.6, right Kan extensions along inclusionsof sieves are right extensions by zero. This suggests the following generalization ofthe cofiber construction to arbitrary pointed derivators.

Related to the span and the cospan we have the fully faithful functors

(8.10) i : [1]→ p and k : [1]→y

classifying the horizontal morphism (0, 0) → (1, 0) and the vertical morphism(1, 0)→ (1, 1), respectively. Combining these functors with the fully faithful inclu-sions ip : p→ and iy : y→ we obtain the fully faithful functors

(8.11) i′ = ip i : [1]→ p→ and k′ = iy k : [1]→y→ .

Definition 8.12. Let D be a pointed derivator.

(i) The cofiber functor cof : D([1])→ D([1]) is defined as

cof : D([1])i∗→ D(p)

(ip)!→ D()(k′)∗→ D([1]).

A further evaluation at 1 ∈ [1] yields C = 1∗ cof : D([1])→ D(1).(ii) The fiber functor fib : D([1])→ D([1]) is defined as

fib : D([1])k!→ D(y)

(iy)∗→ D()(i′)∗→ D([1]).

A final evaluation at 0 ∈ [1] yields F = 0∗ fib : D([1])→ D(1).

Note that i : [1] → p is a sieve while k : [1] →y is a cosieve. As a special case ofCorollary 8.6 it follows that i∗ is right extension by zero and k! left extension by zero.

64 MORITZ GROTH

Given f ∈ D([1]) with underlying diagram X0 → X1 there are thus a cocartesiansquare and a cartesian square in D with respective underlying diagrams

X0f//

X1

cof(f)

Fffib(f)

//

X0

f

0 // Cf, 0 // X1.

Proposition 8.13. For every pointed derivator D there is an adjunction

(cof, fib) : D([1]) D([1]).

Proof. By definition, the functors cof and fib respectively factor as indicated in

D([1])i∗ // D(p)

(ip)!//

i∗oo D()

(iy)∗//

(ip)∗oo D(y)

k∗ //

(iy)∗

oo D([1]);k∗

oo

see (8.10) and (8.11). Thus, there are four adjunctions of which the outer ones‘point in the bad direction’. In order to adress this issue, let us denote by

D(p)ex ⊆ D(p), D(y)ex ⊆ D(y), and D()ex ⊆ D()

the respective full subcategories spanned by the coherent diagrams vanishing atthe lower left corner (0, 1). As a special case of Corollary 8.6 we obtain adjointequivalences of categories

(i∗, i∗) : D([1]) ' D(p)ex and (k∗, k!) : D(y)ex ' D([1]).

Moreover, the fully faithfulness of Kan extensions along fully faithful functors(Proposition 7.25) implies that all four functors (ip)!, (ip)

∗ and (iy)∗, (iy)∗ preserve

the vanishing condition at the lower left corner (0, 1). As an upshot, the functorscof and fib respectively factor as

D([1])i∗ // D(p)ex

(ip)!//

i∗oo D()ex

(iy)∗//

(ip)∗oo D(y)ex

k∗ //

(iy)∗

oo D([1]).k∗

oo

Thus, (cof, fib) is obtained by composing four adjunctions (two of which actuallyare adjoint equivalences), concluding the proof.

Remark 8.14. Given a pointed derivator D we denote by D()cof ⊆ D() thefull subcategory spanned by the cofiber squares, i.e., the cocartesian squareswhich vanish on the lower left corner. Similarly, let D()fib ⊆ D() be the fullsubcategory given by the fiber squares, i.e., the cartesian squares which vanishon the lower left corner. The proof of Proposition 8.13 together with Remark 8.8imply that there are equivalences of categories

D([1]) ' D()cof and D([1]) ' D()fib.

Thus, using coherent formulations, a morphism is simply ‘as good as a (co)fibersquare’. We will come back to this in §8.5; see Proposition 8.36.

In a similar way we can define suspension and loop functors in pointed derivators.Naively, for a pointed derivator D and X ∈ D(1) we would like to set

ΣX = C(X → 0) and ΩX = F (0→ X).

This can be made precise using the functors

0∗ : D(1)→ D([1]) and 1! : D(1)→ D([1]).


In fact, since 0: 1 → [1] is a sieve and 1: 1 → [1] is a cosieve, the above two Kanextension functors are extensions by zero (Corollary 8.6).

For the convenience of the reader, we include a more detailed construction ofthese functors. Let us consider the fully faithful functors

(8.15) i = (0, 0) : 1→ p and k = (1, 1) : 1→y

classifying the initial and the final object, respectively. Postcomposition with thefunctors ip : p→ and iy : y→ yields the fully faithful functors

(8.16) i′ = ip i : 1→ p→ and k′ = iy k : 1→y→ .

Definition 8.17. Let D be a pointed derivator.

(i) The suspension functor Σ: D(1)→ D(1) is defined as

Σ: D(1)i∗→ D(p)

(ip)!→ D()(k′)∗→ D(1).

(ii) The loop functor Ω: D(1)→ D(1) is defined as

Ω: D(1)k!→ D(y)

(iy)∗→ D()(i′)∗→ D(1).

We note again that i : 1 → p is a sieve while k : 1 →y is a cosieve so that i∗ isright extension by zero and k! left extension by zero (Corollary 8.6). As a summaryof these constructions, for X ∈ D(1) there is a cocartesian square and a cartesiansquare in D with respective underlying diagrams

X //

0

ΩX //

0

0 // ΣX, 0 // X.

Proposition 8.18. For every pointed derivator D there is an adjunction

(Σ,Ω): D(1) D(1).

Proof. It follows from the definition as well as equations (8.15) and (8.16) that thefunctors Σ and Ω respectively factor as

D(1)i∗ // D(p)

(ip)!//

i∗oo D()

(iy)∗//

(ip)∗oo D(y)

k∗ //

(iy)∗

oo D(1).k∗

oo

Again, there are four adjunctions of which only the outer ones ‘point in the baddirection’. We denote by


the respective full subcategories spanned by the coherent diagrams vanishing atthe lower left corner (0, 1) and the upper right corner (1, 0). As a special case ofCorollary 8.6 we obtain adjoint equivalences of categories

(i∗, i∗) : D(1) ' D(p)ex and (k∗, k!) : D(y)ex ' D(1).

Since the functors (ip)!, (ip)∗ and (iy)

∗, (iy)∗ preserve the vanishing conditions atthe corners (0, 1), (1, 0), the functors Σ and Ω respectively factor as

D(1)i∗ // D(p)ex

(ip)!//

i∗oo D()ex

(iy)∗//

(ip)∗oo D(y)ex

k∗ //

(iy)∗

oo D(1).k∗

oo

66 MORITZ GROTH

As a composition of two adjoint equivalences and two adjunctions also (Σ,Ω) is anadjunction.

The remaining main goal in this section is to study iterated constructions ofcofibers which, in stable derivators, will lead to canonical triangulations. For thispurpose however we need some basic results concerning (co)cartesian squares whichwe collect in §8.4. Some of these results in turn rely on a basic understanding ofparametrized Kan extensions which we study next.

8.3. Parametrized Kan extensions. The goal of this subsection is to establish aresult making precise that restriction functors and Kan extensions in unrelated vari-ables commute (Proposition 8.19) and to deduce some interesting consequences. Inparticular, this allows us to understand (co)cartesian squares in shifted derivators.

To begin with, let us recall the following observation from ordinary categorytheory. If C is a cocomplete category and B ∈ Cat , then the diagram categoryCB is also cocomplete and colimits in CB are constructed coordinatewise. Moreprecisely, for every small category A ∈ Cat and diagram X : A → CB the colimitcolima∈AX(a) ∈ CB exists and for every b ∈ B there is an isomorphism(

colima∈AX(a))(b) ∼= colima∈A

(X(a)(b)

).

Actually, a typical proof of the cocompleteness is given by showing that the righthand side of the above isomorphisms defines a diagram B → C and that it, togetherwith a suitably defined cocone, yields a colimit colimAX ∈ CB . We sometimes referto the colimits in CB as colimits with parameters or parametrized colimits.It can be shown that every evaluation morphism b∗ : CB → C preserves colimits,i.e., that the corresponding canonical map is an isomorphism.

More generally, as a cocomplete category CB admits left Kan extensions alongfunctors u : A → A′ (Proposition 5.8) which we also refer to as left Kan exten-sions with parameters or parametrized left Kan extensions. It turns outthat evaluation functors also commute with (parametrized) Kan extensions, i.e.,that for every b ∈ B the following square

(CB)Au! //

b∗

(CB)A′

b∗

CAu!

// CA′

@H∼=

commutes up to a certain canonical isomorphism.As a final generalization, we can pass from evaluation functors to general restric-

tion functors. In fact, it turns out that, given a functor v : B → B′, then the inducedrestriction functor v∗ : CB′ → CB preserves colimits and left Kan extensions. Theseresults from ordinary category theory can be obtained by specializing the followingproposition to represented derivators. At the same time, that proposition yieldssimilar results for arbitrary derivators, hence, for example, for homotopy derivatorsof abelian categories or model categories.


Proposition 8.19. For functors v : B → B′ and u : A → A′ between small cate-gories the square

(8.20)

B ×A v×1//

1×u

1

B′ ×A

1×u

B ×A′v×1// B′ ×A′

is homotopy exact, i.e., in every derivator the canonical mate transformations

(1× u)!(v × 1)∗ → (v × 1)∗(1× u)! and (v × 1)∗(1× u)∗ → (1× u)∗(v × 1)∗

are isomorphims.

Proof. We begin by reducing to the special case of u = πA : A → 1. For thispurpose, we consider the following two pastings

B × (u/a′)1×p//

π

B ×A v×1//

1×u

1

B′ ×A1×u

=

B × (u/a′)v×1//

π

1

B′ × (u/a′)1×p//

π

B′ ×A1×u

B1×a′

// B ×A′v×1// B′ ×A′ B

v// B′

1×a′// B′ ×A′

in which the square to the very left and to the very right are obtained from slicesquares be forming the product with a fixed category. We recall from the proof ofProposition 7.32 that such such squares are homotopy exact (see (7.33)).

By (Der2) and the compatibility of mates with respect to pasting, in order toconclude that (8.20) is homotopy exact it suffices to show that the pasting on theleft is homotopy exact. Since these two pastings agree, a further application of thefunctoriality of mates with pasting implies that it suffices to show that the secondsquare from the right is homotopy exact. This completes the reduction to the caseof u = πA : A→ 1.

As an additional reduction we show that is is enough to consider evaluationfunctors instead of more general restriction functors. To this end, let us considerthe pasting on the left in

(8.21)

At×1//

π

1

(B/b)×Ap×1//

π

B ×A v×1//

π

1

B′ ×A

π

=

Avb×1

//

π

1

B′ ×A

π

11

// 1b

// Bv

// B′ 1vb

// B′.

We have to show that in that pasting the right square is homotopy exact. As a slicesquare, we know that the square in the middle is homotopy exact. Moreover, thesquare on the left is induced from the functor t : 1→ (B/b) classifying the terminalobject (b, 1: b→ b) ∈ (B/b). Since right adjoint funtors are final, the left square ishomotopy exact as well (Lemma 7.18). By the functoriality of mates with pastingand (Der2) it hence suffices to show that the pasting is homotopy exact. Since thepasting is easily seen to agree with the commutative square on the right in (8.21),this concludes the reduction to evaluation functors.

68 MORITZ GROTH

Finally, let us consider an object b′ ∈ B′. Our remaining task is to show thatthe commutative square to the left in

Ab′×1

//

π

1

B′ ×A

π

=

At×1//

π

1

(B′/b′)×A

π

p×1//

B′ ×A

π

1b′

// B′ 1 =// 1

b′// B′

is homotopy exact. In the pasting to the right, the right square is a slice squareand hence homotopy exact. The remaining square is induced by t : 1 → (B′/b′)pointing at the terminal object (b′, 1: b′ → b′). Since the pasting on the right iseasily seen to agree with the square to the very left, the finality of right adjoints(Lemma 7.18) and the compatibility of homotopy exact squares with respect topasting (Example 7.17) conclude the proof.

For Kan extensions along fully faithful functors there is the following conse-quence.

Corollary 8.22. Let D be a derivator, let u : A → A′ be fully faithful, and letB ∈ Cat. A coherent diagram X ∈ D(B×A′) lies in the essential image of (1×u)!

if and only if each Xb ∈ D(A′), b ∈ B, lies in the essential image of u!.

Proof. Let us consider the following pasting of commutative squares in which thesquare to the very left is homotopy exact as an instance of (8.20),

A

u

b×1// B ×A

1×u

1×u// B ×A′

=

=

A

u

u // A′

=

b×1// B ×A′

=

A′b×1// B ×A′ =

// B ×A′ A′ =// A′

b×1// B ×A′.

Since 1 × u is fully faithful, X ∈ D(B × A′) lies in the essential image of (1 × u)!

if and only if the adjunction counit (1 × u)!(1 × u)∗ → 1 is an isomorphism onX (Proposition 7.25). We note that this counit is the canonical mate associatedto the second square from the left. Thus, (Der2), the homotopy exactness of thesquare to the very left, and the functoriality of mates with pasting (Lemma 7.8)imply that X lies in the essential image of (1 × u)! if and only if for each b ∈ Bthe canonical mate of the pasting on the left is an isomorphism. Since the abovetwo pastings agree, this is the case if and only if for every b ∈ B the canonicalmate of the pasting on the right is an isomorphism on X. To conclude the proof itsuffices to observe that these canonical mates are the counits u!u

∗ → 1 applied toXb ∈ D(A′), b ∈ B.

Note that we can read X ∈ D(B×A′) as an object in DB(A′) while the objectsXb, b ∈ B, all live in D(A′). In this sense, the essential image of a fully faithfulKan extension functor in a shifted derivator DB can be detected by evaluating atall b ∈ B (see Proposition 7.32 for the construction of Kan extensions in shiftedderivators).

Corollary 8.23. Let D be a derivator and let B ∈ Cat. A square X ∈ DB() iscocartesian if and only if Xb ∈ D() is cocartesian for all b ∈ B.


Proof. This is an immediate application of Corollary 8.22 to the fully faithful func-tor ip : p→ .

We recall that an object X ∈ D([1]) is called an isomorphism if the underlyingdiagram X0 → X1 is an isomorphism.

Corollary 8.24. Let D be a derivator and let B ∈ Cat. The following are equiva-lent for a coherent morphism X ∈ DB([1]).

(i) The morphism X is an isomorphism.(ii) The morphisms Xb ∈ D([1]), b ∈ B, are isomorphisms.

(iii) The morphism X lies in the essential image of 0! : DB(1)→ DB([1]).

Proof. This follows from Example 7.24 and (Der2). Alternatively, we can alsoinvoke Example 7.24 and Corollary 8.22.

8.4. Cartesian and cocartesian squares. In this subsection we extend a fewwell-known results concerning pushouts and pullbacks in ordinary categories tocartesian and cocartesian squares in arbitrary derivators. Recall the definition ofthese squares given in Definition 8.7.

We begin by establishing some terminology. Let D be a derivator and let usconsider a square X ∈ D() with underlying diagram

X(0,0)//

X(1,0)

X(0,1)// X(1,1).

