INTRODUCTION TO THE THEORY OF DERIVATORS
MORITZ GROTH
Abstract. In this book we give an introduction to the theory of derivators, a
purely categorical approach to homological algebra and homotopical algebra,
which was proposed independently by Grothendieck, Heller, Franke, and possibly others. While stable derivators provide an enhancement of triangulated
categories, the theory of derivators is more general in that it also applies to
unstable homotopy theories (like the homotopy theory of topological spaces).
As a slogan, derivators are minimal extensions of the more classical derived
categories of abelian categories or homotopy categories of model categories to
a framework with a well-behaved calculus of homotopy limits and homotopy
Kan extensions. As we shall indicate here, this calculus encodes interesting
constructions arising in various areas of pure mathematics.
Starting with a short review of homological algebra including a discussion
of derived categories, we spend some time giving a thorough motivation for
the definition of a derivator. We use a proof of the result that (strong) stable derivators take canonically values in triangulated categories as a pretext to
study various basic aspects of derivators. As we shall see the calculus of homotopy Kan extensions specializes for example to functorial cone constructions
and functorial (higher) octahedron diagrams.
One advantage of derivators over triangulated categories is that derivators
admit exponentials, i.e., given a derivator and a small category, there is a
corresponding derivator of diagrams of that fixed shape. As a consequence, we
can apply derivators to abstract representation theory, and we conclude this
book by an outlook on such applications. While classically in representation
theory of quivers the focus is on representations over fields, some basic results
extend to representations over rings, schemes, differential-graded algebras, ring
spectra or in arbitrary abstract stable homotopy theories arising in algebra,
geometry, and topology.
Contents
1. Introduction and overview
1.1. From classical derived functors to derived categories
1.2. From derived categories to derivators
1.3. Derivators arising in algebra, geometry, and topology
1.4. Stable derivators as an enhancement of triangulated categories
1.5. Towards abstract representation theory
2. Abelian categories and classical derived functors
2.1. Review of abelian categories
2.2. Classical derived functors
2.3. Group cohomology as a derived limit
3. Derived categories of abelian categories
3.1. Towards derived cokernels
Date: April 25, 2015.
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MORITZ GROTH
3.2. Cones and cofibers
3.3. Derived categories and derived functors
3.4. Derived categories as 2-localizations
3.5. Cones as derived cokernels
3.6. Coherent versus incoherent diagrams
3.7. The case of category algebras
4. Derived categories as triangulated categories
4.1. The homotopy category of an abelian category
4.2. Triangulated categories
4.3. Exact morphisms and classical triangulations
4.4. Beyond triangulated categories
5. Kan extensions
5.1. Motivation and definition
5.2. Final functors
5.3. Pointwise Kan extensions
5.4. Basic properties and first examples
6. Basics on derivators
6.1. Prederivators
6.2. Derivators
6.3. Examples of derivators
6.4. Limits versus homotopy limits
7. Homotopy exact squares and Kan extensions
7.1. The calculus of mates
7.2. Homotopy exact squares
7.3. First applications to Kan extensions
8. Pointed derivators
8.1. Basics on pointed derivators
8.2. Suspensions, loops, cofibers, and fibers
8.3. Parametrized Kan extensions
8.4. Cartesian and cocartesian squares
8.5. Iterated cofiber constructions
9. Stable derivators
9.1. Basics on stable derivators
9.2. The preadditivity of stable derivators
9.3. The additivity of stable derivators
9.4. Morphisms and natural transformations
9.5. Strongness of derivators
9.6. Canonical triangulations in stable derivators
9.7. Exact morphisms of stable derivators
10. Towards abstract representation theory
10.1. Morita equivalences
10.2. Derived equivalences
10.3. Strong stable equivalences
10.4. Abstract tilting theory of An -quivers
10.5. Universal tilting modules
Appendix A. Some category theory
A.1. Adjunctions
A.2. Limits and colimits
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INTRODUCTION TO THE THEORY OF DERIVATORS
A.3. Basic 2-categorical terminology
Appendix B. Examples of derivators
B.1. Represented derivators
B.2. Homotopy derivators of model categories
References
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1. Introduction and overview
This book offers an introduction to the theory of derivators, providing one of
the many different approaches to homological algebra and homotopical algebra.
Starting with the context of homological algebra, one goal of derivators is to address
the following problem: the passage from the category Ch(A) of chain complexes
in an abelian category A to the corresponding derived category D(A) results in
a loss of information. In the theory of derivators, this problem is addressed by
also keeping track of derived categories of diagram categories D(AA ), A ∈ Cat,
and various constructions between them, thereby enhancing the classical derived
categories to a more flexible notion.
A very similar situation arises in homotopy theory where the passage from the
category Top of topological spaces to the homotopy category Ho(Top) should be
suitably refined. In this case, this is achieved by also considering homotopy categories of diagram categories Ho(TopA ), A ∈ Cat, together with suitable functors
between them. Among these functors are the homotopy limit and homotopy colimit functors holimA, hocolimA : Ho(TopA ) → Ho(Top) as well as the more general
homotopy Kan extension functors. These functors encode various interesting constructions including reduced supensions, loop spaces, Borel constructions, homotopy fixed points, and spectrifications of prespectrum objects.
It turns out that, besides these two key examples, there is a plethora of additional situations leading to such ‘systems of diagram categories’ A 7→ D(A). By
definition a derivator simply axiomatizes key formal properties of such systems,
leading to a purely categorical framework to study typical situations arising in algebra, geometry, and topology. In particular, homotopy (co)limits and homotopy
Kan extensions are characterized by ordinary universal properties, making them
accessible to elementary categorical techniques.
Derivators were introduced independently by Grothendieck [Gro], Heller [Hel88],
Franke [Fra96], and possibly others. We refer the reader to the webpage mentioned
in [Gro] for a comprehensive bibliography on the subject. While this book offer an
introductory account of basic aspects of the theory of derivators, we intend to come
back to a more systematic treatment somewhere else.
In the remainder of this introduction we try to fill the above paragraphs with
more life. This section is a rather informal account, while the precise mathematics
only begin in §2. We include some references in the remainder of this introduction;
further references can be found, together with the corresponding details, in the
appropriate sections.
1.1. From classical derived functors to derived categories. To begin with
one typical input leading to derivators, we stick to the framework of homological algebra [Wei94, GM03] and, say, consider categories of modules over rings, categories
of abelian sheaves on topological spaces [?, ?], or more general abelian categories.
4
MORITZ GROTH
Classical homological algebra is the study of such abelian categories and additive
functors between them. Typical examples of such functors are tensor product and
hom functors associated to bimodules over rings and direct and inverse image functors associated to continuous maps between topological spaces.
Let us recall that abelian categories essentially axiomatize key properties of the
calculus of kernels, cokernels, and short exact sequences of module homomorphisms.
Correspondingly, there is the class of exact functors between abelian categories,
i.e., functors preserving short exact sequences (these functors are, in particular,
additive). However, typical examples of additive functors, like the ones mentioned
above, fail to be exact. In many cases the functors under consideration preserve
exactness on one side, say such a functor F : A → B sends short exact sequences
0 → X 0 → X → X 00 → 0 in A to exact sequences F X 0 → F X → F X 00 → 0
in B. Since the kernel of the morphism F X 0 → F X is in general non-trivial, we
are interested in systematic tools to study such kernels, and dually for functors
preserving exactness on the other side.
Classical homological algebra offers various techniques, based on projective, injective, and more general F -adapted resolutions, to define (classical) left and right
derived functors. In the case of a right exact functor F : A → B as above this leads
to additive left derived functors Lk F : A → B, k ≥ 0, such that L0 F ∼
= F . Together
with suitable natural connecting homomorphisms δ = δk these functors associate to
a short exact sequence 0 → X 0 → X → X 00 → 0 in A a natural long exact sequence
δ
. . . → (L1 F )(X) → (L1 F )(X 00 ) → F X 0 → F X → F X 00 → 0
in B. These long exact sequences are convenient for both calculational and theoretical purposes. Moreover, they can be characterized by universal properties in
terms of universal δ-functors [Gro57] as we recall in §2.
Implicitly in the above construction we use the trivial observation that every
additive functor F : A → B yields by a levelwise application an additive functor
F : Ch(A) → Ch(B) between the corresponding categories of chain complexes. One
easily checks that the original functor F : A → B is exact if and only if the induced
functor F : Ch(A) → Ch(B) preserves quasi-isomorphisms. (Let us recall that a
chain map C → D is a quasi-isomorphism if the induced maps Hk C → Hk D,
k ∈ Z, between homology objects are isomorphisms.) The point of this different
characterization is that it suggests the following perspective on derived functors:
try to approximate the functor F : Ch(A) → Ch(B) in a universal way by a functor
which preserves quasi-isomorphisms.
The natural domain for such functors are the classical derived categories of
abelian categories [Ver96, KS06]. Ignoring set-theoretic issues for the time being, let us recall that the derived category D(A) of A is obtained from Ch(A)
by inverting the class of quasi-isomorphisms. The resulting localization functor
γ : Ch(A) → D(A) has the defining universal property that every other functor
Ch(A) → C which sends quasi-isomorphisms to isomorphisms factors uniquely
through γ.
To relate this to classical derived functors, we again consider our right exact
functor F : A → B together with the induced functor F : Ch(A) → Ch(B). In
INTRODUCTION TO THE THEORY OF DERIVATORS
5
gereral, it is not possible to find a dashed arrow in
Ch(A)
γ
D(A)
F
=
@
/ Ch(B)
γ
/ D(B)
making the above diagram commute. As we recall in §3, a left derived functor or
left hyperderived functor is a functor LF : D(A) → D(B) together with a universal
natural transformation LF ◦ γ → γ ◦ F . These desired ‘exact approximations’
(and their duals) exist in typical situations and can be constructed by means of
suitable resolutions. The classical derived functors are recovered by passing to
homology objects and using stalk complexes, i.e., there are natural isomorphisms
(Lk F )(X) ∼
= (Hk ◦ LF )(X), k ∈ Z, where for X ∈ A we also denote the associated
complex concentrated in degree zero by the same symbol.
As an illustration of such left derived functors we establish the following example
in §3: given an arbitrary abelian category A, the classical construction of the cone
of a chain map (together with a suitable universal natural transformation) simply
amounts to constructing the left derived cokernel functor. Thus, denoting the
abelian category of morphisms in A by A[1] , we consider the right exact cokernel
functor F = cok : A[1] → A and obtain the cone functor as the corresponding left
derived functor, Lcok = C : D(A[1] ) → D(A).
1.2. From derived categories to derivators. As we just recalled, the natural
domain for derived functors are derived categories, and these derived categories
come with localization functors γ : Ch(A) → D(A). It is important to be aware of
the fact that the categories Ch(A) and D(A) have rather different formal properties.
While the category Ch(A) is again abelian and hence is, in particular, finitely
complete and finitely cocomplete, the category D(A) is typically rather ill-behaved.
Often the only limits and colimits which exist in D(A) are (finite) products and
coproducts. More importantly, even derived limits and derived colimits can neither
be constructed nor characterized using the derived category D(A) only.
A classical way of adressing these bad properties of D(A) is by endowing them
with more structure. Derived categories are often considered as triangulated categories [Pup67, Ver96, Nee01, HJR10], hence together with a triangulation consisting
of an equivalence Σ : D(A) → D(A) and a class of so-called distinguished triangles.
These triangles are diagrams of the form X → Y → Z → ΣX, and this structure is
then subject to a certain list of axioms; see §4. A typical slogan is that the distinguished triangles in D(A) are certain shadows of short exact sequences in Ch(A).
In fact, they essentially arise from iterations of the cone construction.
The theory of triangulated categories is very sucessful and has been used for a
long time in various areas of pure mathematics, including algebra, algebraic geometry, and topology, and this continues until today. Nevertheless, from the very
beginning on (see already the introduction to [Hel68]) it was obvious that the axioms of a triangulated category have certain defects — a reminiscent of the fact that
the passage from Ch(A) to D(A) results in a loss of information. We conclude §4
by a short discussion of some of these defects. One goal of this book is to show
that these problems are avoided by the theory of stable derivators.
6
MORITZ GROTH
As mentioned above, triangulations consist of additional structure imposed on
certain additive categories, like derived categories of abelian categories. The third
object in a distinguished triangle X → Y → Z → ΣX is often referred to as ‘the
cone’ of the morphism X → Y . A crucial observation however is that in general
these cones do not depend functorially on the morphism in D(A), i.e., in general,
there is no cone functor D(A)[1] → D(A). But, as we already saw, there is a
functorial cone construction C = Lcok : D(A[1] ) → D(A). To put it in words:
there is a functorial cone construction defined on the derived category of the arrow
category but not on the arrow category of the derived category. As an upshot, this
suggests that one should refine the passage A 7→ D(A) by also keeping track of the
category D(A[1] ).
To continue building towards derivators we make the following seemingly picky
observation. Let us recall that the cokernel of a chain map f : X → Y is some
‘colimit type construction’. In fact, while the cokernel is not the colimit of the
diagram X → Y (which would be isomorphic to Y ), the cokernel cok(f ) can be
constructed as the pushout on the right in
X
f
/Y
X
f
/Y
0
X
0
f
/Y
/ cok(f ).
(One easily checks that this reduces to the usual universal property of the cokernel.)
The first above step extends a morphism of chain complexes to a span of chain
complexes by adding the morphism X → 0, while the second step forms the pushout
square. From a more category theoretic perspective, this passage to the span and
the passage to the pushout square (as opposed to the pushout corner only) are
examples of Kan extensions [ML98]. In §5 we recall some basics concerning Kan
extensions, including a few examples and first key properties.
As an upshot, this discussion suggests that in order to also encode a construction
of the cone functor C : D(A[1] ) → D(A) at the level of derived categories, we should,
using suggestive notation, also keep track of the derived categories D(Ap ) and
D(A ) together with suitable functors between them. And for more sophisticated
purposes it will be useful to also remember more general derived categories of
diagram categories, say D(AA ) for arbitrary small categories A. Pursuing this
more systematically, one is led to consider a 2-functor
DA : A 7→ D(AA ),
the derivator DA of the abelian category A, thereby also keeping track of derived
restriction functors. The derived category D(A) is recovered by considering diagrams of trivial shape, and one goal of this book is to show that DA enhances D(A)
to a more flexible notion.
As we discuss in §6, the notion of a derivator axiomatizes key properties of 2functors like DA . For example, derived restriction functors have right adjoints and
left adjoints, given by derived Kan extension functors. As special cases, this yields
derived limits and derived colimits. It turns out that the more general derived Kan
extensions can be calculated pointwise in terms of certain derived limit and derived
colimit expressions, and this property is turned into one of the key axioms of a
derivator.
INTRODUCTION TO THE THEORY OF DERIVATORS
7
1.3. Derivators arising in algebra, geometry, and topology. Before we expand on the content of the remaining sections §§7-10, let us step back a little and see
what we discussed so far. We essentially started with the pair (Ch(A), WA ) consisting of the category of chain complexes together with the class of quasi-isomorphisms
and refined the localization process
−1
(Ch(A), WA ) 7→ D(A) = Ch(A)[WA
].
But pairs of the form (Ch(A), WA ) are only one class of examples of categories
together with a chosen class of morphisms ‘which we would like to treat as isomorphisms’.
For example, in homotopy theory [?] one does not want to distinguish topological
spaces as soon as they are connected by a zig-zag of weak homotopy equivalences.
(Let us recall that a continuous map f : X → Y is a weak homotopy equivalence if
the induced maps πk (X, x0 ) → πk (Y, f (x0 )), k ≥ 0, x0 ∈ X, are bijections.) Denoting by WTop the class of all weak homotopy equivalences, the homotopy category of
spaces Ho(Top) is defined as the localization
−1
(Top, WTop ) 7→ Ho(Top) = Top[WTop
].
Again, the category Ho(Top) lacks the existence of most limits and colimits, and,
moreover, the calculus of homotopy limits, homotopy colimits, and homotopy Kan
extensions [BK72] is not visible to that category alone. Similarly to the previous
situation, this can be fixed by refining the passage to Ho(Top) by considering the
(homotopy) derivator of spaces, a suitable 2-functor
A −1
H oTop : A 7→ Ho(TopA ) = TopA [(WTop
) ].
There is a plethora of additional examples of such 2-functors arising in various
areas of algebra, geometry, and topology. For example in stable homotopy theory
one is interested in the stable homotopy category of spectra [Vog70, Ada74], which
can be enhanced by the (homotopy) derivator of spectra. To mention an additional
algebraic example, stable categories arising in stable module theory or group representation theory can be similarly refined by suitable derivator [?]. And as an
explicit example of the derivator of an abelian category, associated to a scheme X
there is the abelian category of quasi-coherent OX -modules. The corresponding
derivator DX enhances D(X), the derived category of the scheme [?].
More abstractly, it can be shown that every Quillen model category has an
underlying homotopy derivator [Cis03] and a similar result is true for complete and
cocomplete ∞-categories [GPS14]. In this book, we do not assume the reader to
have background knowledge on Quillen model categories [Qui67, Hov99, DS95] or
∞-categories [Joy08, Lur09, Gro10]. Consequently, those readers are asked to take
for granted the existence of derivators of abelian categories, spaces, spectra, and
other examples, and instead to focus on how to work with such a structure.
At the same time we also want to satisfy those readers who know about Quillen
model categories. Since a careful, reasonably self-contained proof that arbitrary
Quillen model categories have homotopy derivators is rather involved we refrain
from including such a proof here. As a compromise, in Appendix B we discuss
the construction of homotopy derivators associated to combinatorial model categories [Gro13]. This class of examples is more easily established and it includes
many interesting examples. In particular, all the above-mentioned explicit examples of derivators can be realized that way.
8
MORITZ GROTH
1.4. Stable derivators as an enhancement of triangulated categories. As
already mentioned, in §6 we define derivators, which can be roughly thought of
as systems of diagram categories coming with restriction functors. One key axiom
for a derivator is that these restriction functors have adjoints on both sides, and
these adjoints are referred to as Kan extension functors. As a special case, every
derivator admits abstract limit and colimit functors. These Kan extension functors
are hence a common generalization of
(i) limits, colimits, and Kan extensions in ordinary complete and cocomplete
categories,
(ii) derived limits, derived colimits, and derived Kan extensions between derived
categories associated to abelian categories, and
(iii) homotopy limits, homotopy colimits, and homotopy Kan extensions between
homotopy categories associated to Quillen model categories or ∞-categories.
It turns out that a good deal of basic manipulation rules for limits and Kan
extensions extends from ordinary category theory to the context of an arbitrary
derivator, and hence applies in the derived and the homotopical situation. The
formalism behind these rules is governed by the notion of a homotopy exact square;
see §7.
A derivator is pointed if it admits a zero object, i.e., an object which is simultaneously initial and final. Typical examples of pointed derivators are derivators
associated to abelian categories and the homotopy derivator of pointed topological spaces. In every pointed derivator, one can define suspensions, loops, cofibers,
and fibers, generalizing the classical constructions from homological algebra and
homotopy theory. In §8 we study some aspects of these functors, including iterated
cofiber constructions. For example, we will see that in every pointed derivator there
are suspension-loops and cofiber-fiber adjunctions, and that a threefold iteration
of the cofiber construction amounts to passing to the suspension. The calculus of
such functors relies on basic facts about the derivator version of pullback squares
and pushout squares, which will also be established in §8.
A pointed derivator is stable if a square is a pullback square if and only if it
is a pushout square. It can be shown that this is the case if and only if the
suspension-loops adjunction is an equivalence. Typical examples of stable derivators are derivators associated to abelian categories and the homotopy derivator of
spectra. In §9 we define stable derivators and establish some basic aspects of their
calculus. With the construction of canonical triangulations in mind, we show that
stable derivators are additive. It turns out that suitable shadows of cofiber sequences induce canonical triangulations on the values of (strong) stable derivators.
Moreover, restriction functors and Kan extension functors can be turned into exact
functors with respect to these triangulations. We also include a short discussion
of morphisms of derivators, and show that, more generally, exact morphisms of
derivators induce exact functors with respect to canonical triangulations.
Stable derivators provide an enhancement of triangulated categories and they
fix some of the typical defects of triangulated categories. For example, at the level
of derivators, cone constructions enjoy universal properties and they are functorial.
Similarly, there are functorial octahedron diagrams and even functorial higher octahedron diagrams. As a further instance, let us recall that in general it is not true
INTRODUCTION TO THE THEORY OF DERIVATORS
9
that diagram categories with values in triangulated categories are again triangulated. The corresponding result is true for stable derivators and, as a consequence,
the theory of stable derivators can be applied to representation theory.
1.5. Towards abstract representation theory. The aim of the final section §10
is to give a short outlook on applications of derivators to abstract representation
theory [GŠ14c, GŠ14b, GŠ14a, GŠ15]. Here the word ‘abstract’ is meant to indicate
the following. While classically in representation theory of quivers the focus is on
representations over fields, it turns out that some basic results extend to the context
of an abstract stable derivator.
To expand on this, we begin by recalling that a quiver Q is simply an oriented
graph and that a representation of Q is simply a diagram Q → Mod(k). Here,
k is an arbitrary field and we abuse notation by also denoting the category freely
generated by Q by the same symbol. For quivers with finitely many vertices there
is an equivalence of categories
Mod(k)Q ' Mod(kQ),
where kQ denotes the (k-linear) path algebra of the quiver Q.
Recall that tilting theory [Hap88, AHHK07] is a derived version of Morita theory [Mor58], studying derived equivalences between algebras. We say that two
quivers Q and Q0 are derived equivalent (over k) if there is an exact equivalence of
derived categories of the corresponding path algebras,
∆
D(kQ) ' D(kQ0 ).
Obviously, in principle, this might depend on the ground field under consideration.
Here we take a different perspective on tilting theory and show that some basic
results are formal consequences of stability. To be more specific, we say that two
quivers Q and Q0 are strongly stably equivalent if for every stable derivator D there
is a natural equivalence
0
DQ ' DQ
between the corresponding derivators of representations. Specializing to the stable
derivator of a field, it follows that strongly stably equivalent quivers are derived
equivalent over arbitrary fields.
However, a priori, it is much more restrictive to be strongly stably equivalent,
since we ask for such equivalences for arbitrary stable derivators. In particular, we
also obtain exact equivalences of derived categories in the context of representations over rings, schemes, differential-graded algebras, ring spectra, or many other
stable homotopy theories arising in algebra, geometry, or topology. To keep this
book at a reasonable length, here we restrict ourselves to only indicating that all
Dynkin quivers of type A of a fixed length are strongly stably equivalent. This is
closely related to the existence of canonical higher triangulations in the sense of
Maltsiniotis [Mal05]; see [GŠ14a]. We conclude §10 with an outlook on additional
results from abstract representation theory.
Prerequisites. Mostly for motivational purposes, some background knowledge
from homological algebra and/or homotopy theory is useful. While not strictly
necessary, it would be helpful if the reader is familiar with the elementary yoga of
triangulated categories. We also assume some basic acquaintance with the language
of category theory. For the convenience of the reader, in Appendix A we collect
Reformulate!
10
MORITZ GROTH
some basics from category theory which a reader with less background might want
to refer to as needed.