The square X ∈ D() is vertically constant if the morphisms X(0,0) → X(0,1) andX(1,0) → X(1,1) both are isomorphisms. There is a similar notion of horizontallyconstant squares and also the combined notion of constant squares.

We now consider the horizontal inclusion (id[1] × 0) : [1] → and the inclusion(0, 0) : 1→ of the initial object.

Corollary 8.25. Let D be a derivator.

(i) The functor (id[1]×0)! : D([1])→ D() is fully faithful and induces an equiv-alence onto the full subcategory of D() spanned by the vertically constantsquares.

(ii) The functor (0, 0)! : D(1)→ D() is fully faithful and induces an equivalenceonto the full subcategory of D() spanned by the constant squares.

Proof. The first statement is immediate from Corollary 8.24 and the second state-ment is a special case of (the dual of) Corollary 7.23.

The horizontal inclusion id[1] × 0: [1]→ factors as ip i : [1]→ p→ .

Lemma 8.26. Let D be a derivator. The functor i! : D([1])→ D(p) is fully faithfuland induces an equivalence onto the full subcategory of D(p) spanned by those Xsuch that X(0,0) → X(0,1) is an isomorphism.

Proof. By Proposition 7.25 the functor i! is fully faithful with essential image thoseX such that ε : i!i

∗(X)→ X is an isomorphism at (0, 1). To reformulate this let us

70 MORITZ GROTH

consider the pasting on the left

1∼= //

1

(i/(0, 1)) //

[1]i //

i

p

1

=

1(0,0)

//

1

z

p

1

11

// 1(0,1)

// p1// p 1

(0,1)// p,

in which the square in the middle is a slice square and the square to the left is givenby an isomorphism. Since these two squares are homotopy exact, the functoriality ofcanonical mates with pasting implies that ε(0,1) is an isomorphism on X if and onlyif the canonical mate associated to this pasting is an isomorphism at X. Since thispasting agrees with the square to the right given by the morphism (0, 0) → (0, 1).The canonical mate associated to this square is X(0,0) → X(0,1), concluding theproof.

The relation of suitably constant squares to (co)cartesian squares is as follows.One of the implications is a derivator version of the classical fact that pushouts ofisomorphisms are isomorphisms.

Proposition 8.27. Let D be a derivator and let X be a square in D such thatX(0,0) → X(0,1) is an isomorphism. The square X is cocartesian if and only ifX(1,0) → X(1,1) is an isomorphism, i.e., the square is vertically constant.

Proof. The inclusion id[1] × 0: [1]→ factors as i : [1]→ p followed by ip : p→ .Hence, there is a canonical isomorphism (id[1]×0)!

∼= (ip)!i!, and, as a consequenceof Corollary 8.25, vertically constant squares are thus cocartesian. Conversely, ifX is a cocartesian square such that X(1,0) → X(1,1) is an isomorphism, then, byLemma 8.26, X lies in the essential image of (ip)! i! ∼= (id[1] × 0)! and is byCorollary 8.25 vertically constant.

Let [2] be the poset (0 < 1 < 2) considered as a category. For 0 ≤ i ≤ j ≤ 2there is the functor ιi,j : [1] → [2] which sends 0 to i and 1 to j. The productcategory [2]× [1] will be denoted by . Given an object X ∈ D(),

X(0,0)//

X(1,0)//

X(2,0)

X(0,1)// X(1,1)

// X(2,1),

by restriction we obtain the left square ι∗01X ∈ D(), the right square ι∗12X ∈ D(),and the composite square ι∗02X ∈ D(). (Here we abused notation and simplywrote ιi,j instead of ιi,j × id[1].)

Cocartesian squares enjoy a composition and cancelation property as made pre-cise by Proposition 8.29. The following lemma is a first step towards that propo-sition. Let A ⊆ be the full subcategory obtained by removing the objects(1, 1), (2, 1) and let B ⊆ be the full subcategory obtained by removing (2, 1)only. Related to these subcategories there are fully faithful inclusion functors

k = j i : A i→ Bj→ .

Lemma 8.28. Let D be a derivator.


(i) The functor i! : D(A) → D(B) is fully faithful and induces an equivalenceonto the full subcategory spanned by all X ∈ D(B) such that ι∗01(X) ∈ D()is cocartesian.

(ii) The functor j! : D(B) → D() is fully faithful and induces an equivalenceonto the full subcategory spanned by all X ∈ D() such that ι∗12(X) ∈ D()is cocartesian.

(iii) The functor k! : D(A) → D() is fully faithful and induces an equivalenceonto the full subcategory spanned by all diagrams X ∈ D() such that bothsquares ι∗01X, ι

∗12X are cocartesian.

Proof. We begin by statement (i). Since i : A→ B is fully faithful, i! is fully faithfuland X ∈ D(B) lies in the essential image if and only if the counit ε : i!i

∗(X)→ Xis an isomorphism at (1, 1) (Proposition 7.25). Using the homotopy exactness ofslice squares and the functoriality of mates with respect to pasting, this is the caseif and only if the canonical mate associated to the pasting on the left in

pι01 //

π

Ai //

i

B

1

=

p1 //

π

pip //

ip

1

ι01 // B

1

1(1,1)

// B1// B 1

(1,1)//

1//

ι01// B

is an isomorphism on X. In the above pasting on the right, the square to the left is aslice square. Since the above two pastings coincide, using again the functoriality ofmates with pasting, the homotopy exactness of slice squares, and Proposition 7.25,we deduce that X lies in the essential image of i! if and only if ι∗01(X) is cocartesian.In fact, the corresponding canonical mate factors as

colimp i∗pι∗01(X)

∼=→ (1, 1)∗(ip)!i∗pι∗01(X)

ε1,1→ (1, 1)∗ι∗01(X)

and is hence an isomorphism if and only if the counit ε1,1 is an isomorphism onι∗01(X) which in turn is the case if and only if the square ι∗01(X) is cocartesian(again by Proposition 7.25).

As for statement (ii), we note that the essential image of j! : D(A) → D(B)consists precisely of those X ∈ D(B) such that the counit ε : j!j

∗(X) → X isan isomorphism at (2, 1) (Proposition 7.25). To re-express this differently let usconsider the pasting on the left in

pι12 //

π

B

π

1 //

Bj//

j

1

=

p1 //

π

pip //

ip

1

ι12 //

1

11// 1

(2,1)//

1// 1

(1,1)//

1//

ι12// ,

and for now let us focus on the two squares to the right. Proposition 7.25, thehomotopy exactness of slice squares, and the compatibility of mates with pastingimply that X lies in the essential image of j! if and only if the canonical mate ofthe pasting of those two squares is an isomorphism on X. We next observe that thefunctor ι01 : p→ B is a right adjoint. In fact, the functor s0 : [2] → [1] determinedby 0, 1 7→ 0 and 2 7→ 1 induces a left adjoint to ι01 : p→ B. Hence, by Lemma 7.18the corresponding square is homotopy and, again by the functoriality of mateswith pasting, we conclude that X lies in the essential image of j! if and only if

72 MORITZ GROTH

the canonical mate of the pasting to the left is an isomorphism on X. Since thatpasting can be rewritten as the above pasting on the right, we can now conclude asin the case of (ii).

One easily combines these two statements to deduce (iii).

This lemma will again show up in §8.5. The composition and cancelation prop-erty is as follows.

Proposition 8.29. Let D be a derivator and let X ∈ D() such that ι∗01X iscocartesian. Then ι∗12X is cocartesian if and only if ι∗02X is.

Proof. We consider X ∈ D() such that the square ι∗01(X) is cocartesian. ByLemma 8.28 and Proposition 7.25 for such a diagram X also the square ι∗12(X) iscocartesian if and only if X lies in the essential image of k! : D(A)→ D() if andonly if the counit ε : k!k

∗(X) → X is an isomorphism at (2, 1). In more detail, byProposition 7.25 the essential image of k! is characterized by ε : k!k

∗ → 1 beingan isomorphism at (1, 1) and (2, 1). Under our assumption the component ε1,1 isalways an isomorphism as the counit factors as

k!k∗(X) ∼= j!i!i

∗j∗(X)ε→ j!j

∗(X)ε→ X

and the component at (1, 1) of the first morphism is an isomorpism by our as-sumption on X (use Proposition 7.25 and Lemma 8.28) while the correspondingcomponent of the second morphism is always an isomorphism (Proposition 7.25).

To reformulate this we consider the pasting on the left in

pι02 //

π

A

π

1 //

Ak //

k

1

=

p 1 //

π

pip //

ip

1

ι02 //

1

11// 1

(2,1)//

1// 1

(1,1)//

1//

ι02// .

Here, the functor ι02 : p→ A is a right adjoint so that the square to the very left ishomotopy exact by Lemma 7.18. In fact, a left adjoint is induced by the functors1 : [2] → [1] determined by 0 7→ 0 and 1, 2 7→ 1. Together with the homotopyexactness of slice squares and the functoriality of mates with pasting this showsthat for our diagram X ∈ D() the square ι∗12(X) is cocartesian if and only if thecanonical mate of the pasting on the left is an isomorphism on X. In the pasting onthe right, the square to the very left is a slice square and one notes that the abovetwo pasting coincide. Using standard arguments we thus conclude that ι∗12(X) iscocartesian if and only if ι∗02(X) is cocartesian.

8.5. Iterated cofiber constructions. In this subsection we apply our above re-sults on cocartesian squares and briefly discuss cofiber sequences. We will also showthat if we iterate the cofiber construction three times then we obtain the suspensionmorphism up to natural isomorphism.

But let us begin with the cofiber sequences themselves.

Definition 8.30. Let D be a pointed derivator. A cofiber sequence in D is adiagram X ∈ D() such that the following two conditions are satisfied.

(i) The squares ι∗01(X), ι∗12(X) ∈ D() are cocartesian.(ii) The diagram vanishes at (2, 0) and (0, 1), X2,0

∼= X0,1∼= 0.


We denote by D()cof ⊆ D() the full subcategory spanned by all cofiber se-quences.

Thus, given a cofiber sequence X, we obtain an underlying incoherent diagramdia(X) : → D(1) looking like

(8.31)

xf//

y //

g

0

0 // z

h// w.

By Proposition 8.29 also the composite square ι∗02(X) is cocartesian and it has anunderlying diagram

x //

0

0 // w.

Let us recall from Definition 8.17 that such squares were used in the constructionof the suspension functor Σ: D(1)→ D(1). In fact, let D()ex ⊆ D() be the fullsubcategory spanned by all cocartesian squares which vanish at (1, 0) and (0, 1).As observed in the proof of Proposition 8.13 there is an equivalence of categoriesD(1) ' D()ex and we just saw that ι∗02(X) ∈ D()ex. It follows from thisobservation that for every cofiber sequence X with underlying diagram (8.31) thereis a canonical isomorphism

(8.32) φ : w ∼= Σx.

Let [3] be the poset (0 < 1 < 2 < 3) and let k : [3] → be the ‘diagonalembedding‘ pointing at the objects (0, 0) < (1, 0) < (1, 1) < (2, 1). Given a cofibersequence X ∈ D()cof , we can pass to the underlying diagram of k∗(X) ∈ D([3])which is an ordinary incoherent diagram [3] → D(1). Combining this diagramwith the canonical isomorphism (8.32) we obtain, as an upshot, for every cofibersequence X as in (8.31) an underlying incoherent cofiber sequence in D(1)

(8.33) xf→ y

g→ zφh→ Σx.

In the context of stable derivators, these incoherent cofiber sequences lead to canon-ical triangulations as we will see in §9.5.

Remark 8.34. In the construction of incoherent cofiber sequences we applied theunderlying diagram functor dia[3] : D([3]) → D(1)[3] and this step results in a lossof information. These incoherent cofiber sequences are some non-canonical shad-ows of certain universal constructions which were applied to a coherent morphism.In particular, given an incoherent cofiber sequence (8.33) it is true that the com-positions x → z and y → Σx are zero morphisms. But at that level we do notremember the reason why these compositions are zero, namely that they belong tocertain cofiber squares (Remark 8.14) in the background (see (8.31)).

As a related remark, in general, it is not possible to canonically reconstruct a co-herent version of the morphism f : x→ y starting with the given incoherent cofibersequence. This is to be seen in contrast to the following result (Proposition 8.36).

74 MORITZ GROTH

We now show that every coherent morphism can be extended to a cofiber se-quence. For this purpose, let A ⊆ again denote the full subcategory obtained byremoving the objects (2, 0), (0, 1) and let k : A→ be the corresponding inclusion.Moreover, let i : [1]→ A classify the horizontal morphism (0, 0)→ (1, 0). For everyderivator D there are related Kan extension functors

(8.35) D([1])i∗→ D(A)

k!→ D().

Proposition 8.36. For every pointed derivator D the functors (8.35) induce anequivalence of categories

D([1]) ' D()cof .

Proof. The functors i : [1]→ A and k : A→ are both fully faithful and hence thesame is true for the Kan extension functors i∗ : D([1])→ D(A), k! : D(A)→ D().Hence, both functors restrict to equivalences onto their essential images and wedeal with both cases individually.

We note first that the functor i is a sieve and that i∗ hence is right extensionby zero (Corollary 8.6). Thus, the essential image of i∗ consists of precisely thoseX ∈ D(A) which vanish at (2, 0) and (0, 1). As for k! : D(A) → D() we alreadyknow by Lemma 8.28 that the essential image of k! consists precisely of thoseX ∈ D() such that both squares ι∗01(X), ι∗12(X) are cocartesian. Since k! is a fullyfaithful Kan extension functor it respects the vanishing condition at the objects(2, 0), (0, 1) ∈ A. In particular, k! restricts to an equivalence between the fullsubcategory D(A)ex of D(A) spanned by all diagrams which vanish at these twoobjects on the one-hand-side and the category D()cof of cofiber sequences on theother-hand-side.

As an upshot, we see that i∗ and k! respectively restrict to equivalences

D([1]) ' D(A)ex ' D()cof ,

concluding the proof.

Thus, if we stay at the level of coherent diagrams, then a morphism in a pointedderivator is simply as good as a cofiber sequence. Compare this observation toRemark 8.14.

We now add one more iteration of the construction of the cofiber. To make thisprecise, let us consider the full subcategory B ⊆ [2] × [2] obtained by removing(0, 2),

(8.37)

(0, 0) //

(1, 0) //

(2, 0)

(0, 1) // (1, 1) //

(2, 1)

(1, 2) // (2, 2).

Similarly to the case of cofiber sequences, we would like to extend a coherent mor-phism X ∈ D([1]) to a diagram of the above shape by first adding certain zeroobjects and then adding cocartesian squares.

To this end, let us denote by A ⊆ B the full subcategory spanned by (0, 0), (1, 0)and (0, 1), (2, 0), (1, 1). Related to this there is the fully faithful functor i : [1]→ A


classifying the horizontal morphism (0, 0) → (1, 0) and the obvious fully faithfulinclusion j : A→ B. At the level of coherent diagrams in a derivator D we obtainthe corresponding Kan extension functors

(8.38) D([1])i∗→ D(A)

j!→ D(B).

Lemma 8.39. Let D be a pointed derivator and let D(B)cof ⊆ D(B) be the fullsubcategory spanned by all X ∈ D(B) satisfying the following conditions.