Acknowledgements. This book is based on various introductory talks on derivators as well as on a course on derivators given by the author at the University
of Bonn during the winter term 2014-2015. I thank the students for their interest and their remarks, comments, and questions during the course. I also thank
Thorge Jensen, Alexander Körschgen, Greg Stevenson, and Jan Šťovı́ček for helpful
comments on earlier versions of this book.
While teaching this course on derivators, the author was supported by the MaxPlanck-Gesellschaft. A good deal of the final writing of the book was done during
a visit at the CRM in Bellaterra. The author is very grateful for the hospitality at
the CRM as well as a research visiting grant from the CRM and the University of
Bielefeld making this visit possible.
2. Abelian categories and classical derived functors
We assume that the reader is familiar with basic homological algebra, including
the construction of classical derived functors and interesting examples. Nevertheless, mostly to set the stage and to smoothen the transition to the theory of
derived categories, here we quickly recall some basics. For more details and examples we refer the reader to the literature; see for example [Rot79], the more
advanced [Wei94, GM03, KS06], and the original [CE99, Gro57].
In §2.1 we recall basic definitions, emphasizing the distinction between properties
and structures. In §2.2 we sketch the construction of classical derived functors and
show that they yield universal δ-functors. In §2.3 we illustrate the notion of derived
limit and derived colimit functors by observing that group cohomology and group
homology, respectively, are special cases of these more general notions.
2.1. Review of abelian categories. To put it as a slogan, homological algebra
is classically the study of abelian categories and derived functors. We begin by
recalling basic definitions concerning abelian categories.
Definition 2.1. A preadditive category is a pointed category with finite biproducts.
Thus, denoting the singleton by ∗, a preadditive category A satisfies the following
three defining axioms.
(i) The category A has a zero object, i.e., an object 0 ∈ A which is both final
and 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 specific, in matrix notation the canonical map in axiom (iii)
is given by
1 0
: X t Y → X × Y.
0 1
INTRODUCTION TO THE THEORY OF DERIVATORS
11
Following standard notation, we write X ⊕ Y for any of X t Y ∼
= X × Y and refer
to it as the biproduct of X and Y .
Remark 2.2.
(i) All axioms of a preadditive category ask for properties. The
only structure is the category itself.
(ii) Any preadditive category A can be canonically endowed with an enrichment in the category AbMon of abelian monoids. In fact, given morphisms
f, g : X → Y , then the sum f + g : X → Y is the composition
f ⊕g
∆
∇
f + g : X → X ⊕ X → Y ⊕ Y → Y,
where ∆ is the diagonal X → X×X and ∇ is the fold map ∇ : Y tY → Y . We
leave it to the reader to check that this defines an abelian monoid structure on
homA (X, Y ) with neutral element 0 : X → 0 → Y and that the composition
is bilinear.
(iii) A closely related perspective on this enrichment is as follows (we will come
back to in §9). In a preadditive category, the fold map ∇ : Y ⊕Y → Y endows
any Y ∈ A with the structure of an abelian monoid object. We leave it as
an exercise to verify that such an abelian monoid structure is equivalent to
specifying a lift of the represented functor homA (−, Y ) : Aop → Set against
the forgetful functor AbMon → Set,
AbMon
:
Aop
/ Set.
homA (−,Y )
Similarly, the diagonal map ∆ : X → X ⊕ X endows any X ∈ A with
the structure of an abelian comonoid object. And one checks that such an
abelian comonoid structure is equivalent to a lift of the corepresented functor
homA (X, −) : A → Set against the forgetful functor AbMon → Set.
Definition 2.3. A preadditive category A is additive if for every X ∈ A the shear
map
1 1
: X ⊕X →X ⊕X
0 1
is an isomorphism.
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 category one can construct the structure of an enrichment in the category Ab of abelian
groups. The good notion of functors between additive categories are additive functors, i.e., functors which preserve zero objects and finite direct sums. Given an
Promise.
12
MORITZ GROTH
additive functor F : A → B it follows that the functor is automatically compatible
with this enrichment, i.e., that the map homA (X, Y ) → homB (F X, F Y ) is a group
homomorphism for every X, Y ∈ A.
Finally, abelian categories are additive categories admitting kernels and cokernels
and such that the first Noether isomorphism theorem is true. In more detail, let us
consider a morphism f : X → Y in an additive category with kernels and cokernels.
(i) The image im(f ) of f is the kernel of the canonical map Y → cok(f ) to the
cokernel, yielding the diagram
im(f ) → Y → cok(f ).
(ii) The coimage coim(f ) of f is the cokernel of the canonical map ker(f ) → X
from 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 ),
f : X → coim(f ) → im(f ) → Y.
In the case of the category of abelian groups A = Ab the map coim(f ) → im(f )
is the usual isomorphism X/ ker(f ) → im(f ) as guaranteed by the first Noether
isomorphism theorem.
Definition 2.5. An abelian category is an additive category satisfying the following 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 additive 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-modules
over a ring R is abelian.
(ii) Given a topological space X, the categories of presheaves or sheaves on X
with values in an abelian category is again abelian.
(iii) Given an abelian category A, the corresponding category Ch(A) of chain
complexes is again abelian.
(iv) A category is abelian if and only if its opposite is abelian.
Q
(v) If Ai , i ∈ I, are abelian categories, then so is the product category i∈I Ai .
In order to state a generalization of Examples 2.6(v) we recall the following
definition.
Definition 2.7. Let C be a category and let A be a small category. The functor category or diagram category C A = Fun(A, C) has as objects the functors
X : A → C and as morphisms the natural transformations α : X → Y .
Lemma 2.8. Let A be an abelian category and let A be a small category. The
functor category AA = Fun(A, A) is again abelian.
Proof. This proof is left as an exercise, the only hint being that limits and colimits
are constructed pointwisely.
INTRODUCTION TO THE THEORY OF DERIVATORS
13
2.2. Classical derived functors. We now focus on additive functors between
abelian categories. Let us recall that a functor F : A → B between additive categories is additive if it preserves zero objects and finite biproducts. As a consequence
there is the following result.
Lemma 2.9. Any additive functor between abelian categories preserves split short
exact sequences.
Proof. This proof is left as an exercise.
While, in general, short exact sequences are not preserved by additive functors,
many functors showing up in nature preserve exactness ‘on one side’, motivating
the following definitions.
Definition 2.10. 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 0 → X → X 00
in A also the sequence 0 → F X 0 → F X → F X 00 is exact.
(ii) The functor F is right exact if for every exact sequence X 0 → X → X 00 → 0
in A also the sequence F X 0 → F X → F X 00 → 0 is exact.
(iii) The functor F is exact if it is left exact and right exact.
Examples 2.11. Let R, S be rings and let M be an R-S-bimodule.
(i) The tensor product M ⊗S − : Mod(S) → Mod(R) is right exact, but, in
general, not left exact.
(ii) The hom functor homR (M, −) : Mod(R) → Mod(S) is left exact, but, in
general, not right exact.
Remark 2.12.
(i) By means of Proposition A.15 one observes that any abelian
category is finitely complete and that it is complete if and only if it admits
products. There is a dual statement concerning colimits and coproducts.
(ii) Proposition A.22 implies that an additive functor between abelian categories
is left exact in the sense of homological algebra (Definition 2.10) if and only
if it is left exact in the sense of category theory (Definition A.21). Together
with Lemma A.23 this yields a formal proof of the positive statements in Examples 2.11.
We next recall the classical construction of left derived functors. Thus, let A, B be
abelian categories, let F : A → B be a right exact functor, and let us for simplicity
assume that A has enough projective objects. Hence, for every object X ∈ A we
can find a projective resolution, i.e., an exact sequence
. . . → P2 → P1 → P0 → X
such that all Pi are projective. Since additive functors preserve zero objects, we
obtain an induced chain complex F P ∈ Ch(B) which is given by
. . . → F P2 → F P1 → F P0 .
A final application of the homology functors Hn : Ch(B) → B, n ≥ 0, concludes the
definition of the (classical) left derived functors Ln F : A → B, i.e., we set
(Ln F )(X) = Hn (F P ),
n ≥ 0.
Since we assume the reader to be familiar with basic homological algebra, we
content ourselves by claiming that the above construction is well-defined, that it can
14
MORITZ GROTH
be extended to additive functors Ln F : A → B, and that, using the right exactness
of F , there is a natural isomorphism L0 F ∼
= F.
As a first justification of the definition of classical left derived functor, let us
recall further that they measure systematically the deviation of F from being left
exact. More precisely, as a consequence of the horseshoe lemma, every short exact
sequence 0 → X 0 → X → X 00 → 0 in A gives rise to a long exact sequence
δ
. . . → (L1 F )(X) → (L1 F )(X 00 ) → F X 0 → F X → F X 00 → 0
Continue here.
in B. In particular, the image im(δ) : (L1 F )(X 00 ) → F X 0 of the connecting homomorphism agrees with the kernel of F X 0 → F X.
An even better justification is given by the observation that classical left derived
functors yield the universal way to measure the deviation from being left exact
(see Theorem 2.15). To make this precise we first recall the following definitions.
Definition 2.13. Let A and B be abelian categories. A homological δ-functor
T from A to B consists of
(i) additive functors Tn : A → B, n ≥ 0, and
(ii) for every short exact sequence 0 → X 0 → X → X 00 → 0 in A of morphisms
δn : Tn (X 00 ) → Tn−1 (X 0 ), n ≥ 1,
such that for every such short exact sequence we obtain a natural long exact sequence
δ
1
. . . → T1 (X) → T1 (X 00 ) →
T0 (X 0 ) → T0 (X) → T0 (X 00 ) → 0.
The naturality means that every morphism of short exact sequences gives rise
to a commutative ladder in B. Thus, the connecting homomorphisms δn of (ii)
assemble to suitable natural transformations.
Definition 2.14.
(i) A morphism of homological δ-functors α : S → T
consists of natural transformations αn : Sn → Tn , n ≥ 0, such that for every
short exact sequence 0 → X 0 → X → X 00 → 0 in A the diagram
Sn (X 00 )
δ
αn
Tn (X 00 )
/ Sn−1 (X 0 )
δ
αn−1
/ Tn−1 (X 0 )
commutes for all n ≥ 1.
(ii) A homological δ-functor T is universal if for every homological δ-functor S
and every natural transformation α0 : S0 → T0 there is a unique morphism
of homological δ-functors α : S → T extending the given α0 .
Here is a justification of the seemingly adhoc construction of classical left derived
functors.
Theorem 2.15. Let F : A → B be a right exact functor between abelian categories
and let A have enough projective objects. The functors Ln F, n ≥ 0, together with
the connecting homomorphisms define a universal homological δ-functor.
Independence?
Proof. We assume that the reader knows that the Ln F together with the connecting
homomorphisms assemble to a homological δ-functor. To prove its universality,
INTRODUCTION TO THE THEORY OF DERIVATORS
15
let T be a homological δ-functor from A to B and let α0 : T0 → F be a natural
transformation.
For n ≥ 1 we assume by induction that a partial morphism of δ-functors consisting of αj , 0 ≤ j < n, has already been constructed and that it is unique. We want
to show that there is a unique way of extending it up to degree n. Given X ∈ A
we choose a short exact sequence
0→K→P →X→0
(2.16)
such that P is a projective object. Since the left derived functors Ln F, n ≥ 1, vanish
on projective objects, by induction assumption we have a commutative (solid arrow)
diagram with exact rows
δ
Tn X
αn−1
∃!
/ (Ln F )X
0
δ
/ Tn−1 P
/ Tn−1 K
/ (Ln−1 F )K
αn−1
/ (Ln−1 F )P.