(i) The diagram X vanishes at (0, 1), (2, 0), and (1, 2).(ii) The restrictions of X to the left square, to the top right square, and to the

bottom right square are cocartesian.

The functors (8.38) induce an equivalence of categories

D([1]) ' D(B)cof .

Proof. Since the proof is similar to the one of Proposition 8.36 we allow ourselvesto be a bit sketchy. Both functors i : [1]→ A and j : A→ B are fully faithful, hencethe same is true for the Kan extension functors i∗ : D([1])→ D(A) and j! : D(A)→D(B). The functor i : [1] → A is a sieve and i∗ is hence right extension by zero(Corollary 8.6) and as a such it induces an equivalence onto the full subcategoryD(A)ex of D(A) spanned by all diagrams which vanish at (0, 1), (2, 0), and (1, 2).

We next analyze the functor j : A→ B and begin by observing that it factors asthe composition of fully faithful inclusions

j : Aj1→ A1

j2→ A2j3→ B

where the individual steps add the objects (1, 1), (2, 1), and (2, 2), respectively. Theleft Kan extension functor j! : D(A) → D(B) is accordingly naturally isomorphicto the composition

D(A)(j1)!→ D(A1)

(j2)!→ D(A2)(j3)!→ D(B).

Similarly to Lemma 8.28, one now observes that each of these three Kan extensionsprecisely amounts to adding a new cocartesian square in the obvious sense.

As an upshot, the functor j! hence amounts to adding three cocartesian squaresand combining this with the first part of the proof we are hence done.

The equivalence D([1]) ' D(B)cof sends a morphism (f : x → y) ∈ D([1]) to acoherent diagram looking like

(8.40)

xf//

y //

g

02

01// z

h //

x′

f ′

03// y′

and making all three squares cocartesian. In this diagram, the objects 01, 02, 03

denote zero objects in D(1) and the subscripts can be ignored for now. We nextshow that the vertical morphism (f ′ : x′ → y′) is naturally isomorphic to Σf .

76 MORITZ GROTH

To make this precise, let us recall from Lemma 8.3 that pointed derivators arestable under shifting. In particular, if D is pointed then so is D [1], and the pointedderivator D [1] comes with a suspension functor Σ: D([1])→ D([1]).

Proposition 8.41. For every pointed derivator D there is a natural isomorphism

cof3 ∼= Σ: D([1])→ D([1]).

Proof. Let B be the category (8.37) and let us consider the functors k : [1] → Band l : [1]→ B classifying the morphisms

(0, 0)→ (1, 0) and (2, 1)→ (2, 2),

respectively. The main step consists of showing that for every Y ∈ D(B)ex thereis a natural isomorphism Σ(k∗Y ) ∼= l∗(Y ). To this end, let Y ∈ D(B)ex withunderlying diagram (8.40). There is a unique functor q : [1] × → B such thatq∗(Y ) ∈ D([1]×) has underlying diagram looking like

(8.42)

x //

f

02

y //

02

01

// x′

f ′

03// y′.

In this diagram, the [1]-coordinate is drawn diagonally while the -coordinate isas before and the decoration of the objects implies that there is at most one suchfunctor (the categories under consideration are posets). We leave it to the readerthat such a functor q : [1]×→ B actually exists, and we note that q satisfies

q(−, (0, 0)) = k : [1]→ B and q(−, (1, 1)) = l : [1]→ B.

The diagram Z = q∗(Y ) ∈ D([1] × ) has the property that both squaresZ0, Z1 ∈ D() are cocartesian. In fact, Z0 is cocartesian as a horizontal pastingof two cocartesian squares while Z1 is a vertical pasting of two cocartesian squares(Proposition 8.29). Thus, by Corollary 8.23, the square Z ∈ D [1]() is cocartesianwhen considered as an object of D [1]. Morever, this cocartesian square vanishesat (1, 0), (0, 1), implying that it qualifies for a construction of Σ: D([1]) → D([1]).More precisely, we obtain a canonical isomorphism

φ : Z1,1∼= Σ(Z0,0).

Unraveling definitions we see that Z1,1 = (1, 1)∗q∗Y = l∗Y and similarly we seethat Z0,0 = (0, 0)∗q∗Y = k∗Y , yielding the desired canonical isomorphism

Σ(k∗Y ) ∼= l∗(Y ).

We leave it to the reader to apply Lemma 8.39 to conclude the proof of thisproposition.

We will see in §9.5 that stable derivators canonically take values in triangulatedcategories. Ignoring signs for the moment, this proposition is a coherent version ofthe following observation. Recall the rotation axiom (T2) for triangulated categories(Definition 4.9) by which a distinguished triangle can be rotated backwards. At thelevel of coherent morphisms this amounts to passing to the cofiber of the coherent


morphism. If we take a distinguished triangle and rotate it three times then weobtain a distinguished triangle which is the (negative of the) suspension of theoriginal triangle. The corresponding statement at the level of coherent morphismsis Proposition 8.41.

9. Stable derivators

Stable derivators are obtained from pointed derivators by imposing the linearitycondition that a square is cocartesian if and only if it is cartesian. This linearityaxiom allows us to establish derivator versions of well-known results from homo-logical algebra and stable homotopy theory, i.e., the study of spectra in the senseof topology. (In the next section we will see less well-known such results.) Stablederivators canonically take values in triangulated categories (based on the incoher-ent cofiber sequences introduced in §8), and stable derivators hence belong to thefamily of different enhancements of triangulated categories.

We begin in §9.1 with the definition of stable derivators and some first conse-quences of these axioms. In §9.2 we show that the values of stable derivators arepreadditive categories and in §9.3 we sketch ideas involved in a proof that thesecategories actually are additive. In §9.4 we include a short digression on morphismsof derivators and natural transformations between such morphisms. We also brieflytalk about adjunctions and equivalences of derivators and related notions. Finally,in §9.5 we show that (strong) stable derivators take values in triangulated categorieswhich are compatible with restriction functors and Kan extension functors.

9.1. Basics on stable derivators. Similarly to the theory of model categoriesand ∞-categories, we obtain stable derivators from pointed ones by imposing thefollowing linearity condition.

Definition 9.1. A pointed derivator D is stable if a square in D is cartesian ifand only if it is cocartesian. A bicartesian square is square which is both cartesianand cocartesian.

Examples 9.2. (i) The homotopy derivator of a Grothendieck abelian categoryis stable. In particular, we obtain stable derivators associated to fields, rings,and schemes.

(ii) The derivator Sp of spectra is stable. In a certain precise sense this is theuniversal example of a stable derivator, namely the free stable derivator gen-erated by the sphere spectrum.

(iii) The homotopy derivator of a stable model category or stable ∞-category isstable, giving rise to a plethora of stable derivators. We refer the reader to[GS14c, Examples 5.5] for a fairly long list of additional examples.

Lemma 9.3. A derivator D is stable if and only if the opposite Dop is stable.

Proof. This is immediate from the definition.

Thus, the duality principle also extends to stable derivators.

Lemma 9.4. Let D be a stable derivator and let B ∈ Cat. The shifted derivatorDB is stable.

Proof. By Lemma 8.3 the shifted derivator DB is pointed, thus it remains to showthat a square in DB is cocartesian if and only if it is cartesian. This follows from

78 MORITZ GROTH

our discussion of parametrized Kan extensions in §8.3. In fact, by Corollary 8.23and its dual we can use the stability of D to see that X ∈ DB() = D(B × ) iscocartesian if and only if all Xb ∈ D(), b ∈ B, are cocartesian if and only if allXb ∈ D(), b ∈ B, are cartesian if and only if X ∈ DB() = D(B×) is cartesian.Thus, DB is also stable.

We begin by a few sanity checks, indicating that Definition 9.1 captures some ofthe expected phenomenons.

Proposition 9.5. Let D be a stable derivator. A morphism f ∈ D([1]) is anisomorphism if and only if Cf ∼= 0 if and only if Ff ∼= 0.

Proof. It follows from the construction of C : D([1]) → D(1) that for every mor-phism f ∈ D([1]) there is a cocartesian square

xf//

y

0 // Cf.

If f is an isomorphism then the same is true for 0 → Cf by Proposition 8.27, es-tablishing one direction. Conversely, let us assume that Cf ∼= 0. Thus, the bottomhorizontal morphism in the above cocartesian square is an isomorphism. Since D isstable, the square is also cartesian and it follows from the dual of Proposition 8.27that f is an isomorphism. By duality this concludes the proof.

Proposition 9.6. Let D be a stable derivator and let X ∈ D(). If two of thesquares ι∗01(X), ι∗12(X), and ι∗02(X) are bicartesian then so is the third one.

Proof. This is immediate from Proposition 8.29 and its dual.

We now establish a derivator version of the 5-lemma. In that result we considertwo cofiber squares X,X ′ ∈ D()cof looking like

x //

y

x′ //

y′

0 // z, 0 // z′

and a morphism X → X ′ with components f : x→ x′, g : y → y′, and h : z → z′.

Proposition 9.7. Let D be a stable derivator and let F : X → X ′ be a morphismof cofiber squares with components f : x → x′, g : y → y′, and h : z → z′. If two ofthe morphisms f, g, and h are isomorphisms then so is the third one.

Proof. Let us assume that f and g are isomorphisms. It follows from axiom (Der2)that the restriction i∗p(F ) is an isomorphism. The fully faithfulness of (ip)! impliesthat also F ∼= (ip)!i

∗p(F ) is an isomorphism hence so is F(1,1) = h. The dual proof

shows that if g, h are isomorphisms then so is f (use that the squares are also fibersquares).

We attack the remaining case and assume that f and h are isomorphisms. Aright Kan extension by zero followed by a left Kan extension allows us to extend


X and X ′ to cofiber sequences X and X ′,

x //

y

// 0

x′ //

y′

// 0

0 // z // w, 0 // z′ // w′,

and to obtain an induced morphism of cofiber sequences F : X → X ′. The restric-tion ι∗0,2(F ) to the cocartesian composite squares is an isomorphism. In fact, it isan isomorphism if we restrict it further along ip : p→ since f is an isomorphismso that axiom (Der2) allows us to conclude. By the first part of this proof it follows

that ι∗0,2(F ) is an isomorphism. Thus, the restriction ι∗1,2(F ) to the squares on theright has the property that the two components at (0, 1), (1, 1) are isomorphisms.Since D is stable, this restriction is a morphism of fiber squares and we henceconclude by the first part of the proof.

Proposition 9.8. For every stable derivator D there is an equivalence

(cof, fib) : D([1]) ' D([1]).

Proof. We note again that the functors cof and fib respectively factor as indicatedin

D([1])i∗ // D(p)

(ip)!//

i∗oo D()

(iy)∗//

(ip)∗oo D(y)

k∗ //

(iy)∗

oo D([1]).k∗

oo

These four adjunctions restrict to equivalences as follows. Let us denote by


the respective full subcategories spanned by the coherent diagrams satisfying thefollowing exactness conditions. In the first two cases we impose the vanishingcondition at (0, 1) while in the third case we consider only the squares which arebicartesian and satisfy this vanishing condition. It follows from Corollary 8.6 andProposition 7.25 that the functors cof and fib respectively factor as

D([1])i∗ // D(p)ex

(ip)!//

i∗oo D()ex

(iy)∗//

(ip)∗oo D(y)ex

k∗ //

(iy)∗

oo D([1]).k∗

oo

and that each individual step is an equivalence.

As already mentioned, we will see in §9.5 that stable derivators D canonicallytake values in triangulated categories. In particular, the underlying category D(1)can be endowed with a triangulated structure. The proposition above establishesthat we can rotate triangles back and forth without a loss of information. And thefollowing proposition gives rise to the suspension equivalences Σ: D(1) → D(1)belonging to the canonical triangulations.

Proposition 9.9. For every stable derivator D there is an equivalence

(Σ,Ω): D(1) ' D(1).

Proof. The proof is very similar to the previous case, the main modification beingthat we impose an additional vanishing condition at the upper right corner (1, 0).The details are left as an exercise to the reader.

80 MORITZ GROTH

We conclude by mentioning the following theorem, offering different characteri-zations of stable derivators. A proof of this result can be found in [GPS14].

Theorem 9.10. The following are equivalent for a pointed derivator D .

(i) The adjunction (Σ,Ω): D(1) D(1) is an equivalence.(ii) The adjunction (cof, fib) : D([1]) D([1]) is an equivalence.

(iii) A square in D is cocartesian if and only if it is cartesian, i.e., D is stable.

9.2. The preadditivity of stable derivators. In this subsection we will see thatthe values D(A), A ∈ Cat , of a stable derivator D are preadditive categories. Asa preparation we generalize the following result from ordinary category theory inwhich we consider a pushout square

∅ //

Y

X // W

in a cocomplete category such that the upper left corner is populated by an initialobject. In this situation the object W is isomorphic to the coproduct X t Y , moreprecisely, the cospan X →W ← Y is a coproduct cocone for the pair (X,Y ).

To extend this to derivators let us consider the functor ((1, 0), (0, 1)) : 1t1→ which factors as compositions of fully faithful functors

(9.11) 1 t 1i→ p ip→ and 1 t 1

j→y iy→ .

Definition 9.12. Let D be a derivator. A coherent cospan X ∈ D(y) is a coprod-uct cocone if it lies in the essential image of j! : D(1 t 1)→ D(y).

Lemma 9.13. For every derivator D the category D(1 t 1) is equivalent to thefull subcategory D()copr ⊆ D() spanned by the cocartesian squares X such thatX(0,0)

∼= ∅. Moreover, a square X lies in D()copr if and only if X(0,0)∼= ∅ and if

the restriction (iy)∗(X) ∈ D(y) is a coproduct cocone.

Proof. The functor k = ((1, 0), (0, 1)) : 1 t 1 → classifying the upper right andthe lower left corner is fully faithful. By Proposition 7.25 the same is true fork! : D(1t 1)→ D() which hence induces an equivalence onto the essential image.Since k factors as indicated in (9.11), there are natural isomorphisms

k!∼= (ip)!i! ∼= (iy)!j!.

All of these functors are fully faithful and these induced factorizations allow us toobtain a different description of the essential image. Using the natural isomorphismk!∼= (ip)!i! we see that X ∈ D() lies in the essential image of k! if and only if

X is cocartesian and X(0,0)∼= ∅. In fact, since i is a cosieve this follows from

Proposition 8.5 and Proposition 7.25. Similarly, using the isomorphism k!∼= (iy)!j!

and the fact that iy is a cosieve it follows from the same two propositions that theessential image of k! consists precisely of those X with X(0,0)

∼= ∅ and such that(iy)∗(X) is a coproduct cocone.

A combination of this lemma with its dual allows us to establish the preadditivityof stable derivators.

Proposition 9.14. Let D be a stable derivator and A ∈ Cat. The category D(A)is preadditive.


Proof. Let D be a stable derivator and A ∈ Cat . The derivator DA is again stable(Lemma 9.4) and there is an isomorphism of categories DA(1) ∼= D(A). Thus, it isenough to show that the underlying category D(1) is preaddive. By Lemma 6.31,the category D(1) admits small (co)products. Hence, in order to conclude it isenough to show that for X,Y ∈ D(1) the canonical map X t Y → X × Y is anisomorphism. To this end, we begin by observing that (Der1) yields a canonicalequivalence of categories D(1 t 1) ' D(1) × D(1). For X,Y ∈ D(1) we denoteby (X,Y ) ∈ D(1 t 1) the object corresponding to the pair (X,Y ) ∈ D(1) × D(1)under this equivalence.