An easy diagram chase shows that there is a unique dashed morphism αn : Tn X →
(Ln F )X such that the square on the left commutes. Thus, the compatibility with
short exact sequences of the form (2.16) implies that there is at most one way of
defining the desired αn : Tn → Ln F . In particular, the morphism is independent of
the choice of (2.16).
As for the existence it remains to show that these unique morphisms assemble
into natural transformations and that they are compatible with arbitrary short
exact sequences (and not only the ones of the form (2.16)). To verify the naturality
let us consider a morphism f : X → Y in A. Such a morphism can be extended to
a 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 following
diagram in which all squares commute with possibly the exception of the naturality
square on the left,
δ
Tn X
'
/ Tn−1 K
(
/ Tn−1 L
Tn Y
(Ln F )X
/ (Ln−1 F )K
'
(Ln F )Y
δ
(
/ (Ln−1 F )L.
A diagram chase shows that in that square the two possibly different compositions
Tn X → (Ln F )Y agree when composed with δ : (Ln F )Y → (Ln−1 F )L. Since again
this morphism δ is a monomorphism, also the square on the left commutes which
is to say that we constructed a natural transformation αn : Tn → Ln F .
16
MORITZ GROTH
It remains to show that αn is compatible with all short exact sequences. For
this purpose, we note that for any short exact sequence 0 → X1 → X2 → X3 → 0
and any short exact sequence 0 → K → P → X3 → 0 such that P is a projective
object we can extend the identity morphism id : X3 → X3 to a morphism of short
exact sequences, yielding a commutative diagram
0
/K
/P
/ X3
0
/ X1
/ X2
/ X3
/0
=
/ 0.
Similarly to the previous step, associated to this morphism we obtain a diagram
δ
Tn X3
=
(
/ Tn−1 K
)
/ Tn−1 X1
Tn X3
(Ln F )X3
/ (Ln−1 F )K
=
'
(Ln F )X3
δ
(
/ (Ln−1 F )X1
in which all squares with possibly the exception of the front square commute (by
induction assumption and by the previous steps). Precomposing the two morphisms
Tn X3 → (Ln−1 F )X1 with id : Tn X3 → Tn X3 , a short diagram chase implies that
also the front face commutes, concluding the proof.
Dualizing these constructions we are led to right derived functors of left exact
functors F : A → B. In this case, assuming the existence of enough injective objects
in A, every object X ∈ A has an injective resolution 0 → X → I 0 → I 1 → . . ..
If we denote the truncated cochain complex by I, then the n-th (classical) right
derived functor Rn F is defined by
(Rn F )(X) = H n (F I),
n ≥ 0.
These classical right derived functors assemble to universal cohomological δ-functors
in the obvious sense.
For variants of these results and a discussion of examples and applications we
refer the reader to the literature. Here we only mention that in the case of Examn
ples 2.11 we obtain the torsion products TorR
n (M, N ) and Ext-groups ExtR (M, N )
as typical examples of classical left derived and right derived functors, respectively.
2.3. Group cohomology as a derived limit. In this subsection we observe a
close relation between group (co)homology and certain derived limit and derived
colimit functors. This is meant to be a first illustration that derived (co)limit
functors encode interesting constructions. The reader who is not familiar with
group (co)homology can safely skip most of this subsection.
Recall from Lemma 2.8 that for every abelian category A and every small category A the functor category AA is again abelian. Assuming A to have products
in case A is not finite, the limit functor limA : AA → A exists and is part of an
adjunction
(∆A , limA ) : A AA ,
INTRODUCTION TO THE THEORY OF DERIVATORS
17
showing that limA is a left exact functor (see Lemma A.23 and Remark 2.12). If we
moreover assume that AA has enough injective objects (which follows for example
from A having enough injective objects), then we can apply the techniques of §2.2
in order to obtain (classical right) derived limit functors
Rn limA : AA → A,
n ≥ 0.
Making dual assumptions on A, we obtain an adjunction (colimA , ∆A ) : AA A
and associated (classical left) derived colimit functors
Ln colimA : AA → A,
n ≥ 0.
Here we are only interested in the category A = Ab of abelian groups. The
category of abelian groups is complete and cocomplete. Moreover, it can be shown
that all functor categories AbA have enough injectives and projectives so that the
derived (co)limit functors exist.
In our special situation we do not need this general statement since we can argue
more directly. Given a discrete group G, there is an associated category, again
denoted by G, which has one object ∗ only and such that G is the corresponding
endomorphism monoid,
homG (∗, ∗) = G.
We recall that the integral group ring ZG on G is defined as follows. The
underlying abelian group is the free abelian group on G. Denoting by eg the
generator associated to g ∈ G, the ring multiplication is the bilinear extension
of the assignment
eg · eg0 = egg0 , g, g 0 ∈ G.
The reader easily checks that there is an equivalence of categories
(2.17)
AbG ' Mod(ZG),
so that AbG clearly has enough injectives and projectives. We will not distinguish
notationally objects corresponding to each other under this equivalence.
The following two constructions apply to modules M ∈ Mod(ZG) and are central
to the theory of (co)homology of groups. Here we simplify notation and write
gm = eg m, g ∈ G, m ∈ M .
(i) The abelian group of invariants is M G = {m ∈ M | gm = m, g ∈ G} and
this defines a functor (−)G : Mod(ZG) → Ab.
(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 the
coinvariant functor is right exact. A different way of seeing this is as follows. We
observe that the integral group ring ZG comes with an augmentation map
X
X
: ZG → Z :
ri egi 7→
ri ,
which is easily seen to be a ring homomorphism. Restriction of scalars on the left
and on the right applied to the regular bimodule Z ZZ yields
ZG ZZ
and
Z ZZG ,
respectively. We refer to these G-actions as trivial G-actions.
Lemma 2.18. Let G be a discrete group.
18
MORITZ GROTH
(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 Z
is endowed with the trivial right G-action.
Proof. This proof is left as an exercise.
Together with Examples 2.11 this also yields the claimed exactness properties.
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 2.19. Let G be a discrete group, let M ∈ Mod(ZG), and let n ≥ 0.
(i) The n-th group cohomology of G with coefficients in M is the abelian
group
H n (G; M ) = Rn (−)G (M ) ∼
= ExtnZG (Z, M ).
(ii) The n-th group homology of G with coefficients in M is the abelian group
∼ TorZG (Z, M ).
Hn (G; M ) = Ln (−)G (M ) =
n
The equivalences (2.17) offer a different perspective on group (co)homology. To
make it precise, we give the following definition, which is a rather obvious weakening
of the assumption that two functors are naturally isomorphic.
Definition 2.20. Let F1 : C1 → D1 and F2 : C2 → D2 be functors. The functors
F1 , F2 are equivalent if there are equivalences φ : C1 ' C2 , ψ : D1 ' D2 and a
natural isomorphism ψ ◦ F1 ∼
= F2 ◦ φ,
C1
φ '
C2
F1
/ D1
∼
=
' ψ
F2
/ D2 .
The point is that equivalent functors are often equally good for many practical
purposes. For example, many properties of functors (like being fully faithful, essentially surjective, preserving certain (co)limits) are common to equivalent functors.
Proposition 2.21. Let G be a discrete group.
(i) The invariants functor (−)G : Mod(ZG) → Ab and limG : AbG → Ab are
equivalent.
(ii) The coinvariants functor (−)G : Mod(ZG) → Ab and colimG : AbG → Ab
are equivalent.
Proof. More precisely, we show that for every discrete group G the following diagram commutes up to natural isomorphisms
Mod(ZG)
(−)G
(−)G
Ab d
z
∼
=
colimG
'
AbG
∼
=
limG
%
9 Ab.
INTRODUCTION TO THE THEORY OF DERIVATORS
19
In this diagram the vertical functor is the equivalence (2.17). The claim concerning
the triangle on the right follows from the general construction of limits in terms
of equalizers and products; see (A.16). In fact, in this particular case, the general
construction specializes to the equalizer of abelian groups
Y limG M ∼
M ,
= eq M ⇒
g∈G
where one map is the diagonal map while the other map is m 7→ (gm)g∈G . Clearly,
this equalizer is naturally isomorphic to the invariants M G ∈ Ab.
Similarly, using the explicit construction of colimits in (A.17), one observes that
also the triangle on the left commutes up to a natural isomorphism.
Corollary 2.22. Let G be a discrete group and let n ≥ 0.
(i) The functor H n (G; −) : Mod(ZG) → Ab and Rn limG : AbG → Ab are equivalent.
(ii) The functor Hn (G; −) : Mod(ZG) → Ab and Ln colimG : AbG → Ab are
equivalent.
Proof. This is purely formal consequence of Proposition 2.21 and the details are
left to the reader.
Thus, there is a very close relation between group cohomology functors H n (G; −)
and derived limit functors Rn limG and, similarly, between group homology functors
Hn (G; −) and derived colimit functors Ln colimG . Of course, we do not 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 abstract derived
(co)limit functors specialize to well-known constructions which are of independent
interest.
In the remainder of this book we indicate that suitable combinations of derived
limit functors (and, more generally, derived Kan extension functors) encode interesting constructions, and that such constructions are 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 abelian
category [?]. These derived categories can be thought of as rather refined invariants
of the given abelian categories [?]. Important special cases are derived categories
of rings [Hap88] and derived categories of schemes [?, ?], which respectively show
up prominently in representation theory and algebraic geometry. While in this
section we consider derived categories as plain categories only (without additional
structure), in §4 we briefly discuss the classical triangulations on derived categories.
In §3.1 we observe that cokernel functors in general fail to be exact, but that they
are exact on monomorphisms. Following the typical reasoning from homological
algebra, this suggest to ‘redefine’ the cokernel, thereby obtaining cone and cofiber
functors (see §3.2). In §3.3 we recall the definition of derived categories and the
corresponding notion of derived functors. In §3.4 we show that derived categories,
although defined as localizations, enjoy the stronger universal property of being
2-localizations. This allows us to conclude in §3.5 that for every abelian category
the cone functor is the left derived cokernel functor. In general, these cone functors
do not factor through the morphism category of the derived category. To put
20
MORITZ GROTH
this into context, in §3.6 we emphasize that one has to distinguish between derived
categories of diagram categories and diagram categories of derived categories. Using
again the property of a 2-localization, we observe that these categories are related
by underlying diagram functors. In §3.7 we illustrate these underlying diagram
functors in the context of category algebras (like group algebras, path algebras, and
incidence algebras) and observe that for semisimple coefficients they are equivalent
to homology functors. This shows that underlying diagram functors in general
discard relevant information.
3.1. Towards derived cokernels. Let A be an abelian category and let A be a
small category. In §2.3 we mentioned first examples of such categories such that
the limit and colimit functors
limA , colimA : AA → A
exist but fail to be exact. The existence of the adjunction (colimA , ∆A ) implies that
colimA is right exact, but, in general colimA is not left exact, and dually for limA .
In this subsection we discuss exactness properties of (co)products and (co)kernels.
We already observed that abelian categories are finitely cocomplete and that
arbitrary finite colimits can be constructed from finite coproducts and cokernels,
and dually. Hence we consider these two types of constructions independently. Let
us recall that a category is discrete if all morphisms are identity morphisms. Given
a discrete
Q category D and an arbitrary category C, there is a canonical isomorphism
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
M
: AD → A
d∈D
is exact.
Proof. Given an abelian category A, finite coproduct and finite product functors
are part of adjunctions
a
Y
(
, ∆D ) : AD A
and
(∆D ,
) : A AD ,
d∈D
`
d∈D
Q
showing that d∈D is right exact and d∈D is
preadditivity of
` left exact.
L The Q
abelian categories yields natural isomorphisms d∈D ∼
= d∈D ∼
= d∈D , implying
that finite biproducts are exact.
Hence, finite coproducts and finite products in abelian categories are exact. Of
course this lemma already applies to preadditive categories but here we focus on
abelian categories.