Associated to (X,Y ) ∈ D(1 t 1) we construct a coherent diagram of shape[2]× [2]. Here, [2] is again the poset (0 < 1 < 2) and the product category [2]× [2]hence looks like

(0, 0) //

(1, 0) //

(2, 0)

(0, 1) //

(1, 1) //

(2, 1)

(0, 2) // (1, 2) // (2, 2).

Related to [2]× [2] there are categories and fully faithful functors

1 t 1i→ A

j→ Bk→ [2]× [2]

which are defined as follows.

(i) The category A ⊆ [2] × [2] is the full subcategory spanned by the objects(2, 0), (2, 1) and (0, 2), (1, 2). (Thus, A is isomorphic to [1] t [1].) We definethe functor i : 1 t 1→ A to be the cosieve classifying (1, 2) and (2, 1).

(ii) The category B ⊆ [2]× [2] is the full subcategory obtained from A by addingthe object (2, 2) and the functor j : A→ B is the obvious fully faithful inclu-sion. Note that this inclusion is a sieve.

(iii) Finally, the functor k : B → [2]×[2] is also the obvious fully faithful inclusion.

Associated to these fully faithful functors there are by Proposition 7.25 the fullyfaithful Kan extension functors

(9.15) D(1 t 1)i!→ D(A)

j∗→ D(B)k∗→ D([2]× [2]).

Moreover, since i is a cosieve and j a sieve it follows from Corollary 8.6 that i! andj∗ are left and right extension by zero, respectively. To understand the remainingfunctor k∗ we note that k : B → [2] × [2] factors as a composition of four fullyfaithful inclusions,

k : Bk1→ B1

k2→ B2k3→ B3

k4→ [2]× [2],

which are obtained by adding the objects (1, 1), (1, 0), (0, 1), and (0, 0) one by onein turn. Using arguments similar to the proof of Lemma 8.28, the reader checksthat each of the corresponding right Kan extension functors precisely amounts toadding a new cartesian square.

As an upshot, we have shown that (9.15) induces an equivalence between D(1t1)and the full subcategory D([2] × [2])ex ⊆ D([2] × [2]) spanned by all diagramsQ ∈ D([2]× [2]) satisfying the following exactness properties.

(i) The diagram Q vanishes at the three corners (0, 2), (2, 0), and (2, 2).

82 MORITZ GROTH

(ii) The diagram Q makes the four squares cartesian.

Now, let (X,Y ) ∈ D(1t1) and let Q ∈ D([2]× [2])ex be the corresponding coherentdiagram. The diagram Q looks like

(9.16)

Z //

1

X ′ //

2

0

Y ′ //

3

B //

4

Y

0 // X // 0.

Considering the squares 2 , 4 only, Proposition 9.6 and (the dual of) Proposi-tion 8.27 imply that the composite square is cartesian and that the structure mapX ′ → X is an isomorphism. Similarly, we obtain that the structure map Y ′ → Y isan isomorphism and also that the upper left corner in (9.16) is populated by a zero

object Z ∼= 0. An application of (the dual of) Lemma 9.13 to 4 implies that B is

the product of X,Y . Finally, since D is stable the square 1 is also cocartesian andLemma 9.13 hence implies that B is the coproduct of X ′, Y ′. A final combinationof these observations with the isomorphisms X ′ ∼= X and Y ′ ∼= Y concludes theproof.

9.3. The additivity of stable derivators. In this subsection we mention ingre-dients involved in a proof that stable derivators take values in additive categories.Let us recall from §2.1 (see, in particular, Lemma 2.4) that it suffices to show thatthe abelian monoid structures on sets of morphisms in the preadditive categoriesD(A), A ∈ Cat , actually are group structures.

To this end, let D be a pointed derivator and let X ∈ D(1). Mimicking a classicalconstruction from topology, one can show that the loop object ΩX ∈ D(1) can beturned into a group object. The corresponding multiplication map

(9.17) ∗ = ∗X : ΩX × ΩX → ΩX, X ∈ D(1),

is an abstraction of the classical concatenation of loops in a pointed topologicalspace. We do not include the construction of these concatenation maps hereand instead refer the reader to [Gro13]. Nevertheless we collect the following resultmaking precise that loop objects are group objects.

Theorem 9.18. Let D be a pointed derivator and let X ∈ D(1). The concatenationmap (9.17) endows ΩX with the structure of a group object.

While the precise definition of the concatenation (9.17) is not relevant here, wewill later need some hint where the inversion map comes from. This hint is obtainedfrom the following heuristics in the context of pointed topological spaces.

Let (X,x0) be a pointed topological space. Then a typical model for the loopspace is given by the space of maps [0, 1] → X which send the boundary pointsto x0 (the topology is the compact-open topology). Here it is more convenient touse a different description of the same homotopy type. To begin with, let PX bethe space of paths [0, 1] → X starting at x0 and let ev1 : PX → X be the mapwhich evaluates a path at its target 1 ∈ [0, 1]. We define the loop space ΩX to be


the pullback

(9.19)

ΩXp2 //

p1

PX

ev1

PXev1

// X.

Thus, up to homeomorphism, ΩX is the space of maps [−1, 1] → X which sendthe boundary points −1, 1 to the base point x0. The base point of this space is theconstant map [−1, 1]→ X with value x0.

In this model for the loop space, the inversion of loops ι : ΩX → ΩX is given by areparametrization via the reflection at the origin 0 ∈ [−1, 1] and this reparametriza-tion can be obtained by interchanging the two copies of PX in the pullback square(9.19). More formally, the outer commutative square in the diagram

ΩXp1

))p2

""

ΩXp2//

p1

PX

ev1

PXev1

// X

induces by the universal property of the pullback square a canonical dashed mor-phism making everything commute. The reader easily checks that this morpism isthe above reparametrization ι.

To mimick this in arbitrary pointed derivators, let

σ : y→ybe the swap symmetry which interchanges the objects (1, 0) and (0, 1). Givena pointed derivator D and X ∈ D(1) it can be shown that this swap symmetryinduces a loop inversion ι : ΩX → ΩX, yielding the inverses for the group structureof Theorem 9.18. In more detail, ι is essentially the canonical mate limy σ

∗ → limyassociated to

y σ //

π

y

π

1 =// 1

;C1

evaluated on the coherent cospan (1, 1)!(X) ∈ D(y). More precisely, it is thecombination of this mate with the canonical isomorphism σ∗(1, 1)!(X) ∼= (1, 1)!(X),i.e., we define the loop inversion map as

(9.20) ι : ΩX ∼= limy

(1, 1)!(X) ∼= limyσ∗(1, 1)!X ∼= lim

y(1, 1)!X ∼= ΩX.

Theorem 9.21. Let D be a stable derivator and let A ∈ Cat. The category D(A)is additive.

Proof. By Lemma 9.4 it is enough to show that the underlying category D(1) isadditive. By Proposition 9.14 the category D(1) is preadditive and we have toshow that the abelian monoid structure on homD(1)(Y, Y ), Y ∈ D(1), is a groupstructure. The stability of D implies by Proposition 9.9 that there is an equivalence

84 MORITZ GROTH

of categories (Σ,Ω): D(1) ' D(1), yielding a canonical isomorphism η : Y ∼= ΩΣY .Thus, it suffices to consider the case of loop objects ΩX. In this case one canshow that the loop inversion map (9.20) is an additive inverse to id : ΩX → ΩX,concluding the proof by Lemma 2.4. We refer the reader to [Gro13] for moredetails.

We conclude this subsection by a remark sketching an alternative proof of theadditivity of stable derivators.

Remark 9.22. (i) Abstracting a classical result concerning the homotopy cate-gory Ho(Top∗) of pointed topological spaces, one can show that in arbitrarypointed derivators 2-fold loop objects are abelian group objects. More pre-cisely, let D be a pointed derivator and let X ∈ D(1). The object Ω2X canbe endowed with two structures of a group object, respectively coming withmultiplication morphisms

∗1 : Ω2X × Ω2X → Ω2X and ∗2 : Ω2X × Ω2X → Ω2X.

These morphisms are abstractions of the loop concatenations with respect todifferent sphere coordinates in topology. One can show that these two groupstructures agree and actually are abelian group structures.

(ii) The previous point yields a different proof of the additivity of stable derivators(Theorem 9.21). Again, it suffices to show that the preadditive category D(1)is additive. The preadditivity implies that every object X ∈ D(1) can beessentially uniquely endowed with the structure of a commutative monoidobject, and by Lemma 2.4 it remains to show that this unique commutativemonoid structure is a group structure. Since D is stable there is a naturalisomorphism X ∼= Ω2Σ2X and the concatenation morphism on 2-fold loopobjects turns X into an abelian group object. Thus, the unique abelianmonoid structures are abelian group structures, concluding the proof.

9.4. Morphisms and natural transformations. In this subsection we definemorphisms of derivators and natural transformations between such morphisms.These definitions are special cases of the more general concepts of pseudo-naturaltransformations and modifications.

Definition 9.23. A morphism of prederivators is a pseudo-natural transfor-mation.

Let us unravel this definition. If D and E are prederivators, then a morphismF : D → E consists of

(i) functors FA : D(A)→ E (A), A ∈ Cat , and(ii) natural isomorphisms γF,u = γu : u∗ FB → FA u∗ for u : A→ B in Cat .

This datum has to satisfy the following three coherence properties.

(a) For A ∈ Cat the transformation γ1A: FA → FA is the identity.


(b) For Au→ B

v→ C in Cat the equation γvu = γuv∗ · u∗γv holds,

D(C)(vu)∗

//

FC

D(A)

FA

=

D(C)v∗ //

FC

D(B)u∗ //

FB

D(A)

FA

E (C)(vu)∗

// E (A)

?G∼=

E (C)v∗// E (B)

u∗//

@H∼=

E (A).

@H∼=

(c) For functors u1, u2 : A → B and a natural transformation α : u1 → u2 wehave the equality γu2 · (α∗FB) = (FAα

∗) · γu1 .

Thus, a morphism of prederivators is given by a collection of functors which arecompatible with restrictions up to specified, coherent isomorphisms. A morphismof derivators is simply a morphism of underlying prederivators. We say that amorphism is strict if all γu are identity transformations (which is to say that wehave a 2-natural transformation).

Examples 9.24. (i) Let F : C → D be a functor between ordinary categories.Associativity of composition of functors implies that the induced functorsCA → DA : X 7→ F X commute with restriction functors on the nose.Hence, they assemble to a strict morphism of represented prederivators

y(F ) : y(C)→ y(D).

(ii) Let D be a prederivator and let v : B → B′ be in Cat . The restrictionfunctors (v × 1A)∗ : D(B′ × A) → D(B × A), A ∈ Cat , assemble to a strict

restriction morphism v∗ : DB′ → DB . In particular, there are evaluationmorphisms b∗ : DB → D , b ∈ B.

(iii) Left Quillen functors between combinatorial model categories have associatedleft derived morphisms between homotopy derivators, and similarly for rightQuillen functors; see §B.2 for more details.

Definition 9.25. Let F,G : D → E be morphisms of prederivators. A naturaltransformation φ : F → G is a modification.

Thus, such a φ : F → G consists of natural transformations φA : FA → GA whichfor each u : A→ B make the following diagram commute

u∗FBγF,u

//

u∗φB

FAu∗

φAu∗

u∗GB γG,u

// GAu∗.

The same definition also applies to derivators. The reader easily defines compositionlaws for morphisms and natural transformations of (pre)derivators and checks thatthese yield 2-categories.

Notation 9.26. We write PDER for the 2-category of prederivators, morphisms,and natural transformations and DER for the 2-category of derivators, morphisms,and natural transformations. Thus, the 2-category DER of derivators is a full sub-2-category of PDER. Often we simplify notation and drop some of the decorationsof the components of morphisms and natural transformations of (pre)derivators.

86 MORITZ GROTH

Examples 9.27. (i) Let F, F ′ : C → D be functors between ordinary categoriesand let φ : F → F ′ be a natural transformation. For A ∈ Cat there is the2-functor (−)A : CAT → CAT and φ hence yields natural transformationsφA : FA → F ′A of functors CA → DA, defining a natural transformationy(φ) : y(F )→ y(F ′). In fact, the passage to represented prederivators definesan embedding

y : CAT → PDER.

By restriction, there is a similar embedding of the 2-category of complete andcocomplete categories into the 2-category DER of derivators.

(ii) Let D be a prederivator, let u, v : A → B be in Cat , and let φ : u → v be anatural transformation. The transformations φ× 1: u× 1C → v× 1C inducea natural transformation φ∗ : u∗ → v∗ of strict morphisms DB → DA. Infact, the following stronger result is true: the passage to shifted prederivatorsdefines a 2-functor

(−)(−) : Catop × PDER → PDER : (A,D) 7→ DA.

In particular, for every morphism F : D → E and B ∈ Cat there is an inducedshifted morphism

(9.28) FA : DA → EA.

Recall that a morphism of derivators is a pseudo-natural transformation comingwith structure isomorphisms. We can use the isomorphisms γ−1

u : FAu∗ → u∗FB

and γu : u∗FB → FAu∗,

D(B)u∗ //

FB

∼=

D(A)

FA

D(B)u∗ //

FB

D(A)

FA

E (B)u∗// E (A), E (B)

u∗// E (A),

@H∼=

in order to talk about morphisms of derivators which preserve certain Kan exten-sions.

Definition 9.29. (i) A morphism of derivators F : D → E preserves left Kanextensions along u : A→ B if the canonical mate transformation

(9.30) u!FAη→ u!FAu

∗u!γ−1u→ u!u

∗FBu!ε→ FBu!

is an isomorphism. A morphism of derivators is cocontinuous if it preservesarbitrary left Kan extensions.

(ii) A morphism of derivators F : D → E preserves right Kan extensionsalong u : A→ B if the canonical mate transformation

FBu∗η→ u∗u

∗FBu∗γu→ u∗FAu

∗u∗ε→ u∗FA

is an isomorphism. A morphism of derivators is continuous if it preservesarbitrary right Kan extensions.

Specializing to Kan extensions along πA : A → 1, there are obvious variants ofmorphisms preserving limits of a fixed shape or arbitrary limits, and dually.


Example 9.31. Let F : C → D be a functor between complete and cocompletecategories and let A ∈ Cat . If we consider the induced morphism of derivatorsy(F ) : y(C) → y(D) then the canonical mate (9.30) associated to u = πA : A → 1can be identified with the canonical map colimA F → F colimA defined in (A.19)(the reader is invited to verify this as an exercise).

Lemma 9.32. A morphism of derivators is cocontinuous if and only if it preservescolimits.


As a consequence one can deduce the following.

Remark 9.33. Let C and D be complete and cocomplete categories. A functorF : C → D is cocontinuous in the usual sense of Definition A.21 if and only if thestrict morphism y(F ) : y(C)→ y(D) is cocontinous in the sense of Definition 9.29.

Our discussion of parametrized Kan extensions in §8.3 immediately establishesthe following result.

Proposition 9.34. Let D be a derivator and let v : B → B′ be a functor. Therestriction morphism v∗ : DB′ → DB is continuous and cocontinuous.

Proof. By duality it suffices to show that restriction morphisms are cocontinuous.Recall from Proposition 7.32 the construction of left Kan extensions in shiftedderivators. From this it follows that, given a functor u : A → A′, we have to showthat a certain canonical map

(1× u)!(v × 1)∗ → (v × 1)∗(1× u)!

is an isomorphism. But unraveling definitions the reader checks that this is preciselythe canonical mate expressing that the square (8.20) is homotopy exact and wehence conclude by Proposition 8.19.

As in ordinary category theory, examples of continuous or cocontinuous mor-phisms come from adjunctions. To this end we recall from §A.1 that an adjunctioncan be specified by two functors and two natural transformations satisfying thetriangular identities (A.3). This description allows us to extend to concept of anadjunction to arbitrary 2-categories.

Definition 9.35. An adjunction of derivators is an adjunction internal to the2-category DER.

Thus, an adjunction consists of morphisms L : D1 → D2 and R : D2 → D1

together with natural transformations η : 1 → RL and ε : LR → 1 satisfying thetriangular identities

1 = εL · Lη : L→ L and 1 = Rε · ηR : R→ R.

This general notion can be re-expressed in more elementary terms.

Proposition 9.36. (i) A morphism of derivators is a left adjoint if and only ifeach component is a left adjoint and the morphism is cocontinuous.

(ii) A morphism of derivators is a right adjoint if and only each component is aright adjoint and the morphism is continuous.

Proof. We refer the reader to [Gro13] for a proof.

88 MORITZ GROTH

The proof of this result shows the following. Let L : D1 → D2 be a cocontin-uous morphism such that all components LA : D1(A) → D2(A) are left adjoints.Then one can choose right adjoint functors RA : D2(A)→ D1(A) and these can beassembled to a right adjoint morphism of derivators R : D2 → D1.

We specialize this to the following situation. Let D be a derivator and let usconsider a restriction morphism v∗ : DB′ → DB which is cocontinuous by Proposi-tion 9.34. By (Der3) each component of v∗ has a left adjoint (v × 1)! which hence

assemble to a left Kan extension morphism of derivators v! : DB → DB′ .

Corollary 9.37. Let D be a derivator and let u : A → B be a functor. There areadjunctions of derivators

(u!, u∗) : DA DB and (u∗, u∗) : DB DA.

In particular, the left Kan extension morphism u! : DA → DB is cocontinuous andthe right Kan extension morphism u∗ : DA → DB is continuous.

Definition 9.38. An equivalence of derivators is an equivalence internal to the2-category DER.

Thus, an equivalence consists of morphisms F : D1 → D2 and G : D2 → D1

together with natural isomorphisms η : 1 ∼= GF and ε : FG ∼= 1. In that case wealso say that the morphism F or G alone is an equivalence. Similarly to the caseof adjunctions there is the following description of equivalences in more elementaryterms.

Proposition 9.39. A morphism of derivators is an equivalence if and only if eachcomponent is an equivalence.

We conclude this subsection with the following remark.

Remark 9.40. In earlier sections of this course we established results about Kanextension functors in arbitrary derivators or suitable classes of derivators. If we takethe philosophy of derivators more serious, then we should not content ourselveswith such results concerning categories of coherent diagrams. In fact, based onthe shifting operation of derivators (Proposition 7.32), we should instead aim forsimilar such results about derivators of coherent diagrams. More specifically, in thecontext of homotopy derivators of model categories we would obtain statementsabout homotopy theories of diagrams which, in general, are stricter than merestatements about homotopy categories of diagrams.

It turns out that with a bit more work one can show that basically all our aboveresults extend to statements about Kan extension morphisms of derivators. Forexample, given a derivator D and a fully faithful functor u : A → B, then theKan extension morphisms u!, u∗ : DA → DB are fully faithful and hence induceequivalences onto their essential images.

These refined versions are central to our applications in abstract representationtheory in the final section, and in that section we freely use these derivator versionsof our earlier results. (In a more systematic treatment one would hence right awayestablish these results in their derivator version while for pedagogic reasons werefrained from doing so here.)

9.5. Canonical triangulations in stable derivators. In this subsection we con-struct canonical triangulations on strong stable derivators. Recall from Lemma 6.29


that for every prederivator D and categories A,B ∈ Cat there is the partial under-lying diagram functor

diaB,A : D(B ×A)→ D(A)B ,

making a coherent diagram incoherent in the B-direction.

Definition 9.41. A derivator is strong if dia[1],A : D([1] × A) → D(A)[1] is fulland essentially surjective for every A ∈ Cat .

Note that we do not ask that the functor is faithful, so the partial underlyingdiagram functors dia[1],A are not asked to be equivalences.

Examples 9.42. (i) Represented derivators are strong. In fact, in that case allpartial underlying diagram functors are equivalences.

(ii) Homotopy derivators of model categories are strong (see §B.2 for a proof).Thus, typical derivators arising in homological algebra and homotopy theory(like the ones mentioned in Examples 6.23) are strong.

(iii) A derivator D is strong if and only if Dop is strong.(iv) If D is a strong derivator and if B ∈ Cat , then the shifted derivator DB is

also strong.

Remark 9.43. The precise form of Definition 9.41 depends on the author. Forexample, Heller [Hel88] asks that the functors diaF,A : D(F × A) → D(A)F arefull and essentially surjective for all finite free categories F and all A ∈ Cat . Itcan be shown that the derivators mentioned in Examples 9.42 all satisfy this morerestrictive axiom.

The property of being strong is important when one wants to relate propertiesof stable derivators to structure on its values. This is illustrated in the proof of thefollowing theorem.

As a preparation recall from §8.5 the construction of (coherent) cofiber sequencesin a pointed derivator D . By Proposition 8.36 there is an equivalence of categoriesD([1]) ' D()cof sending a morphism to its cofiber sequence. Every coherentcofiber sequence has an underlying incoherent cofiber sequence, yielding the functorD()cof → D(1)[3] defined in (8.33). In the case of a stable derivator we denotethe following composition by

(9.44) tria : D([1]) ' D()cof → D(1)[3]

and refer to tria(X), X ∈ D([1]), as the standard triangle associated to X. Atriangle x → y → z → Σx in D(1) is distinguished if it is naturally isomorphicto a standard triangle.

Passing to shifted derivators this also defines a class of distinguished triangles onD(A), A ∈ Cat . In fact, with D also the shifted derivator DA is stable (Lemma 9.4)and the underlying category of DA is canonically isomorphic to D(A).

Theorem 9.45. Let D be a strong, stable derivator and A ∈ Cat. The suspensionfunctor Σ: D(A) → D(A) together with the above class of distinguished trianglesturn D(A) into a triangulated category.

Proof. Passing to shifted derivators we may assume that A = 1. By Theorem 9.21the category D(1) is additive and the suspension functor Σ: D(1) → D(1) is anequivalence by Proposition 9.9. It remains to show that the above class of distin-guished triangles satisfies the axioms (T1)-(T4) of Definition 4.9.

90 MORITZ GROTH

(T1): Let x ∈ D(1) and let us consider the corresponding identity morphismid: x→ x as an object in D(1)[1]. Since D is strong, the underlying diagram functorD([1]) → D(1)[1] is essentially surjective and we can find a coherent morphismX ∈ D([1]) and an isomorphism dia[1](X) ∼= id (one can also avoid using thestrength here by restricting along π : [1] → 1 or by invoking Example 7.24). Tosee that x → x → 0 → Σx is distinguished the reader easily checks that it sufficesto show that the third object in tria(X) is a zero object. And this claim followsimmediately from Proposition 9.5 since the third object in tria(X) is isomorphic toCX which is trivial as X is an isomorphism.

By definition the class of distinguished triangles is closed under isomorphismsand for axiom (T1) it hence remains to show that every morphism f : x → yin D(1) extends to a distinguished triangle. By assumption the diagram functorD([1]) → D(1)[1] is essentially surjective and we can hence find X ∈ D([1]) anda natural isomorphism dia[1](X) ∼= f . The reader checks that this isomorphismtogether with the distinguished triangle tria(X) yields the desired distinguishedtriangle extending f (using once more that the class of distinguished triangles isclosed under isomorphisms).

(T2): We leave it to the reader to verify that it suffices to establish the followingclaim: if we rotate backwards a standard triangle then we obtain a distinguishedtriangle. Since our distinguished triangles are obtained from two iterations of thecofiber construction, here it is convenient to consider coherent diagrams encodingthree iterations of cofiber constructions as it was already done in §8.5 (we suggestthe reader to recall the statement of Lemma 8.39 as well as Proposition 8.41 togetherwith its proof).

By this lemma, every coherent morphism X = (f : x → y) ∈ D([1]) can beextended to a coherent diagram looking like (8.40),

(9.46)

xf//

y //

g

02

01// z

h //

x′

f ′

03// y′,

vanishing as indicated and making all squares cocartesian (we can ignore the sub-scripts in the objects 01, 02, 03 for now). Note that the cofiber sequence consistingof the two top squares in (9.46) is used to construct the standard triangle tria(f).In more detail, one considers the composition of these two squares, looking like

(9.47)

x //

02

01// x′,

and the associated identification φ : x′ ∼= Σx in order to obtain the standard triangle

(9.48) xf→ y

g→ zφh→ Σx.


We next collect a detailed description of the standard triangle tria(g). To beginwith we note that if we reflect the two squares on the right in (9.46) in order todraw them horizontally, then we obtain the cofiber sequence on the left in

(9.49)

yg//

z

h

// 03

y //

03

02// x′

f ′// y′, 02

// y′,

which can hence be used to calculate tria(g). In fact, the cocartesian square on theright in (9.49) obtained by composing the cocartesian squares on the left yields anidentification ψ : y′ ∼= Σy and tria(g) hence looks like

(9.50) yg→ z

h→ x′ψf ′→ Σy.

It remains to show that this triangle is isomorphic to the rotation of (9.48). Tothis end, we recall that f ′ is naturally isomorphic to Σf . Since we need some detailsabout this identification, we repete from the proof of Proposition 8.41 that there isa coherent cube (8.42),

x //

f

02

y //

02

01

// x′

f ′

03// y′,

making the back face and the front face cocartesian. And the natural isomorphismf ′ ∼= Σf is obtained from this cube. Thus, in more detail, we use the identificationsx′ ∼= Σx and y′ ∼= Σy induced from the cocartesian squares

(9.51)

x //

02

y //

02

01// x′, 03

// y′.

Since the square on the left is precisely the square (9.47), the first identificationis again the morphism φ : x′ ∼= Σx showing up in (9.48). However, if we comparethe square on the right in (9.51) to the square on the right in (9.49), then we seethat they differ by a restriction along the automorphism swapping (1, 0) ↔ (0, 1).It follows from the sketch proof of Theorem 9.21 that the second identificationinduced by (9.51) hence is −ψ : y′ ∼= Σy, where ψ is as in (9.50). To summarize,the natural isomorphism f ′ ∼= Σf has components

x′f ′//

φ ∼=

y′

−ψ∼=

ΣxΣf// Σy.

92 MORITZ GROTH

This finally gives us the desired isomorphism of triangles

y

=

g// z

=

h // x′

φ

ψf ′// Σy

=

yg// z

φh// Σx

−Σf// Σy

from the standard triangle tria(g) (see (9.50)) to the triangle which is obtainedfrom tria(f) (see (9.48)) by rotation.

(T3): We leave it to the reader to show that this axiom follows rather directlyfrom the assumption that D is strong.

(T4): It remains to establish the octahedron axiom, saying that for every pairof composable morphisms in D(1) there is an associated octahedron diagram. Wefirst show that every coherent pair of composable morphisms X ∈ D([2]) with

underlying diagram f3 = f2 f1 : xf1→ y

f2→ z gives rise to such a diagram. Choosinga slightly different notation than in Definition 4.9, we have to show that there is acommutative diagram

(9.52)

xf1 //

=

yg1 //

f2

0

uh1 //

f4

1

Σx

=

xf3 // z

g2

g3 //

2

vh3 //

g4

3

Σx

Σf1

w

h2

=//

4

w

h4

h2

// Σy

ΣyΣg1

// Σu

such that the rows and colums are distinguished triangles, i.e., such that

(i) xf1→ y

g1→ uh1→ Σx,

(ii) yf2→ z

g2→ wh2→ Σy,

(iii) xf3→ z

g3→ vh3→ Σx, and

(iv) uf4→ v

g4→ wh4→ Σu

are distinguished triangles. Thus, these four distinguished triangles have to be

constructed such that the squares 0 − 4 in (9.52) all commute.This can be achieved by considering coherent diagrams looking like

(9.53)

xf1 //

yf2 //

z //

0

0 // u //

v //

x′ //

0

0 // w // y′ // u′.


In more detail, let B ⊆ [4] × [2] be the full subcategory obtained by removing theobjects (4, 0), (0, 2). The functor [2] → B classifying the composable morphisms

(0, 0) → (1, 0) → (2, 0) factors as a composition [2]i→ A

j→ B of fully faithfulfunctors where A ⊆ B contains the objects (0, 0), (1, 0), (2, 0) and the four objectspopulated by zeros in (9.53). Similarly to Lemma 8.39 one checks that the fully

faithful functors D([2])i∗→ D(A)

j!→ D(B) induce an equivalence

(9.54) D([2]) ' D(B)cof ,

where D(B)cof ⊆ D(B) is the full subcategory spanned by all diagrams

(i) which vanish at the four objects (0, 1), (3, 0), (1, 2), (4, 1) and(ii) which make all squares cocartesian.

Using implicitly certain identifications obtained by similar methods as in §8.5, thisshows that every X ∈ D([2]) with underlying diagram f3 = f2 f1 : x → y → zgives rise to a coherent diagram Q ∈ D(B)cof with underlying diagram

(9.55)

xf1 //

yf2 //

g1

z //

g3

0

0 // uf4 //

vh3 //

g4

Σx //

Σf1

0

0 // wh2

// ΣyΣg1

// Σu.

(Of course the notation is slightly abusive in that the respective squares in theunderlying diagram are no pushout squares anymore; the decoration is only meantto indicate that they come from coherent cocartesian squares.) A ‘diagram chase’in (9.53) combined with the identifications made in (9.55) yields the desired four

distinguished triangles satisfying the above five relations 0 − 4 . To begin with, ifone considers the top three cocartesian squares and glues the two to the right, thenone obtains the first distinguished triangle together with the relation h3f4 = h1,

settling equation 1 . Let us next consider the copy of [2] × [2] in the middle ofthe diagram and compose vertically. This gives the second distinguished triangle

together with the relation g2 = g4g3, hence taking care of 2 . If we again considerthe top three cocartesian squares but this time glue the two squares to the left,then we construct the third distinguished triangle. For the remaining distinguishedtriangle we consider the bottom three cocartesian squares and compose the two to

the right. This also yields the relation h4 = (Σg1)h2, hence making 4 commute.

Finally, in order to show that also the squares 0 and 3 commute it sufficesto consider the middle square in the top and bottom row of (9.53), respectively.This completes the construction of an octahedron diagram for a coherent pair ofcomposable morphisms.