Remark 3.2. Infinite (co)products in abelian categories are, in general, not exact.
However, one can impose additional axioms on abelian categories guaranteeing
first of all the existence of infinite coproducts or infinite products and, second, that
infinite coproducts or infinite products are exact. In this order, attributing credit
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, including
a discussion of examples of abelian categories satisfying these stronger axioms; see
[Gro57] or [Fai73, §14].
INTRODUCTION TO THE THEORY OF DERIVATORS
21
We now turn to the case of (co)kernels which, in general, happen to fail to be
exact. To begin with, as a special case of Lemma 2.8, associated to an abelian
category A there is the abelian category A[1] of arrows in A. Here, [1] is the
partially ordered set [1] = (0 < 1), considered as a category. Thus, objects in A[1]
are morphisms X0 → X1 in A while a morphism X → Y is a commutative square
/ Y0
X0
X1
/ Y1 .
Since limits and colimits in A[1] are constructed levelwise (see again the proof
of Lemma 2.8 which was left as an exercise), a short exact sequence in A[1] corresponds precisely to a morphism of short exact sequences in A, i.e., a commutative
diagram
/ X0
/ Y0
/ Z0
/0
0
(3.3)
g
f
0
h
/ Y1
/ X1
/ Z1
/ 0,
such that the rows are short exact sequences.
Lemma 3.4. Let A be an abelian category.
(i) The kernel functor ker : A[1] → A is left exact.
(ii) The cokernel functor cok : A[1] → A is right exact.
(iii) In general, the functors ker, cok : A[1] → A are not exact.
Proof. Since both functors are additive, the first two statements are simply a reformulation of the snake lemma. In fact, given a short exact sequence of morphisms
as in (3.3), 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 any non-zero object X ∈ A
and the corresponding short exact sequence of morphisms
0
/0
/X
0
/X
/X
id
/X
/0
/0
/ 0.
id
id
In fact, by the snake lemma we obtain an induced exact sequence
∼
=
0 → 0 → 0 → X → X → 0 → 0 → 0,
showing that the kernel is not right exact and that the cokernel is not left exact.
We now reformulate the failure of exactness using a slightly different perspective.
Definition 3.5. Let A be an abelian category and let f : X → Y be a morphism
∼
in Ch(A). The chain map f is a quasi-isomorphism, in notation f : X → Y, if it
induces isomorphisms in homology,
∼
=
Hk (f ) : Hk (X) → Hk (Y ),
k ∈ Z.
22
MORITZ GROTH
The class of all quasi-isomorphisms is denoted by W = WA .
We recall that any additive functor F : A → B induces an additive functor
F : Ch(A) → Ch(B) between the corresponding categories of chain complexes, obtained by applying F in all degrees.
Lemma 3.6. An additive functor F : A → B between abelian categories is exact if
and only if the induced functor F : Ch(A) → Ch(B) preserves quasi-isomorphisms.
Proof. This proof is left as an exercise.
This motivates the following definition.
Definition 3.7. Let A, B be abelian categories. A functor F : Ch(A) → Ch(B) is
exact or homotopy-invariant if it preserves quasi-isomorphisms.
Lemma 3.6 is to be seen in contrast with the following result.
Lemma 3.8. Let F : A → B be an additive functor between abelian categories. The
induced functor F : Ch(A) → Ch(B) preserves chain homotopies and, in particular,
chain homotopy equivalences.
Proof. This proof is left as an exercise.
For later applications of Lemma 3.6 to limit and colimit functors, it is convenient to have a refined version of Lemma 2.8. Therein we consider the following
isomorphism of categories. Given a diagram of chain complexes X : A → Ch(A),
one easily checks that
Yk (a) = X(a)k ,
k ∈ Z, a ∈ A,
defines a chain complex Y ∈ Ch(AA ), and conversely. Moreover, this obviously
yields an isomorphism of categories
Ch(A)A ∼
= Ch(AA ) :
(3.9)
0
X ↔ Y.
00
Given two functors X , X : A → Ch(A) and a morphism X 0 → X 00 in Ch(A)A ,
i.e., a natural transformation α : X 0 → X 00 , we say that α is a levelwise quasiisomorphism if the chain maps αa : X 0 (a) → X 00 (a), a ∈ A, are quasi-isomorphisms.
A
The class of all levelwise quasi-isomorphims is denoted by WA
.
Lemma 3.10. Let A be an abelian category and let A be a small category.
(i) The functor category AA = Fun(A, A) is abelian.
A
(ii) The isomorphism of categories (3.9) identifies the class WA
of levelwise quasiisomorphisms with the class WAA of quasi-isomorphisms.
Proof. This proof is left as an exercise.
Let us recall that kernels and cokernels in Ch(A) are calculated degreewise. Thus,
Lemma 3.4, Lemma 3.6, and Lemma 3.10 together imply the following warning.
Warning 3.11. Let A be an abelian category. In general, the functors
cok, ker : Ch(A)[1] → Ch(A)
are not homotopy-invariant, i.e., they do not send levelwise quasi-isomorphisms to
quasi-isomorphisms.
INTRODUCTION TO THE THEORY OF DERIVATORS
23
Despites this warning, there are the following positive statements concerning
levelwise quasi-isomorphisms X → Y in Ch(A)[1] , i.e., commutative squares of
chain complexes as in
f
X0
(3.12)
∼
Y0
/ X1
g
∼
/ Y1 .
Lemma 3.13. Let A be an abelian category and let (3.12) be a levelwise quasiisomorphism in Ch(A)[1] .
(i) If f and g are monomorphisms, then the induced map cok(f ) → cok(g) is a
quasi-isomorphism.
(ii) If f and g are epimorphisms, then the induced map ker(f ) → ker(g) is a
quasi-isomorphism.
Proof. We give a proof of (i) and observe that there is a morphism of short exact
sequences of chain complexes
0
/ X0
f
∼
0
/ Y0
/ X1
∼
g
/ Y1
/ cok(f )
/0
/ cok(g)
/ 0.
The induced long exact sequences in homology and the 5-lemma allow us to conclude
the proof of (i). The case of (ii) is dual.
Thus, the cokernel and the kernel functors are exact on certain morphisms of
chain complexes. The idea behind the construction of derived (co)kernel functors
is now simply to quasi-isomorphically approximate arbitrary morphisms of chain
complexes by these good ones.
3.2. Cones and cofibers. In this subsection we recall some classical constructions
from homological algebra. This includes the seemingly adhoc construction of the
cone functor, a certain ‘corrected version’ of the cokernel functor. While a first
justification of this construction is provided by Proposition 3.23, a more conceptual
one will be given in §3.5. Everything can be dualized to yield a similar ‘correction’
of the kernel functor.
To motivate these constructions we include the following different description of
the cokernel. Let f : X0 → X1 be a morphism in an abelian category A. We note
that the cokernel of f can also be constructed by considering pushout squares
X0
(3.14)
0
f
/ X1
/ cok(f ).
In fact, the universal property of this pushout square is easily checked to reduce to
the usual universal property of the cokernel. Thus, although the cokernel cok(f ) is
not the colimit of the diagram f : X0 → X1 (which would be isomorphic to X1 ) it
24
MORITZ GROTH
is the colimit of the span (0 ← X0 → X1 ). We will come back to this intermediate
step later.
Now, as recalled in Warning 3.11, in general the cokernel functor is not exact.
Using this different description of the cokernel, a classical way of fixing this problem
is by replacing the morphism X0 → 0 in (3.14) quasi-isomorphically by a nice
inclusion, namely by the inclusion i : X0 → CX0 of X0 in its cone CX0 . We
quickly recall this construction and, since we also have a use for further variants of
such constructions, we allow us to include a minor digression.
Digression 3.15. Let FZ denote the category of finitely generated, free abelian
groups and let Chb (FZ ) be the category of bounded chain complexes in FZ . We
recall that for every additive category A there is a two-variable functor
(3.16)
⊗ : Chb (FZ ) × Ch(A) → Ch(A)
which is additive in both variables separately.
To begin with there is a similar biadditive action ⊗ : FZ ×A → A which is already
determined by asking that there is a natural isomorphism Z⊗X ∼
= X, X ∈ A. Thus,
for a free abelian group F of rank r, there is a natural isomorphism F ⊗ X ∼
= r · X,
where r · X denotes the r-fold direct sum X ⊕ . . . ⊕ X.
This action can be extended to chain complexes by the following ‘convolution
type construction’. Given chain complexes F ∈ Chb (FZ ) and X ∈ Ch(A) we define
the underlying graded object of F ⊗ X ∈ Ch(A) by
M
(F ⊗ X)k =
Fp ⊗ Xq , k ∈ Z.
p+q=k
(This direct sum is finite by the boundedness assumption on F .) The differentials
(F ⊗ X)k → (F ⊗ X)k−1 are obtained from those on F and X using the usual
Koszul sign convention. (Recall that this convention says that, in differential graded
algebra, ‘whenever the order of two homogeneous elements of degree p and q is
swapped, a sign (−1)pq has to appear’.)
We illustrate the action (3.16) by the following examples.
e cell ([0, 1]) ∈ Chb (FZ ) be the reduced cellular chain
Examples 3.17.
(i) Let I = C
complex of the interval endowed with the CW structure consisting of two
0-cells and one 1-cell only. More explicitly, I0 = I1 = Z, the differential
I1 → I0 is the identity, and all Ik , k 6= 0, 1, vanish. We call I the interval
and 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) is
the cone functor. (If you did not see a definition of C before, then you can
simply take this construction as a definition and we suggest as an exercise to
unravel it.) The notation is of course motivated from similiar constructions
for pointed topological spaces, where smashing with the interval yields the
(reduced) cone construction.
e cell ({0, 1}) ∈ Chb (FZ ) be the reduced cellular chain complex of the
(ii) Let S 0 = C
0-sphere endowed with the CW structure conisting of two 0-cells only. Thus,
S 0 is the stalk complex Z. For X ∈ Ch(A) there is a natural isomorphism
S0 ⊗ X ∼
= X.
(iii) The continuous map {0, 1} → [0, 1] yields a map S 0 → I in Chb (FZ ) and
hence a natural transformation S 0 ⊗ − → I ⊗ − of functors Ch(A) → Ch(A).
INTRODUCTION TO THE THEORY OF DERIVATORS
25
Under the above natural isomorphisms, for X ∈ Ch(A) this yields the usual
natural inclusion i : X → CX.
(iv) Let S 1 ∈ Chb (FZ ) be the reduced cellular chain complex of the 1-sphere
considered as a quotient CW complex [0, 1]/{0, 1}. Thus, S 1 is the abelian
group Z considered as a complex concentrated in degree 1. For X ∈ Ch(A)
there is a natural isomorphism S 1 ⊗ 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 the
vertical morphism on the left in (3.14) by X0 → CX0 and hence consider the
pushout diagram
(3.18)
/ X1
f
X0
cof(f )
i
/ Cf.
CX0
Definition 3.19. Let A be an abelian category and let f : X0 → X1 be in Ch(A).
(i) The chain complex Cf in (3.18) is the cone of f .
(ii) The chain map cof(f ) : X1 → Cf in (3.18) is the cofiber of f .
It is immediate that both the cone and the cofiber are functorial in f , i.e., we
have functors
C : Ch(A)[1] → Ch(A)
and
cof : Ch(A)[1] → Ch(A)[1] .
Lemma 3.20. Let A be an abelian category and let X ∈ Ch(A).
(i) The cone CX is contractible.
(ii) The cone of an isomorphism is contractible.
Proof. Since the interval I is contractible and the construction (3.16) is additive in
the first variable it follows from Lemma 3.8 that I ⊗ X ∼
= CX is contractible. If
f : X0 → X1 is an isomorphism, then the construction (3.18) yields an isomorphism
CX0 ∼
= Cf . Thus, by the first part, Cf is contractible.