Let us now consider a pair of composable morphisms x→ y → z in the underlyingcategory D(1) and show that it has an associated octahedron diagram. The readereasily checks that it suffices to construct a coherent diagram X ∈ D([2]) such thatthe underlying diagram is naturally isomorphic to x → y → z. If we use thestronger variant of strength (see Remark 9.43) then this is immediate since [2] is afinite, free category. The reader who does not want to use this stronger axiom is

94 MORITZ GROTH

invited to come up with an independent construction of such a natural isomorphismdia(X) ∼= (x→ y → z), using Definition 9.41 only.

The triangulations of the theorem are referred to as canonical triangulations.As a special case the theorem yields canonical triangulations on underlying cate-gories of strong, stable derivators. We illustrate this by a few examples.

Examples 9.56. (i) If A is an abelian category, such that the homotopy derivatorDA exists (for example, if A is a Grothendieck abelian category; see Defini-tion 6.18), then the underlying category DA(1) ∼= D(A) is a triangulatedcategory. Taking up again Examples 6.19, Theorem 9.45 thus reproduces theclassical triangulations on derived categories of fields, rings, and schemes.

(ii) If we apply Theorem 9.45 to the stable derivator of spectra (6.24), then weobtain the classical triangulation on the stable homotopy category SHC.

(iii) More generally, homotopy derivators of stable model categories and sta-ble ∞-categories are strong, stable derivators. Applied to such examples,Theorem 9.45 reproduces the triangulations on homotopy categories of sta-ble model categories ([Hov99, §7]) and homotopy categories of stable ∞-categories ([Lur11, §1]).

We conclude this subsection by a short remark on the octahedron axiom (relatedto this see also Remark 8.34).

Remark 9.57. Let us recall the main steps of the proof of the octahedron axiom(T4) for the canonical triangulations from Theorem 9.45. Let D be a strong, stablederivator and let X ∈ D([2]) be a coherent pair of composable morphisms. As-sociated to X we obtain a coherent diagram looking like (9.53) satisfying certainexactness and vanishing properties. More precisely, there is an equivalence of cate-gories (9.54), and one might refer to the target category D(B)cof as the categoryof coherent octahedron diagrams. Thus, as long as we stay at the level ofcoherent diagrams, having a pair of composable morphisms is equivalent to havingan octahedron diagram.

Now, given a coherent octahedron diagram X ∈ D(B)cof , making some identifi-cations as indicated in (9.55) we obtain an incoherent octahedron diagram lookinglike (9.52), and this step amounts to a loss of information since we pass to un-derlying diagrams. If we denote the shape of an incoherent octahedron diagram(9.52) by O, then these two steps hence amount to the functorial construction ofoctahedron diagrams

octa: D([2]) ' D(B)cof → D(1)O.

If we start with an incoherent pair of composable morphisms instead, then we firsthave to lift against dia[2] : D([2])→ D(1)[2], and this step destroys the functoriality.

This reminds us of the discussion of the non-functoriality of the constructionof cones and distinguished triangles at the level of triangulated categories; see thebeginning of §4.4. Also in these cases we can obtain functorial such constructionsif we stay at the level of coherent diagrams as made precise by Proposition 8.36and the functor tria : D([1]) → D(1)[3] defined in (9.44). Again, if we insteadbegin with incoherent morphisms, then we first have to lift against the functordia[1] : D([1]) → D(1)[1], resulting in the loss of functoriality. These two similar


observations can be summarized by the following two diagrams

D([1])' //

tria%%

dia[1]

D()cof

D([2])' //

octa%%

dia[2]

D(B)cof

D(1)[1] D(1)[3], D(1)[2] D(1)O,

in which the vertical functors result in a loss of information.

9.6. Exact morphisms of stable derivators. Our next goal is to show thatthe triangulations D(A), A ∈ Cat , of Theorem 9.45 are compatible with restrictionfunctors u∗ : D(B) → D(A) and Kan extension functors u!, u∗ : D(A) → D(B).This actually is a special case of a more general result concerning exact morphismsof (strong) stable derivators.

Recall from Definition 9.29 what it means that a morphism of derivators preservescertain Kan extensions. Specializing this to Kan extensions along πA : A → 1 forsuitable A we obtain the following definition.

Definition 9.58. (i) A morphism of derivators is left exact if it preserves ter-minal objects and cartesian squares.

(ii) A morphism of derivators is right exact if it preserves initial objects andcocartesian squares.

(iii) A morphism of derivators is exact if it is left exact and right exact.

The following is immediate from Remark 9.33 and Proposition A.22.

Example 9.59. A functor F : C → D between complete and cocomplete categoriesis left exact in the sense of Definition A.21 if and only if the induced morphism ofderivators y(F ) : y(C)→ y(D) is left exact in the sense of Definition 9.58.

Lemma 9.60. (i) A morphism of stable derivators is left exact if and only if itis right exact if and only if it is exact.

(ii) Let (F,G) : D E be an adjunction between stable derivators. The mor-phisms F,G both are exact.

Lemma 9.61. Let F : D → E be a morphism of derivators, let u : A → A′, andlet B ∈ Cat. If F preserves left Kan extensions along u, then F also preserves leftKan extensions along 1× u : B ×A→ B ×A′.

Proof. TODO.

Let us recall from Examples 9.27 that the shifting operation also applies tomorphisms of derivators (see (9.28)). A different way of phrasing Lemma 9.61 isby saying that if F : D → E preserves left Kan extensions along u then so does theshifted morphism FB : DB → E B .

Corollary 9.62. Let F : D → E be an exact morphism of stable derivators and letB ∈ Cat. The induced morphism F : DB → E B is again exact.

Proof. This is immediate from Lemma 9.61 and its dual.

Recall from Definition 4.17 that an exact functor between triangulated categoriesis an additive functor endowed with exact structure.

96 MORITZ GROTH

Proposition 9.63. Let F : D → E be an exact morphism of strong, stable deriva-tors and let A ∈ Cat. The functor FA : D(A) → E (A) can be turned into an exactfunctor with respect to canonical triangulations.

Proof. By Corollary 9.62 we may assume without loss of generality that A = 1.We begin by showing that the underlying functor F1 : D(1) → E (1) is additive.TODO.

Corollary 9.64. Let D be a strong, stable derivator and let u : A→ B be in Cat.The functors u∗ : D(B) → D(A), u! : D(A) → D(B), and u∗ : D(A) → D(B) canbe turned into exact functors with respect to canonical triangulations.

Proof. By Corollary 9.37 there are adjunctions of strong, stable derivators

(u!, u∗) : DA DB and (u∗, u∗) : DB DA,

given by restriction and Kan extension morphisms. It follows from Lemma 9.60that these three morphisms are exact and the underlying functors can hence beturned into exact functors (Proposition 9.63). We note that these underlyingfunctors respectively are the functors u∗ : D(B) → D(A), u! : D(A) → D(B), andu∗ : D(A)→ D(B), concluding the proof.

Remark 9.65. Given a strong, stable derivator D , by Theorem 9.45 there are canon-ical triangulations on D(A), A ∈ Cat , and by Proposition 9.63 the restriction func-tors u∗ : D(B) → D(A) can be turned into exact functors. There is also such anobservation for natural transformations.

To make this precise, we recall the following definition. Let T , T ′ be trian-gulated categories and let (F, σ), (F ′, σ′) be exact functors T → T ′. A naturaltransformation α : F → F ′ is exact if the diagram

FΣσ∼=//

αΣ

ΣF

Σα

F ′Σσ′

∼= // ΣF ′

commutes. Given a strong, stable derivator D , functors u, v : A→ B, and a trans-formation α : u→ v, one can show that the natural transformation α∗ : u∗ → v∗ isexact in this sense.

A concise way of summarizing this is as follows. Triangulated categories, exactfunctors, and exact natural transformations are assembled in a (again very large)2-category T riaCAT of triangulated categories. Every strong, stable derivatorD : Catop → CAT admits a lift against the forgetful 2-functor T riaCAT → CAT ,

T riaCAT

Catop

D//

∃D88

CAT .

10. Towards abstract representation theory

10.1. Tilting theory and strong stable equivalences. Given a ring R we de-note by Mod(R) the abelian category of R-modules. Attributing credit to [?] thereis the following definition.


Definition 10.1. Two rings R,S are Morita equivalent if there is an equivalenceof categories Mod(R) ' Mod(S).

It can be shown that such an equivalence is necessarily given by the tensor prod-uct with an (R,S)-bimodule. In representation theory people are often interestedin derived categories of rings and algebras. The derived analogue of Definition 10.1is the following.

Definition 10.2. Two rings R,S are derived equivalent if there is an exact

equivalence D(R)∆' D(S).

We now specialize to particular types of rings, namely path algebras of finitequivers. Let Q be a finite quiver and let k be a field. We recall that the pathalgebra kQ is constructed as follows. The underlying k-vector space has as basisthe set of all paths in Q. More precisely, if Qn, n ≥ 0, denotes the set of all pathsof length precisely n, then we set

kQ =⊕n≥0

kQn.

The multiplication is the bilinear extension of the concatenation of two such paths(the product of two basis elements is set to be zero if the source and targets donot match appropriately). A key feature about this k-algebra is that there is anequivalence of categories

(10.3) Mod(kQ) ' Mod(k)Q.

Here, Q also denotes the free category associated to the oriented graph Q, andrepresentations can hence equivalently be specified by Q-shaped diagrams of vectorspaces. Of course, instead of working over a field k we can consider a ring R andobtain a similarly defined path algebra RQ and a corresponding equivalence (10.3).

Definition 10.4. Let Q,Q′ be finite quivers and let R be a ring. The quivers Q,Q′

are derived equivalent over R if there is an exact equivalence

D(RQ)∆' D(RQ′).

Note that in principle the property of being derived equivalent might depend onthe ring R. Classically, such derived equivalences are often established over fields.To mention one such result we recall the following definition.

Definition 10.5. A quiver is acyclic if it admits no non-trivial, oriented loops. Aquiver is a tree if it admits no non-trivial, unoriented loops.

A reorientation of a quiver Q is a quiver Q′ obtained from Q by changing theorientation of some edges. For example the quivers

1→ 2→ 3 and 1← 2→ 3

are reorientations of each other. We leave it to the reader to come up with a directargument showing that the corresponding path algebras over fields are not Moritaequivalent. Nevertheless, it turns out that these quivers are derived equivalent overarbitrary fields. In fact, this is only a special case of the following more generalresult of Happel.

Theorem 10.6. Let T be a finite tree and let T ′ be a reorientation of T . Thequivers T, T ′ are derived equivalent over arbitrary fields.

98 MORITZ GROTH

By a purely combinatorial argument the task is reduced to the following situa-tion. Let T be a finite tree and let q0 ∈ T be a source, i.e., a vertex q0 such thatall edges adjacent to it start at q0. The reflected tree T ′ of T at the source q0 isobtained by changing the orientation of all edges starting at q0 and thereby turningthe source q0 into a sink. Now, in order to establish Theorem 10.6 it suffices to showthat finite trees and their reflections at sources and sinks are derived equivalent overarbitrary fields.

It turns out that such derived equivalences even exist more generally. The notionof a reflection at a soure or a sink makes perfectly well sense for arbitrary quivers.And in the case of acyclic quivers there is the following result of Happel [Hap87].

Theorem 10.7. Let Q be a finite, acylic quiver and let Q′ be the reflection of Q ata source or a sink. The quivers Q,Q′ are derived equivalent over arbitrary fields.

In fact, such derived equivalences can be explicitly constructed as derived reflec-tion functors in the sense of Bernsteın, Gel′fand, and Ponomarev (see [BGP73] and[Hap87]). Using the formalism of derivators, one can show that such equivalencesexist in much broader generality. To this end we begin by a minor reformulation ofDefinition 10.4. As a special case of Lemma 3.7 we observe that for a finite quiver Qthere is an isomorphism of categories

Ch(Mod(R)Q) ∼= Ch(R)Q

which sends quasi-isomorphisms to levelwise quasi-isomorphisms. If we combinethis with a degreewise application of (10.3) then we obtain an equivalence of cate-gories

Ch(RQ) ' Ch(R)Q,

again sending quasi-isomorphisms to levelwise quasi-isomorphisms. As an upshot,if two quivers Q,Q′ are derived equivalent over a ring R, then there is the tophorizontal and the two vertical equivalences

(10.8)

D(RQ)' //

'

D(RQ′)

'

Ho(Ch(R)Q) '// Ho(Ch(R)Q

′),

hence inducing the bottom equivalence. Thus, to put it in words, the fact that twoquivers are derived equivalent over R means that if we pass to the correspondinghomotopy theories of diagrams in Ch(R) then the associated homotopy categoriesare equivalent. In some cases one can obtain similar results for homotopy theoriesof representations in arbitrary stable homotopy theories in the sense of the followingdefinition (see [GS14c]).

Definition 10.9. Two quivers Q,Q′ are strongly stably equivalent if for everystable derivator D there is an equivalence of derivators

DQ ' DQ′

which is natural with respect to exact morphisms of derivators.

We include a few comments about the definition.

Remark 10.10. (i) This definition of course makes perfectly well sense for smallcategories A,A′ as well, but here we focus on quivers.


(ii) If two finite quivers Q,Q′ are strongly stably equivalent then they are alsoderived equivalent over arbitrary rings. In fact, considering the special caseof the stable derivator DR of a ring (Examples 9.2), we obtain an equivalence

of derivators DQR ' DQ′

R . Passing to underlying categories this yields an

equivalence of categories DQR (1) ' DQ′

R (1) which is exact with respect tothe canonical triangulations of Theorem 9.45. Combining this with (10.8) wededuce that Q,Q′ are derived equivalent over R. (One can also invoke the

equivalence of derivators DQR ' DRQ; see Examples 7.31.)

(iii) However, the notion of being strongly stably equivalent is a priori much morerestrictive for the following three reasons.(a) First, simply by choosing other examples of stable derivators (Exam-

ples 9.2) and passing to underlying categories we obtain further exactequivalences of triangulated categories of representations. For example,considering the stable derivator DX of a scheme X we obtain equiva-lent derived categories of representations in quasi-coherent OX -modules.Similarly, if we pass to the stable derivator DA of a differential-gradedalgebra A, then we get exact equivalences of triangulated categories ofdifferential-graded representations. Moreover, there are further such re-sults for spectral representations, simply by passing to the stable deriva-tor of spectra or the stable derivator DE of a —say— symmetric ringspectrum E. Still more generally, we can consider homotopy derivatorsassociated to arbitrary stable model categories or stable ∞-categories,concluding that there are exact equivalences of triangulated categories ofabstract representations of the quivers under consideration. (We againrefer the reader to [GS14c, Examples 5.5] for a fairly long list of addi-tional examples.)

(b) Second, by the very definition we ask for equivalences of derivators asopposed to merely asking for exact equivalences of the underlying trian-gulated categories. Recall that, in general, it is a stronger statement tohave an equivalence of homotopy theories and not only an equivalence ofhomotopy categories. Thus, there are equivalences of homotopy theoriesof abstract representations of the quivers under consideration.

(c) Third, these equivalences are compatible with respect to exact mor-phisms. In fact, by the very definition suitably restricted corepresentedcopresheaves (−)Q, (−)Q

′are asked to be equivalent. In particular, the

various equivalences listed above are compatible with (derived) restric-tions of scalars, inductions and coinductions of scalars, (derived) tensorand hom functors, as well as localization and colocalization functors.