To prove the homotopy invariance of the cone construction we collect the following lemma.
Lemma 3.21. Let A be an abelian category and let us consider a square
X0
i
p
j
X2
/ X1
q
/X
in A. The square is a pushout square if and only if the sequence
(i,j)t
(p,−q)
X0 → X1 ⊕ X2 → X → 0
is exact.
Proof. This proof is left as an exercise.
26
MORITZ GROTH
In the special case of the defining pushout square (3.18) of the cone, we hence
obtain a short exact sequence
0 → X0 → X1 ⊕ CX0 → Cf → 0,
(3.22)
since the first map is clearly also a monomorphism (see Examples 3.17(iii)). The
reader easily checks that this short exact sequence is functorial in the morphism
f : X0 → X1 . While this construction allows us to give a partial justification of the
cone construction (see again Lemma 3.6), a better one will be given in §3.5 (see
Theorem 3.42).
Proposition 3.23. 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 (3.12) such that the vertical
maps are quasi-isomorphisms, i.e., a levelwise quasi-isomorphism f → g. Associated
to such a levelwise quasi-isomorphism we obtain (by the functoriality of (3.22)) a
morphism of short exact sequences
0
/ X0
/ X1 ⊕ CX0
/ Cf
/0
/ Y1 ⊕ CY0
/ Cg
/ 0.
∼
0
/ Y0
Since the objects CX0 , CY0 have trivial homology objects (Lemma 3.20), any
chain map between them is a quasi-isomorphism. As finite direct sums of quasiisomorphisms are again quasi-isomorphisms, in the above diagram also the second
vertical morphism is a quasi-isomorphism. The long exact sequence in homology
and the 5-lemma imply that the morphism Cf → Cg is a quasi-isomorphism, concluding the proof.
3.3. Derived categories and derived functors. As a preparation for the definition of derived categories, let us reconsider the construction of classical derived
functors between abelian categories. Thus, let F : A → B be a right exact functor
between abelian categories and let A have enough projective objects. The key steps
in the construction of the Lk F, k ≥ 0, consisted of choosing projective resolutions
(3.24)
. . . → P2 → P1 → P0 → X
for X ∈ A, applying F to the truncated chain complex
P = (. . . → P2 → P1 → P0 ),
INTRODUCTION TO THE THEORY OF DERIVATORS
27
and then passing to homology. A different way of saying that (3.24) is a resolution
is to say that the commutative diagram
..
.
..
.
P2
/0
P1
/0
(3.25)
P0
/X
defines a quasi-isomorphism : P → X where X also denotes the associated complex concentrated in degree 0, i.e., the complex (. . . → 0 → X → 0 → . . .) ∈ Ch(A).
This simple trick of rewriting the horizontal diagram (3.24) in a vertical fashion as
in (3.25) is an important step. In fact, it suggests that, secretely in the construction
of classical derived functors, we already passed from objects in the abelian category
(like modules over a ring) to chain complexes. From a more abstract perspective
this amounts to passing from a good old category to a ‘homotopy theory’, namely
to the ‘homotopy theory’ (Ch(A), WA ) consisting of the category Ch(A) together
with the class W = WA of quasi-isomorphisms. We say a bit more about ‘abstract
homotopy theories’ in §6.3 and Appendix B.
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 reasons
we would like to invert the quasi-isomorphisms WA in Ch(A). For example this
would allow us to identify all projective resolutions of a fixed object X ∈ A.
Definition 3.26. Let A be an abelian category. The derived category D(A) of
A is the localization of Ch(A) at the class WA of quasi-isomorphisms,
−1
D(A) = Ch(A)[WA
].
By the very definition, the derived category is hence a pair (D(A), γ) consisting
of a category D(A) and a localization functor γ : Ch(A) → D(A) which sends quasiisomorphisms to isomorphisms and which is initial with this property. Thus, every
functor F : Ch(A) → C which sends quasi-isomorphisms to isomorphisms factors
uniquely through D(A),
Ch(A)
γ
D(A).
∀F
/ C,
<
F (WA ) ⊆ IsoC ,
∃!
Warning 3.27. There is a typical warning in the context of such localizations,
namely, that, in general, such localizations do not exist (at least not in a fixed
universe). A different way of saying this is that such localizations can always be
constructed (see for example [GZ67]), but, in general, the resulting categories are
Promise.
Add more motivational
comments?
28
MORITZ GROTH
not locally small, i.e., there might be proper classes of morphisms between certain
objects. 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 be
justified.
(ii) One can restrict attention to the large class of Grothendieck abelian categories
since in that case all these set-theoretic problems happen to disappear. See
for example [Fai73, §14] for a discussion of Grothendieck abelian categories.
(iii) The language of Quillen model categories [Qui67, Hov99, DS95] is an abstract
framework which, among many other things, allows us to deal with such
problems. As soon as the pair (Ch(A), WA ) can be extended to a Quillen
model category, these set-theoretic problems automatically disappear. And,
in fact, in the case of Grothendieck abelian categories such model structures
always exist; see [Hov01].
(iv) If one imposes certain boundedness conditions on the chain complexes and
assumes the existence of sufficently many injective or projective objects, then
one can show that corresponding derived categories are equivalent to quotient categories of suitable categories of chain complexes. In particular, these
categories are again locally small categories.
For an actual example of such set-theoretic problems we refer to an old example of
Freyd (see the reprint [Fre03]) as discussed in [Kra10, Example 4.15].
For the remainder of this section we assume that the derived categories under
consideration exist. (Thus, we essentially follow approach (i).)
Examples 3.28.
(i) Let R be a ring and let Mod(R) be the Grothendieck abelian
category of R-modules. The derived category of the ring R is the derived
category
D(R) = D(Mod(R)) = Ch(R)[WR−1 ].
In particular in the case of finite-dimensional algebras over fields, such derived
categories have been studied a lot in representation theory; see for example [?,
?, ?].
(ii) Let X be a scheme, let Mod(X) be the category of quasi-coherent OX modules, and let Ch(X) = Ch(Mod(X)) be the corresponding category of
unbounded cochain complexes. The derived category of the scheme X
is the derived category
−1
D(X) = D(Mod(X)) = Ch(X)[WX
].
References in this case include [?, ?, ?, ?].
Corollary 3.29. Let A, B be abelian categories and let F : Ch(A) → Ch(B) be an
exact functor. There is a unique functor F : D(A) → D(B) such that
Ch(A)
F
γ
D(A)
/ Ch(B)
γ
F
/ D(B)
commutes. This applies, in particular, if F comes from an exact functor F : A → B.
INTRODUCTION TO THE THEORY OF DERIVATORS
29
Proof. Since the functor γ ◦ F : Ch(A) → D(B) sends quasi-isomorphisms to isomorphisms this is immediate from the defining universal property of the localization
functor γ : Ch(A) → D(A) (see Definition 3.26). The special case coming from an
exact functor A → B is simply Lemma 3.6.
A first trivial example comes from the homology functors. Given an abelian
category A, we denote by AZ the category of Z-graded objects in A. (Thus, we
consider Z as a discrete category and AZ is the corresponding diagram category.)
Taking all homology functors Hk : Ch(A) → A, k ∈ Z, at once we obtain a functor
H∗ : Ch(A) → AZ .
Example 3.30. Let A be an abelian category. The functor H∗ : Ch(A) → AZ sends
quasi-isomorphisms to isomorphisms and hence induces a functor H∗ : D(A) → AZ .
The following example is important.
Example 3.31. Let A be an abelian category. The cone C : Ch(A)[1] → Ch(A)
induces by Proposition 3.23 and Lemma 3.10 a functor
C : D(A[1] ) → D(A).
Thus, to emphasize, there is a functorial cone construction defined on the derived
category 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 exact
or right exact. In such situations one has to work harder in order to obtain certain
universal induced functors at the level of derived categories.
Definition 3.32. Let A, B be abelian categories and let F : Ch(A) → Ch(B) be
a functor. A left derived functor of F is a pair (LF, ) consisting of a functor
LF : D(A) → D(B) and a natural transformation : (LF )γ → γF such that for
every other such pair (G : D(A) → D(B), α : Gγ → γF ) there is a unique natural
transformation α0 : G → LF such that
α = ◦ (α0 γ) : Gγ → (LF )γ → γF.
Thus, a left derived functor can be depicted by a square populated by a natural
transformation as in
F /
Ch(B)
Ch(A)
γ
D(A)
@H
LF
γ
/ D(B)
and the universal property of a left derived functor can be illustrated as follows
Ch(A)
γ
D(A)
/ Ch(B)
F
α
@H
G
γ
/ D(B)
=
γ
D(A)
/ Ch(B)
F
Ch(A)
@H
LF
⇑α0
γ
)
5 D(B).
G
Remark 3.33.
(i) The idea behind the definition is that the left derived functor
(LF, ) is a universal approximation of F by an exact functor.
30
MORITZ GROTH
(ii) Since derived functors are defined by a universal property, once one knows
that derived functors exist, then they are unique up to a canonical natural
isomorphism. The existence of derived functors is often settled using suitable
resolutions.
(iii) There is the dual notion of a right derived functor consisting of a functor
RF : D(A) → D(B) and a universal natural transformation η : γF → (RF )γ
such that (RF, η) is suitably initial.
(iv) We warn the reader that Definition 3.32 differs slightly from the one found in
typical books on homological algebra; see [Wei94, GM03]. While in those references in the definition of a left derived functor both F and LF are assumed
to be exact functors of triangulated categories (see §4), here we define derived
functors as it is typically done in the theory of Quillen model categories.
In §3.5 we show that the functor C : D(A[1] ) → D(A) constructed in Example 3.31 together with a suitable natural transformation is a left derived functor of
the cokernel cok : Ch(A[1] ) → Ch(A) (see Theorem 3.42). As a preparation we establish in §3.4 that derived categories enjoy a seemingly stronger universal property
than that one of a localization.
3.4. Derived categories as 2-localizations. Derived categories of abelian categories are defined as suitable localizations (Definition 3.26) and as such they satisfy
a universal property. In this subsection we show that they actually enjoy a stronger
universal property, namely that derived categories are 2-localizations. This seemingly technical result has interesting consequences as we see here and in §§3.5-3.6.
We begin collecting the following lemma.
Lemma 3.34. Let C be a category, let A be a small category, and let X, Y : A → C
be functors. There is a bijection between
(i) the set of natural transformations α : X → Y ,
(ii) the set of functors H : A × [1] → C such that H(−, 0) = X and H(−, 1) = Y ,
and
(iii) the set of functors K : A → C [1] such that 0∗ K = X and 1∗ K = Y .
Proof. This proof is left to the reader.
Let us consider an abelian category A together with the localization functor
γ : Ch(A) → D(A). For every category C, precomposition of functors with γ yields
a map
γ ∗ : hom(D(A), C) → hom(Ch(A), C).
If we denote by homW(Ch(A), C) ⊆ hom(Ch(A), C) those functors which send quasiisomorphisms to isomorphisms, then the defining universal property of γ is that for
all categories C precomposition with 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 this
purpose, we denote by
FunW(Ch(A), C) ⊆ Fun(Ch(A), C)
the full subcategory of the functor category spanned by all functors which send
quasi-isomorphisms to isomorphisms.
INTRODUCTION TO THE THEORY OF DERIVATORS
Lemma on path objects
31
Proposition 3.35. Let A be an abelian category. The localization γ : Ch(A) →
D(A) is a 2-localization, i.e., for every category C precomposition with γ induces
an isomorphism of categories
∼
=
γ ∗ : Fun(D(A), C) → FunW(Ch(A), C).
In particular, for every pair of functors F 0 , G0 : D(A) → C there is a bijection
γ ∗ : hom(F 0 , G0 ) → hom(F 0 γ, G0 γ).
Proof. By definition of the localization, γ ∗ is bijective on objects and it hence
suffices to show that γ ∗ is also fully faithful. Let F, G ∈ FunW(Ch(A), C) with
corresponding factorizations F = F 0 γ, G = G0 γ and let α : F → G be a natural transformation. Note that α : F → G is equivalently specified by the functor
α̃ : Ch(A) → C [1] defined by
α̃(X) = (αX : F X → GX).
If w : X → Y is a quasi-isomorphism, then, since F, G send quasi-isomorphisms to
isomorphisms, the vertical morphisms in
FX
∼
= Fw
FY
/ GX
α
Gw ∼
=
/ GY,
α
are isomorphisms, showing that α̃ ∈ FunW(Ch(A), C [1] ). The universal property of
γ implies that there is a unique functor αe0 : D(A) → C [1] such that
α̃
Ch(A)
/ C [1]
I
γ
D(A)
e0
∃!α
commutes. But αe0 now corresponds to a natural transformation α0 : F 0 → G0
defined by
0
αe0 (X) = (αX
: F 0 X → G0 X), X ∈ D(A).
The reader easily checks that this is the unique natural transformation F 0 → G0
satisfying α0 γ = α.
As a first illustration of this 2-categorical universal property we collect the following result. This result makes precise the slogan that ‘exact functors (Definition 3.7)
do not have to be derived’; see again Corollary 3.29.
Corollary 3.36. Let A, B be abelian categories, let F : Ch(A) → Ch(B) be exact,
and let F : D(A) → D(B) be the unique induced functor such that
Ch(A)
F
γ
D(A)
commutes.
/ Ch(B)
γ
F
/ D(B)
32
MORITZ GROTH
(i) The pair (F : D(A) → D(B), id : F ◦ γ → γ ◦ F ) is a left derived functor of F .
(ii) The pair (F : D(A) → D(B), id : γ ◦F → F ◦γ) is a right derived functor of F .
Proof. We take care of statement (i), the case of (ii) is dual. To this end, let
(G : D(A) → D(B), α : G ◦ γ → γ ◦ F )
be an instance of a pair among which a left derived functor is supposed to be terminal (see Definition 3.32). Since γ ◦ F = F ◦ γ, the natural transformation α reads
as α : G ◦ γ → F ◦ γ. By Proposition 3.35 there is a unique natural transformation
α0 : G → F such that
α = α0 γ = id ◦ (α0 γ),
i.e., we have
α
γ
/ Ch(B)
F
Ch(A)
D(A)
@H
Ch(A)
γ
=
/ D(B)
G
id
γ
D(A)
/ Ch(B)
F
@H
F
⇑α0
γ
)
5 D(B).
G
Thus, (F, id) enjoys the universal property of a left derived functor.
Using that γ : Ch(A) → D(A) is a 2-localization, in the above proof we could
avoid the explicit construction of a natural transformation between functors defined
on derived categories. This is convenient since, in general, morphisms in derived
categories are a bit tricky to understand. We will see a further instance of such a
situation in the next subsection.
3.5. Cones as derived cokernels. We now take a closer look at the cone construction C : Ch(A[1] ) → Ch(A) as defined via the pushout square (3.18). Clearly,
CX0 → 0 induces a morphism of spans (CX0 ← X0 → X1 ) → (0 ← X0 → X1 ),
and hence an induced morphism of pushout diagrams
/ X1
f
X0
=
#
=
%
/ X1
X0
/ Cf
CX0
$ / cok(f ).
" 0
The exact sequence of Lemma 3.21 is easily seen to be natural in the pushout
diagram, hence we obtain a natural morphism of exact sequences
0
(3.37)
/ X1 ⊕ CX0
/ X0
= ∼
X0
∼
f
/ X1
and, in particular, a natural transformation
(3.38)
φ : Cf → cok(f ).
/ Cf
/0
/ cok(f )
/ 0,
INTRODUCTION TO THE THEORY OF DERIVATORS
33
Remark 3.39. We observe that the first two components of the natural morphism
of exact sequences (3.37) define a natural levelwise quasi-isomorphism
/ X1 ⊕ CX0
X0
(3.40)
= ∼
X0
∼
/ X1 .
f
Moreover, the domain of this quasi-isomorphism is a monomorphism and we know
by Lemma 3.13 that the cokernel preserves quasi-isomorphisms between monomorphisms. We refer to this by saying that (3.40) is a functorial resolution on
Ch(A[1] ) which is adapted to the functor cok : Ch(A[1] ) → Ch(A). It follows that
the morphism φ : Cf → cok(f ) is a quasi-isomorphism for all monomorphisms f .
As a preparation for the theorem we recall that, by Example 3.31, the cone
induces a functor C : D(A[1] ) → D(A). The natural transformation (3.38) yields a
natural transformation : C ◦ γ → γ ◦ cok given by
(3.41)
γφ
: C ◦ γ = γ ◦ C → γ ◦ cok.
We also collect the corresponding dual statement concerning the fiber functor
F : D(A[1] ) → D(A). Since its construction is dual to that of C : D(A[1] ) → D(A),
we leave the details to the reader.
Theorem 3.42. Let A be an abelian category.
(i) The cone functor C : D(A[1] ) → D(A) together with defined in (3.41) is a
left 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 a
right derived functor of the kernel ker : Ch(A[1] ) → Ch(A).
Proof. We have to show that (C, ) satisfies the universal property of Definition 3.32.
For this purpose let G : D(A[1] ) → D(A) be a functor and let α : G ◦ γ → γ ◦ cok
be a natural transformation,
Ch(A[1] )
γ
D(A[1] )
cok
AI
G
/ Ch(A)
γ
/ D(A).
We claim that it is enough to show that there is a unique natural transformation
ψ : G ◦ γ → C ◦ γ such that
α = ◦ ψ : G ◦ γ → C ◦ γ → γ ◦ cok.
In fact, by Proposition 3.35 we then deduce that there is a unique natural transformation α0 : G → C such that ψ = α0 γ, which immediately yields the desired
universal property of (C, ).
34
MORITZ GROTH
f
In order to construct such a transformation ψ we consider (X0 → X1 ) ∈ Ch(A[1] )
and associate to it the following commutative diagram
Gγ(X0 → X1 ⊕ CX0 )
∼
=
α
∼
=
Gγ(X0 → X1 )
/ γcok(X0 → X1 ⊕ CX0 ) o
α
/ γcok(X0 → X1 ) o
Cγ(X0 → X1 ⊕ CX0 )
∼
=
Cγ(X0 → X1 ).
In this diagram, the vertical morphisms are induced by the resolutions adapted
to the cokernel (see Remark 3.39) and it hence follows that the two outer vertical
morphisms are isomorphisms. Moreover, both squares commute by naturality of α
and , respectively. Finally, the top horizontal morphism
= γφ : Cγ(X0 → X1 ⊕ CX0 ) → γcok(X0 → X1 ⊕ CX0 )
is an isomorphism since X0 → X1 ⊕ CX0 is a monomorphism; see again Remark 3.39. Thus, the above commutative diagram shows us that there is at most
one such natural transformation ψ. These morphisms Gγ(f ) → Cγ(f ) actually assemble to a natural transformation Gγ → Cγ as one can check from the naturality
of the above diagram in (f : X0 → X1 ).
Note that Proposition 3.35 again allowed us to avoid the explicit construction of
a natural transformation between functors defined on derived categories.
Remark 3.43. We include a short philosophical remark concerning the construction
of the cone as a derived version of the cokernel. The cokernel is a categorical
construction which amounts to ‘collapsing’ the image of a morphism. As observed
in Warning 3.11 the cokernel is not an exact or homotopy invariant construction in
that 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 the
defining pushout square (3.18). In fact, thinking geometrically, by glueing a cone
on the image of the morphism we add the potential of collapsing it since we can
now push it to the apex of the cone. Similar comments also apply to other derived
(co)limit constructions and this picture also extends to the context of topological
spaces and homotopy (co)limits.
3.6. Coherent versus incoherent diagrams. In this subsection we briefly discuss differences between coherent and incoherent diagrams. The main point of the
theory of derivators is that many interesting constructions are available at the level
of coherent diagrams while, in general, this is not the case for incoherent diagrams.
To already get used to the terminology from the theory of derivators, we make the
following definition. Let us recall that given an abelian category B the localization
functor γ : Ch(B) → D(B) can be chosen to be the identity on objects. Hence,
objects in the derived category are again simply chain complexes.
Definition 3.44. Let A be an abelian category and let A be a small category.
(i) An object in D(AA ) is a coherent diagram of shape A.
(ii) An object in D(A)A is an incoherent diagram of shape A.
Thus, a coherent diagram is simply a chain complex X ∈ Ch(AA ), i.e., a chain
complex of functors A → A. Under the isomorphism Ch(AA ) ∼
= Ch(A)A from
INTRODUCTION TO THE THEORY OF DERIVATORS
35
Lemma 3.10 this corresponds to a diagram X : A → Ch(A). The adjective ‘coherent’ is meant to indicate that this is a strictly commutative diagram in the sense
that it preserves compositions and identies on the nose.
In contrast to this, an incoherent diagram is a functor X : A → D(A). Let us
think of the derived category as being realized as a quotient category, i.e., objects
are suitably nice chain complexes and morphisms are chain homotopy classes. An
incoherent diagram then is merely a homotopy-commutative diagram. For example,
for every pair of composable morphisms f : a0 → a1 and g : a1 → a2 in A it follows
that the triangle on the right in the diagram
? a1
g
f
a0
X(f )
gf
/ a 2 ,
X(a0 )
X(a1 )
:
'
X(gf )
X(g)
$
/ X(a2 ),
commutes up to an unspecified chain homotopy. (In that triangle we of course
abused notation and wrote X(f ), X(g), and X(gf ) respectively for representatives
of the corresponding chain homotopy classes.) Now, in general, it is not possible
to replace a homotopy-commutative diagram up to quasi-isomorphism by a strictly
commutative one.
Put differently, the two categories D(AA ) and D(A)A are honestly different
categories. As an additional application of Proposition 3.35 we observe next that
these categories are related by a functor D(AA ) → D(A)A . Since the construction
of this functor is a bit abstract, we discuss some specific examples in §3.7.
Let us denote by 1 the terminal category consisting of one object ∗ and its
identity morphism only. For any category A the evaluation at this unique 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
a
1
'
7 A,
b
which we again denote by f .
Lemma 3.45. Let A be an abelian category, let A be a small category, and let
a ∈ A. The evaluation functor a∗ : Ch(AA ) → Ch(A) sends quasi-isomorphisms to
quasi-isomorphisms. Hence, there is a unique evaluation functor a∗ : D(AA ) →
D(A) such that the following diagram commutes,
Ch(AA )
a∗
γ
D(AA )
/ Ch(A)
γ
a∗
/ D(A).
Proof. This is immediate from Lemma 3.10 and Definition 3.26.
36
MORITZ GROTH
Given a morphisms of coherent diagrams g : X → Y in D(AA ) we simplify
notation by writing
ga : Xa → Ya , a ∈ A,
for the image of g under a∗ : D(AA ) → D(A).
Now, every morphism f : a → b in A induces a transformation f ∗ : a∗ → b∗
between the corresponding evaluation functors a∗ , b∗ : AA → A. A levelwise application gives rise to a natural transformation f ∗ : a∗ → b∗ between chain-level
evaluation functors a∗ , b∗ : Ch(AA ) → Ch(A).