With these remarks in mind, let us mention the following generalization of The-orem 10.6; see [GS14b].

Theorem 10.11. Let T be a finite tree and let T ′ be a reorientation of T . Thequivers T, T ′ are strongly stably equivalent.

In §10.2 we give a sketch proof in the special case of Dynkin quivers of type A.Similar techniques can also be used to take care of trees with at most one branchingpoint, i.e., vertex of valences at least three. With more refined techniques one canmimic the classical construction of reflection functors in the sense of Bernsteın,Gel′fand, and Ponomarev [BGP73] in order to obtain the strong stable equivalences

100 MORITZ GROTH

of Theorem 10.11; see [GS14b]. Theorem 10.7 can be refined to yield the followingresult.

Theorem 10.12. Let Q be a finite, acylic quiver and let Q′ be the reflection of Qat a source or a sink. The quivers Q,Q′ are derived equivalent over arbitrary fields.

In fact, in [GS15] there will be the construction of abstract reflection functorsestablishing this result. The techniques of that paper apply even more generally,yielding additional examples of strong stable equivalences. A further specializingto derivators of rings and fields gives rise to new examples of derived equivalencesof category algebras, generalizing Theorem 10.7.

10.2. Abstract tilting theory of An-quivers. In this subsection we give a sketchproof that different Dynkin quivers of type A of the same length are strongly stablyequivalent. These observations are closely related to the construction of canonicalhigher triangulations on (strong) stable derivators.

To begin with let us recall that a Dynkin quiver of type A is simply a finitezig-zag, i.e., an oriented graph obtained by endowing the unoriented graph

1 2 . . . n− 1 n

with an arbitrary orientation. More precisely, fixing the length n this way we obtainthe An-quivers. Thus, the goal is to see that all An-quivers are strongly stablyequivalent. As a warmup let us consider the four different An-quivers

1→ 2→ 3, 1← 2→ 3, 1→ 2← 3, and 1← 2← 3.

As a preparation for the proofs in this subsection we suggest the reader to againtake a look at Remark 9.40.

Proposition 10.13. All A3-quivers are strongly stably equivalent.

Proof. Let D be a stable derivator and let us show that the relevant stable derivatorsof representations in D are naturally equivalent. To begin with, since the twocategories (1→ 2→ 3) and (1← 2← 3) are isomorphic these quivers are stronglystably equivalent.

We next show that the quivers p= (1← 2→ 3) and y = (1→ 2← 3) are stronglystably equivalent. To this end, we consider the functors ip : p→ and iy : y → and the associated fully faithful Kan extension morphisms (ip)! : Dp → D and(iy)∗ : Dy → D. The essential image of (ip)! consists precisely of those X ∈ D(A)such that every Xa ∈ D(), a ∈ A, is cocartesian (Corollary 8.22) and similarly forthe essential image of (iy)∗. Since D is stable these two essential images coincideand will be denoted by D,ex ⊆ D. As an upshot we obtain equivalences ofderivators

Dp ' D,ex ' Dy,

showing that these two A3-quivers are strongly stably equivalent.It remains to show that the quivers [2] ∼= (1 → 2 → 3) and p= (1 ← 2 → 3)

are strongly stably equivalent. Let X = (X0 → X1 → X2) ∈ D [2] be an abstractrepresentation of [2]. Then such an equivalence is obtained by passing to the cofiberof (X0 → X1). In more detail, us consider the full subcategory B ⊆ [2]×[1] obtained


by removing the final object (2, 1),

(0, 0) //

(1, 0) //

(2, 0)

(0, 1) // (1, 1).

The functor [2]→ B classifying (0, 0)→ (1, 0)→ (2, 0) factors as a composition offully faithful functors

[2]i1→ A1

j1→ B,

where A1 is obtained from B by also removing the object (1, 1). Associated tothese functors there are fully faithful Kan extension morphisms (i1)∗ : D [2] → DA1

and (j1)! : DA1 → DB . Since i1 is a sieve, the right Kan extension morphism (i1)∗is right extension by zero (Corollary 8.6). Moreover, the essential image of (j1)! iseasily checked to consist of precisely those coherent diagrams such that the obvioussquare is cocartesian. As an upshot, (j1)!(i1)∗ : D [2] → DB induces an equivalenceonto the full subderivator of DB spanned by all Z satisfying the following exactnessproperties.

(i) The diagram Z makes the obvious square cocartesian.(ii) The diagram Z vanishes at the lower left corner, Z(0,1)

∼= 0.

Let us now consider the functor p→ B classifying (1, 1) ← (1, 0) → (2, 1). Thisfunctor factors as a composition of fully faithful functors

pi2→ A2

j2→ B,

where A2 is obtained from B by also removing the object (0, 0). Using simi-lar arguments as above and an additional cofinality argument, one checks that(j2)∗(i2)! : Dp → DB induces an equivalence onto the full subderivator of DB

spanned by all Z satisfying the following exactness properties.

(i) The diagram Z makes the obvious square cartesian.(ii) The diagram Z vanishes at the lower left corner, Z(0,1)

∼= 0.

Note that since D is stable in both cases the essential images coincide (Corol-lary 8.22), and let us denote it by DB,ex ⊆ DB . As an upshot, the above construc-tions yield equivalences of derivators

D [2] ' DB,ex ' Dp,

showing that the A3-quivers [2] and p are strongly stably equivalent. Since beingstrongly stably equivalent is an equivalence relation, this concludes the proof.

Similar techniques can be used to also show that all An-quivers for a fixed n ≥ 1are strongly stably equivalent. It will pay off to approach these constructions a bitmore systematically. To motivate the construction, let us emphasize that in the caseof A3-quivers, once the dust settles, everything essentially boiled down to expandingour representations by passing to (co)fibers and adding certain bicartesian squares.In fact, similar arguments were already used in the proof of Theorem 9.45, namelyin the proof of the octahedron axiom by means of the construction of coherentoctahedron diagrams; see (9.55) and Remark 9.57.

It turns out that one convenient strategy consists of expanding coherent diagramslooking like (9.55) doubly infinitely by adding more zeros and bicartesian squares

as we explain next. From now on, let us write ~An = [n−1] for the linearly oriented

102 MORITZ GROTH

An-quiver. Given an abstract representation X = (x → y → z) ∈ D~A3 , suitable

combinations of Kan extensions yield a coherent diagram looking like

(10.14)

· · · 0

##

0

0

!!

0

""

· · ·<<

!!

Ωw

==

z

@@

Σx

<<

""

Σu

<<

""· · · Ωv

<<

""

y

AA

v

>>

Σy

<<

""

· · ·==

""

x

>>

!!

u

AA

w

<<

""

Σz

<<

""· · · 0

;;

0

@@

0

==

0

<<

· · ·

The defining exactness properties of such coherent diagrams are that they vanishon the boundary stripes and that all squares are bicartesian. We denote the cor-responding shape by M3 ∈ Cat and note that there is the fully faithful inclusion

functor ~A3 →M3 pointing at the objects populated by (x→ y → z). More gener-ally, for n ≥ 1 there is a similar category Mn ∈ Cat and a fully faithful inclusion

functor ~An → Mn. Note that this inclusion functor factors as a composition offully faithful functors

~Ani1→ B1

i2→ B2i3→ B3

i4→Mn,

where the respective full subcategories of Mn are obtained as follows.

(i) The category B1 is obtained from the image of ~An → Mn by adjoining allobjects on the boundary stripes sitting below this image.

(ii) The category B2 is obtained from B1 by adding the remaining objects belowthis image.

(iii) The category B3 is obtained from B2 by adjoining the remaining objects onthe boundary stripes.

Given a derivator D , associated to these inclusions there are the following fullyfaithful Kan extension morphisms,

(10.15) D~An

(i1)∗→ DB1(i2)!→ DB2

(i3)!→ DB3(i4)∗→ DMn .

Theorem 10.16. Let D be a stable derivator and let n ≥ 1. The fully faithful Kan

extension morphisms (10.15) induce an equivalence of derivators D~An ' DMn,ex

where DMn,ex ⊆ DMn is the full subderivator spanned by all coherent diagramsZ ∈ DMn satisfying the following exactness properties.

(i) The diagram Z vanishes on all objects in the boundary stripes.(ii) The diagram Z makes all squares bicartesian.

Moreover, this equivalence is natural with respect to exact morphisms of derivators.

Proof. The idea of the proof is obvious. The first morphism in (10.15) is rightextension by zero, the second one amounts to adding cocartesian squares in onedirection, the third morphism is left extension by zero, and the fourth morphismadds cartesian squares in the other direction. Thus, in stable derivators the essentialimage is as claimed. For a detailed proof see [GS14a, §4].

One reason for us to consider these shapes Mn is the following observation.

Besided the inclusions ~An → Mn there are many additional embeddings Q → Mn


for arbitrary An-quivers Q of the same length. And related to these embedding oneconstructs, for every stable derivator D , similar equivalences of derivators

DQ ' DMn,ex,

which again are natural with respect to exact morphisms of stable derivators. Notethat in this case as well as in the equivalence of Theorem 10.16 an inverse equiva-lence is simply given by restriction along the respective embedding.

To illustrate this, in the case of n = 3, let us consider the inclusion p→ M3

pointing at the objects populated by (u← y → z) in (10.14). Similar constructionsas in (10.15) induce natural equivalences Dp ' DM3,ex. If we combine this withthe equivalence of Theorem 10.16, then we obtain natural equivalences

D~A3 ' DMn,ex ' Dp,

showing that ~A3 and p are strongly stably equivalent. The reader easily checks thatthis reproduces the strong stable equivalence constructed in the proof of Proposi-tion 10.13.

Theorem 10.17. For every fixed n ≥ 1 all An-quivers are strongly stably equiva-lent.

Proof. Let Q,Q′ be two An-quivers of the same length and let D be a stable deriva-tor. Then we can choose embeddings iQ : Q → Mn and iQ′ : Q

′ → Mn as above.Related to these embedding, the abovementioned more general version of Theo-rem 10.16 yields natural equivalences

DQ ' DMn,ex ' DQ′ ,

showing that Q,Q′ are strongly stably equivalent.

An additional reason for us to consider the shapes Mn ∈ Cat is that they showup in the definition of strong triangulations in the sense of Maltsiniotis; see [Mal05].The rough idea is as follows. A strong triangulation on an additive category Aconsists of a self-equivalence Σ: A ' A and chosen classes of distinguished n-triangles. Here, an n-triangle in A is a diagram Mn → A together with suit-ably natural isomorphisms making precise that ‘suspensions show up as intended’;see (10.14) to get an idea. This structure has to satisfy a list of compatibilityaxioms, including relations between distinguished n-triangles for varying n.

Thus, 2-triangles look like a more refined version of classical triangles and 3-triangles like refined octahedron diagrams; see Remark 9.57. The distinguishedn-triangles for n ≥ 4 can be thought of as ‘higher octahedron diagrams’. A preciseformulation and a proof of the following result can be found in [GS14a, §13].

Theorem 10.18. Let D be a strong stable derivator and let A ∈ Cat. The categoryD(A) admits a canonical strong triangulation.

As in the case of triangulations (see Proposition 9.63 and Remark 9.65) one canshow that these strong triangulations are natural with respect to exact morphismsand that a strong stable derivator D : Catop → CAT lifts against the forgetfulfunctor from a certain 2-category of strong triangulated categories, exact functors,and exact transformation. We again refer the reader to [GS14a, §13] for moredetails.

10.3. Universal tilting modules. Will be added soon.

104 MORITZ GROTH

Appendix A. Some category theory

In this section we review some basics from category theory and refer the readerto [ML98, Bor94a, Bor94b] for more details.

A.1. Adjunctions. Let us begin by recalling that an adjunction (L,R) : C Dbetween two categories C,D consists of two functors L : C → D and R : D → Ctogether with the choice of an isomorphism

(A.1) φ = φc,d : homD(Lc, d) ∼= homC(c,Rd)

which is natural in c ∈ C and d ∈ D. There are various equivalent ways of encodingsuch a datum, one of them uses the notion of adjunction (co)units. Specializing(A.1) to objects d = Lc and applying φ to the identity morphisms 1: Lc → Lc weobtain a natural transformation η : 1 → RL, the (adjunction) unit. Dually, ap-plying φ−1 to identity morphisms 1: Rd→ Rd we obtain a natural transformationε : LR→ 1, the (adjunction) counit.

These two transformations are not unrelated since they are constructed using theisomorphisms (A.1) and their inverses. First, let us note that we can reconstruct φfrom R and η. In fact, the naturality of φ implies that for every f : Lc → d thereis a commutative diagram

homD(Lc, Lc)φ//

f∗

homC(c,RLc)

(Rf)∗

homD(Lc, d)φ// homC(c,Rd).

Tracing 1: Lc → Lc through the diagram and using the definition of η we seethat φ(f) = (Rf)∗(ηc) = Rf ηc. Similarly, the inverse isomorphism φ−1 can bereconstructed from L and ε, and we have the commutative diagram

(A.2)

homC(RLc,Rd)

η∗

((

homD(Lc, d)

R

66

φ//homC(c,Rd)

φ−1

oo

Lvv

homD(Lc, LRd).

ε∗

hh

In particular, starting with any morphism Lc → d or c → Rd an application ofthe four maps in the clockwise direction gives us back the same morphism. If wespecialize this to the identity morphisms 1: Lc → Lc and 1: Rd → Rd, then wededuce the following two relations between η and ε,

(A.3)

LLη//

=++

LRL

εL

RηR//

=++

RLR

Rε

L, R,

called the triangular identities. It can be shown that an adjunction is determinedby the datum of two functors L : C → D, R : D → C and two natural transformationsη : 1→ RL, ε : LR→ 1 such that the triangular identities are satisfied.


In these notes we denote adjunction units generically by η and adjunction counitsby ε. If we want to emphasize that there are different adjunctions in a given context,then we use obvious notational variations like η′, ε′. We recall the following factabout adjunctions.

Lemma A.4. Let (L,R) : C D be an adjunction.

(i) The left adjoint L is fully faithful if and only if the unit η : 1 → RL is anatural isomorphism. An object d ∈ D lies in the essential image of L if andonly if the counit εd : LRd→ d is an isomorphism.

(ii) The right adjoint R is fully faithful if and only if the counit ε : LR → 1 is anatural isomorphism. An object c ∈ C lies in the essential image of R if andonly if the unit ηc : c→ RLc is an isomorphism.

(iii) The adjunction (L,R) is an adjoint equivalence if and only if L and R arefully faithful if and only if η and ε are natural isomorphisms.

It is immediate from (A.1) that adjunctions can be composed. Given adjunc-tions (L′, R′) : C D and (L′′, R′′) : D E then composing the respective ad-junction isomorphisms we obtain an adjunction (L,R) : C E with L = L′′L′ andR = R′R′′. The adjunction unit and counit of this composite adjunction can becalculated as

η : 1η′→ R′L′

R′η′′L′→ R′R′′L′′L′ and ε : L′′L′R′R′′L′′ε′R′′→ L′′R′′

ε′′→ 1.

Natural transformations between left adjoint functors are closely related to nat-ural transformations between associated right adjoint functors.

Lemma A.5. Let (L,R), (L′, R′) : C D be adjunctions and let α : L → L′ be anatural transformation. There is a unique natural transformation α′ : R′ → R suchthat the following diagram commutes

(A.6)

homD(Lc, d)∼=φ//

OO

α∗

homC(c,Rd)OO

(α′)∗

homD(L′c, d) ∼=

φ′// homC(c,R

′d).

This defines a bijection between natural transformations α : L → L′ and naturaltransformations α′ : R′ → R.

Proof. Let us choose c = R′d and trace the counit ε′d : L′R′d → d through thediagram. This implies α′d = φ(ε′d αR′d), showing that there is at most one suchnatural transformation. Note that using the description of the adjunction isomor-phism φ in terms of R and η as in (A.2) we obtain α′ = Rε′ RαR′ ηR′ and thisshows that we actually constructed a natural transformation α′.

Natural transformations α and α′ making (A.6) commute are conjugate or totalmates. The proof and its converse show that conjugate natural transformationsα, α′ determine each other by means of the relations

(A.7) α′ = Rε′ RαR′ ηR′ : R′ → RLR′ → RL′R′ → R

and

(A.8) α = εL′ Lα′L′ Lη′ : L→ LR′L′ → LRL′ → L′.

106 MORITZ GROTH

The uniqueness of conjugate transformations implies that the construction is com-patible with compositions and identity transformations.

Corollary A.9. If α : L → L′ and α′ : R′ → R are conjugate natural transforma-tions, then α is a natural isomorphism if and only if α′ is a natural isomorphism.

A more systematic way of putting this is as follows. Given two categories C,D,let LAdj(C,D) be the category of left adjoint functors from C to D. An object inLAdj(C,D) is a left adjoint L : C → D and morphisms are natural transformationsbetween left adjoints. Dually, we define the category RAdj(C,D).

Corollary A.10. Let C and D be categories. There is an equivalence of categoriesLAdj(C,D) ' RAdj(D, C)op.

Proof. We describe a functor LAdj(C,D) → RAdj(D, C)op. For each left adjointfunctor L : C → D we choose a right adjoint R : D → C. Given these choices, forevery pair of left adjoint functors L,L′ the passage to conjugate transformationsdefines a bijection between natural transformations L→ L′ and R′ → R, which, asalready observed, is compatible with compositions and identities. Thus, we obtaina fully faithful functor LAdj(C,D) → RAdj(D, C)op. Clearly this functor is alsoessentially surjective and hence an equivalence.

Thus, the specification of a left adjoint is essentially equivalent to the specifi-cation of a right adjoint. We will later need the following more precise statementalong these lines.

Lemma A.11. Let (L,R, η, ε) : C D and (L′, R, η′, ε′) : C D be adjunctionswith the same right adjoint R. There is a unique natural isomorphism α : L → L′

making the following diagrams commute

1η//

η′ ))

RL

∼= Rα

LRε //

αR ∼=

1

RL′, L′R.ε′

JJ

Proof. The existence of a unique natural isomorphism α making the diagram onthe left commute is immediate from the definition of units as certain initial objects.In order to check that also the remaining triangle commutes it is enough to showthat the two compositions have the same image under the natural isomorphismφ : homD(LRd, d) ∼= homC(Rd,Rd). By definition of the adjunction counit ε wehave to show that φ(ε′ αR) = 1. By (A.2) we know that φ(ε′ αR) is given by

RηR→ RLR

RαR→ RL′RRε′→ R.

Since we already showed that Rα η = η′, we see that φ(ε′ αR) = Rε′ η′R whichis the identity 1: R→ R by a triangular identity (A.3).

A.2. Limits and colimits. We assume the reader to have some basic familiar-ity with limits and colimits, including a discussion of terminal objects, products,pullbacks and the dual notions of initial objects, coproducts, and pushouts. Nev-ertheless we recall some key definitions and results about (co)limits, mainly toestablish notation and to prepare the ground for homotopical generalizations asdiscussed in the main body of the text.


Definition A.12. Let A be a small category, let C be a category, and let X : A→ Cbe a functor. A cone on X is a pair (l, α) consisting of an object l ∈ C andmorphisms αa : l → Xa, a ∈ A, such that for every morphism f : a → a′ in A thediagram

lαa //

αa′

Xa

f∗

Xa′

commutes. A morphism of cones (l, α)→ (l′, α′) is a morphism l→ l′ in C suchthat

l

αa

l′α′a

// Xa

commutes for every a ∈ A.

With the obvious composition law and identity morphisms this defines the cat-egory of cones on X. Passing to the formal dual, there is the notion of a cocone(c, β) on X : A→ C. In this case the maps βa : Xa→ c are compatible in the sensethat

Xaβa //

f∗

c

Xa′βa′

>>

commutes for every morphism f : a → a′. Defining morphisms of cocones in theobvious way, associated to a functor X : A→ C there are the two categories

cone(X), cocone(X) ∈ CAT ,

both coming with obvious forgetful functors cone(X)→ C and cocone(X)→ C.Universal examples of (co)cones deserve particular names.

Definition A.13. Let A be a small category, let C be a category, and let X : A→ Cbe a functor.

(i) A limit of X is a terminal object limAX of cone(X).(ii) A colimit of X is an initial object colimAX of cocone(X).

Thus, to emphasize, such a limit is a pair consisting of an underlying object,also denoted by limAX, together with a universal cone which we also refer to as alimiting cone. Similarly, we abuse notation and write colimAX for the underlyingobject of a colimit and refer to the corresponding cocone as a colimiting cocone.

As always with universal construction, if (co)limits exist, then they are unique upto unique isomorphisms and this uniqueness implies functoriality. In the followingproposition we write ∆ = ∆A : C → CA for the diagonal functor which associatesto x ∈ C the constant diagram ∆(x) : A→ C with value x.

Proposition A.14. Let A be a small category and let C be a category.

108 MORITZ GROTH

(i) If every X : A→ C has a limit, then the assignment X 7→ limAX extends toa limit functor limA : CA → C which is right adjoint to ∆A,

(∆A, limA) : C CA.(ii) If every X : A→ C has a colimit, then the assignment X 7→ colimAX extends

to a colimit functor colimA : CA → C which is left adjoint to ∆A,

(colimA,∆A) : CA C.

For particular choices of small categories A, there are special names for theresulting limits and colimits, namely,

(i) final objects and initial objects if A is empty,(ii) products and coproducts if A is discrete (all morphisms are identities),(iii) equalizers and coequalizers if A = (0⇒ 1), and(iv) pullbacks for limits over A = ((0, 1) → (1, 1) ← (1, 0)) and pushouts for

colimits over A = ((0, 1)← (0, 0)→ (1, 0)).

We next recall that these are the basic building blocks for arbitrary (co)limits inthe following precise sense. A category is complete or cocomplete if it admitslimits or colimits of all small diagrams, respectively.

Proposition A.15. (i) A category is complete if and only if it admits equalizersand (small) products.

(ii) A category is cocomplete if and only if it admits coequalizers and (small)coproducts.

Proof. By duality it is enough to take care of the first statement. The task is toshow that an arbitrary limit can be obtained by combining products and equalizers.To this end, let A be a small category and X : A → C such that C admits smallproducts and equalizers. We consider the following pair of parallel morphisms

d0, d1 :∏a∈A

Xa→∏

f : a0→a1

Xa1

in C. Using the universal property of the product∏f : a0→a1 Xa1, in order to define

the morphisms d0, d1 it suffices to specify the respective compositions with theprojections onto the factor Xa1 associated to an arbitrary f : a0 → a1. In the caseof d0 this composition is chosen to be∏

a∈AXa→ Xa0

f∗→ Xa1

while in the case of d1 we simply take the projection∏a∈A

Xa→ Xa1.

The underlying object of the limit limAX is now defined to be the equalizer ofthese two morphisms,

(A.16) limAX = eq

( ∏a∈A

Xa⇒∏

f : a0→a1

Xa1

).

Using the universal cone belonging to this equalizer, we can define the universalcone of limAX to have components

limAX →

∏a∈A

Xa→ Xa′, a′ ∈ A,


where the second morphism is the projection onto the a′-component. We leave itto the reader to verify that this is a limiting cone for X.

In the case of colimits the key step consists of forming the coequalizer

(A.17) colimAX = coeq( ∐f : a0→a1

Xa0 ⇒∐a∈A

Xa)

of two similarly defined morphisms d0, d1 :∐f : a0→a1 Xb→

∐a∈AXa. The details

are left to the reader.

This proposition admits a few variants, one of which we include here. A categoryis finitely complete if it admits finite limits, i.e., limits of diagrams defined onfinite categories (recall that a category is finite if it has finitely many objects andmorphisms only).

Corollary A.18. The following are equivalent for a category C.

(i) The category C is finitely complete.(ii) The category C admits finite products and equalizers.

(iii) The category C admits terminal objects and pullbacks.

Proof. The equivalence of the first two statements follows as above, so that it issuffices to show that terminal objects and pullbacks generate finite products andequalizers. Since equalizers are special cases of pullbacks it remains to constructfinite products. The reader easily checks that products of two objects are obtainedby forming pullbacks over terminal objects, and the case of finite products hencefollows by induction.

We conclude this section by a short discussion of the preservation of (certain) lim-its by functors. Let us consider a functor X : A→ C admitting a colimit colimAXand let F : C → D be a functor such that also F X : A → D admits a colimitcolimA F X. If we apply F to the colimiting cocone of X, then for every f : a→ a′

in A there is a commutative diagram

F (Xa)

// F (colimAX)

F (Xa′)

88

in D, which is to say that we obtain a cocone on F X. The universality of thecolimiting cocone of F X implies that there is a unique morphism

(A.19) colimA(FX)→ F (colimAX)

compatible with these two cocones. We refer to this morphism and a dually definedmorphism

(A.20) F (limAX)→ limA(FX)

as the canonical morphisms.Let F : C → D be a functor between categories admitting colimits of shape A. We

say that F preserves colimits of shape A if for every X : A → C the canonicalmorphism colimA(FX)→ F (colimAX) is an isomorphism.

Definition A.21. (i) A functor between complete categories is continuous ifit preserves all limits.

110 MORITZ GROTH

(ii) A functor between cocomplete categories is cocontinuous if it preserves allcolimits.

(iii) A functor between finitely complete categories is left exact if it preservesfinite limits.

(iv) A functor between finitely cocomplete categories is right exact if it preservesfinite colimits.

(v) A functor between finitely complete and finitely cocomplete categories is ex-act if it is left exact and right exact.

Proposition A.15 and Corollary A.18 have the following variants for functors.

Proposition A.22. (i) A functor between complete categories is continuous ifand only if it preserves equalizers and products.

(ii) A functor between finitely complete categories is left exact if and only if itpreserves equalizers and finite products if and only if it preserves pullbacksand terminal objects.

The following lemma provides a large class of (co)continuous functors.

Lemma A.23. (i) A right adjoint functor between complete categories is con-tinuous and hence left exact.

(ii) A left adjoint functor between cocomplete categories is cocontinuous and henceright exact.

Appendix B. Examples of derivators

B.1. Represented derivators. Will be added soon.

B.2. Homotopy derivators of model categories. In this subsection we includea detailed proof that combinatorial model categories have underlying homotopyderivators. The proof relies on the non-trivial results that diagram categories incombinatorial model categories can be endowed with both the projective and theinjective model structures.

Let M be a Quillen model category and let A ∈ Cat . In general, it is not truethat we can endow the diagram categoryMA with a model structure such that theweak equivalences are precisely the levelwise weak equivalences, i.e., those naturaltransformations f : X → Y : A→M such that all components fa : Xa → Ya, a ∈ A,are weak equivalences in M. However, if one imposes additional conditions on M,then there are two such model structures for arbitrary A ∈ Cat . We begin by theone which is adapted to the study of homotopy colimits and homotopy left Kanextensions.

Definition B.1. LetM be a model category, let A ∈ Cat , and let X,Y : A→M.

(i) A morphism f : X → Y is a projective fibration if it is levelwise fibration.(ii) A morphism f : X → Y is a projective weak equivalence if it is a levelwise

weak equivalence.(iii) A morphism f : X → Y is a projective cofibration if it has the LLP with

respect to acyclic projective fibrations.

If these three classes define a model structure onMA, then we refer to it as theprojective model structure. For the notion of a cofibrantly generated modelstructure we refer the reader to [Hov99]. Let us only mention that most modelstructure showing up in nature (like homotopy theory, homological algebra, and


higher category theory) enjoy this property, so that the following result often ap-plies.

Theorem B.2. Let M be a cofibrantly generated model category and let A ∈ Cat.The projective model structure exists on MA.

The category MA endowed with the projective model structure will be denotedby MA

proj. Recall that the category of simplicial sets can be endowed with thecofibrantly generated Kan–Quillen model structure. In this case, the correspondingprojective model structures on diagram categories were first established in [].

Projective model structures have good functorial properties. For the study ofhomotopy left Kan extensions we observe the following.

Proposition B.3. LetM be a cofibrantly generated model category and let u : A→B be in Cat. The adjunction (u!, u

∗) : MAproj MB

proj is a Quillen adjunction. In

particular, (colimA,∆A) : MAproj M is a Quillen adjunction.

Proof. We have to show that u∗ : MB → MA preserves projective fibrations andacyclic projective fibrations which is immediate since both classes are defined lev-elwise.

Passing to left derived functors, this proposition yields homotopy left Kan ex-tensions and homotopy colimits. Dualizing Definition B.1 we obtain the following.

Definition B.4. LetM be a model category, let A ∈ Cat , and let X,Y : A→M.

(i) A morphism f : X → Y is an injective cofibration if it is levelwise cofibra-tion.

(ii) A morphism f : X → Y is an injective weak equivalence if it is a levelwiseweak equivalence.

(iii) A morphism f : X → Y is an injective fibration if it has the RLP withrespect to acyclic injective cofibrations.

As in the projective case, if the above three classes define a model structure onMA, then we refer to it as the injective model structure and denote the resultingmodel category by MA

inj. One of the first examples was established by Heller in

[] where he shows that diagram categories in simplicial sets admit the injectivemodel structure. In general, as of this writing, we only know that injective modelstructures exist on diagram categories in combinatorial model categories. Let usrecall that a model category is combinatorial if

(i) the model structure is cofibrantly generated and if(ii) the underlying category is locally presentable.

We refer the reader to [Gro10, §3.2] for a short introduction to combinatorial modelcategories and additional references. Here, we only mention that many modelcategories arising in nature are combinatorial.

Theorem B.5. Let M be a combinatorial model category and let A ∈ Cat. Theinjective model structure exists on MA.

Injective model structures enjoy the following dual functorial properties.

Proposition B.6. Let M be a combinatorial model category and let u : A → Bbe in Cat. The adjunction (u∗, u∗) : MB

inj MAinj is a Quillen adjunction. In

particular, (∆A, limA) : MMAinj is a Quillen adjunction.

112 MORITZ GROTH

Proof. The functor u∗ : MB → MA preserves injective cofibrations and acyclicinjective cofibrations since both classes are defined levelwise.

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114 MORITZ GROTH

MPIM, Vivatsgasse 7, 53111 Bonn, Germany

E-mail address: [email protected]

notes on derivators - groth

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