Lemma 3.46. Let A be an abelian category, let A be a small category, and let
f : a → b be a morphism in A. There is a unique induced natural transformation
f ∗ : a∗ → b∗ between a∗ , b∗ : D(AA ) → D(A) such that γf ∗ = f ∗ γ,
a∗
Ch(AA )
⇓f ∗
)
5 Ch(A)
b∗
γ
D(AA )
γ
)
5 D(A).
a∗
⇓f ∗
b∗
Proof. This is immediate from Proposition 3.35 and Lemma 3.45.
Given a coherent diagram X ∈ D(AA ), we denote the corresponding component
of the natural transformation f ∗ : a∗ → b∗ by
Xf : Xa → Xb ,
which is a morphism in D(A). As always we see that uniqueness implies functoriality. Given morphisms f : a → b and g : b → c, the uniqeness implies the equations
Xg ◦ Xf = Xg◦f
and
Xida = idXa .
Thus, associated to every coherent diagram X ∈ D(AA ) we obtain an incoherent
diagram diaA (X) : A → D(A) defined by
diaA (X) : A → D(A),
a 7→ Xa ,
f 7→ Xf .
We refer to diaA (X) as the underlying diagram of X.
Finally, let us again consider a morphism g : X → Y in D(AA ). Then there is
a natural transformation diaA (g) : diaA (X) → diaA (Y ) between the corresponding
incoherent diagrams. In fact, the component of diaA (g) at an object a ∈ A is simply
the morphism
diaA (g)a = ga : Xa → Ya .
We leave it to the reader to check that diaA (g) is a natural transformation. The
functoriality of this construction is summarized in the following proposition.
Proposition 3.47. Let A be an abelian category and let A be a small category.
The assignments X 7→ diaA (X) and g 7→ diaA (g) define a functor
diaA : D(AA ) → D(A)A
Proof. This proof is left as an exercise.
We refer to this functor as the underlying diagram functor.
INTRODUCTION TO THE THEORY OF DERIVATORS
37
Warning 3.48.
(i) Just to re-emphasize, the domain and the target categories
of diaA are very different. While the domain is the derived category of a
diagram category the target is a diagram category of the derived category.
In general, diaA is not an equivalence and it is important to distinguish
between these two categories. Intuitively speaking, the functor diaA takes
a strict diagram to the underlying homotopy-commutative diagram, thereby
discarding important information.
(ii) As a special case let us consider A = [1] and the associated underlying diagram functor dia[1] : D(A[1] ) → D(A)[1] from the derived category of the arrow
category to the arrow category of the derived category. One slogan concerning
defects of triangulated categories (see §4) is that cone constructions are not
functorial at that level. In the case of derived categories, this statement takes
the more precise form that the cone functor C : D(A[1] ) → D(A) (see Example 3.31) does not factor through dia[1] : D(A[1] ) → D(A)[1] , i.e., in general,
there is no dashed arrow
D(A[1] )
C
/ D(A)
:
dia[1]
D(A)[1]
@
making the diagram commute. In general, the functorial cone construction
only exists at the level of coherent diagrams.
While we come back to the above warning in §9.5, in the following subsection we
illustrate the construction of underlying diagram functors. In particular, this yields
examples of underlying diagram functors which are far from being equivalences.
Promise!
3.7. The case of category algebras. Since the construction of underlying diagram functors (Proposition 3.47) was fairly abstract, in this subsection we want to
identify them in more specific situations. These special cases illustrate a few fairly
typical features of underlying diagram functors.
To begin with we collect a lemma about the homology functor (Example 3.30) in
the context of semisimple abelian categories, i.e., abelian categories in which
every short exact sequence splits.
Lemma 3.49. For every semisimple abelian category A the homology functor
H∗ : D(A) → AZ is an equivalence of categories.
Proof. We refer the reader to [GM03, §] for a discussion of this lemma.
As a special case, given a semisimple ring R (for example, R could be a field k),
the homology functor is an equivalence of categories
H∗ : D(R) ' Mod(R)Z .
This observation will allow us to identify particular instances of underlying diagram
functors with suitable homology functors.
Let A be a small category with finitely many objects only and let R be a ring.
The category algebra RA is the following R-algebra. The underlying R-module
of RA is free with basis given by the morphisms in A,
M
RA =
R.
f : a0 →a1
Check ref!
38
MORITZ GROTH
Let us denote by ef ∈ RA the generator corresponding to the morphism f : a0 → a1 .
The multiplication is essentially the R-bilinear extension of the composition in A,
i.e., it is the bilinear extension of the assignment
egf , s(g)=t(f),
(eg , ef ) 7→
0
, otherwise.
It is easy to see that Σa∈A eida ∈ RA is a neutral element for this multiplication
(and this is the reason why we assumed A to have finitely many objects only).
Having a module over a category algebra RA is as good as having a diagram
X : A → Mod(R). More
precisely, given such a diagram X, the reader checks
L
that the direct sum
a∈A X(a) ∈ Mod(R) actually can be turned into an RAmodule. Moreover, this construction extends to a functor and there is the following
statement about this functor.
Proposition 3.50. Let A be a category with finitely many objects only and let R
be a ring. There is an equivalence of categories
M
Mod(R)A ' Mod(RA) : X 7→
X(a).
a∈A
Proof. This proof is left as an exercise. As a hint we suggest to take a look at the
idempotent elements eida ∈ RA for all a ∈ A in order to establish the essential
surjectivity.
As general references for representations of category algebras and small categories
we mention [?, ?]. Here we content ourselves by illustrating the notion of a category
algebra by the following examples.
Examples 3.51. Let R be a ring.
(i) Any discrete group G can be considered as a category with one object only
(as in §2.3). In this case the category algebra is simply the group algebra RG
and the proposition reproduces the equivalence of categories
Mod(R)G ' Mod(RG).
(ii) Let us recall that a quiver Q is simply an oriented graph, hence a quadrupel
Q = (Q0 , Q1 , s, t : Q1 → Q0 ) given by a set of vertices, a set of edges, and
source and target maps. Abusing notation, we also denote by Q the corresponding path category, i.e., the category freely generated by the quiver Q
(this can be thought of as a many object version of the construction of the free
monoid on a set). Assuming Q0 to be finite, the category algebra RQ is the
path algebra of the quiver and the proposition specializes to the equivalence
Mod(R)Q ' Mod(RQ).
(iii) Let P = (P0 , ≤) be a partially ordered set. We abuse notation and denote
by P the category with P0 as set of objects and morphisms given by
∗ , x ≤ y,
homP (x, y) =
∅ , otherwise.
Under the assumption that P0 is finite, in this case the category algebra RP
is the incidence algebra of P and the proposition yields an equivalence of
categories
Mod(R)P ' Mod(RP ).
INTRODUCTION TO THE THEORY OF DERIVATORS
39
Proposition 3.50 describes an equivalence of abelian categories. Every such
equivalence induces a derived equivalence in the following sense.
Lemma 3.52. Let F : A ' B be an equivalence of abelian categories A, B. Then
applying F levelwisely induces equivalences
F : AZ ' B Z ,
F : Ch(A) ' Ch(B),
and
F : D(A) ' D(B).
Proof. This proof is left as an exercise.
We want to apply this lemma to the situation of Proposition 3.50. Thus, let A
be a small category with finitely many objects only and let R be a ring. The
equivalence Mod(R)A ' Mod(RA) induces the two vertical equivalences in the
diagram
D(Mod(R)A )
(3.53)
diaA
/ D(R)A
H∗
/ (Mod(R)Z )A
/ (Mod(R)A )Z
'
'
D(RA)
∼
=
H∗
/ Mod(RA)Z .
Lemma 3.54. For every small category A with finitely many objects only and for
every ring R the diagram (3.53) commutes up to a canonical natural isomorphism.
Proof. Unraveling definitions, this is an immediate consequence of the fact that
homology is additive. In more detail, using the explicit description of the equivA
alence in Proposition 3.50 we see
L that if we trace X ∈ D(Mod(R) ) clockwisely
through (3.53) then we obtain L
a∈A H∗ (Xa ). Tracing the object X counterclockwisely through (3.53) we get H∗ ( a∈A Xa ). Since homology commutes with finite
direct sums this shows that the two compositions are canonically naturally isomorphic.
In the semisimple case there is the following corollary (see again Definition 2.20).
Corollary 3.55. Let A be a small category with finitely many objects only and let R
be a semisimple ring. The functor diaA : D(Mod(R)A ) → D(R)A is equivalent
to the homology functor H∗ : D(RA) → Mod(RA)Z . In particular, diaA is an
equivalence if and only if H∗ is an equivalence.
Proof. Since R is semisimple, the homology functor H∗ : D(R) → Mod(R)Z is an
equivalence of categories by Lemma 3.49. Hence the same is true for the induced
functor H∗ : D(R)A → (Mod(R)Z )A and the first claim follows since (3.53) commutes up to natural isomorphism (Lemma 3.54). Finally, it is easy to see that
if two functors F1 , F2 are equivalent in the sense of Definition 2.20 then F1 is an
equivalence if and only if F2 is an equivalence.
If we specialize to the case of a field and take up again Examples 3.56 then we
obtain the following examples.
Examples 3.56. Let k be a field.
(i) For every discrete group G the functor diaG : D(Mod(k)G ) → D(k)G is equivalent to the homology functor
H∗ : D(kG) → Mod(kG)Z .
40
MORITZ GROTH
(ii) For every quiver Q with only finitely many objects the underlying diagram
functor diaQ : D(Mod(k)Q ) → D(k)Q is equivalent to the homology functor
H∗ : D(kQ) → Mod(kQ)Z .
(iii) For every finite, partially ordered set P the underlying diagram functor
diaP : D(Mod(k)P ) → D(k)P is equivalent to the homology functor
H∗ : D(kP ) → Mod(kP )Z .
These examples illustrate the following two points.
Remark 3.57.
(i) In general, underlying diagram functors are far from being
equivalences. In fact, an abstract way to see this is as follows. The derived
categories in Examples 3.56 are at the same time triangulated categories
(see §4) and abelian. But any such category is necessarily semisimple — a
property which in general is not enjoyed by group algebras, path algebras,
and incidence algebras. (We will later take a closer look at specific examples of underlying diagram functors, showing more explicitly that one has to
distinguish between coherent and incoherent diagrams.)
(ii) Even if we are in one of the few cases in which derived categories admit
(co)limits different from (co)products, then derived (co)limits and categorical
(co)limits of course do not match up to an application of underlying diagram
functors.
As a specific example, let k be a field and let G be a discrete group. Since
the derived category D(k) is equivalent to Mod(k)Z it admits colimits of
shape G. However, the diagram
D(Mod(k)G )
diaG
/ D(k)G
/ (Mod(k)Z )G
colimG
LcolimG
D(Mod(k))
'
H∗
/ Mod(k)Z
does not commute up to a natural isomorphism. Since LcolimG is equivalent
to the group homology functor H∗ (G; −) (see again §2.3), by Examples 3.56
the previous diagram commutes up to a natural isomorphism if and only if
this is the case for
D(Mod(kG))
H∗
(−)G
H∗ (G;−)
D(Mod(k))
/ Mod(kG)Z
H∗
/ Mod(k)Z .
Now, consider any G and M ∈ Mod(kG) such that the group homology
Hn (G; M ) is non-trivial for some n > 0. Then (H∗ M )G is the vector space
MG concentrated in degree zero, while this is not the case for the image
of M under the remaining composition. In particular, neither of the above
diagrams commutes up to a natural isomorphism.
INTRODUCTION TO THE THEORY OF DERIVATORS
41
4. Derived categories as triangulated categories
TODO: proof-read and
expand this section.
Derived categories are obtained from categories of chain complexes by inverting
the class of quasi-isomorphisms. It is important to note that these two categories
(the category of chain complexes and the derived category) behave quite differently.
(i) Categories of chain complexes are again abelian and as such they are finitely
complete and finitely cocomplete. In contrast to this, in general, the only
limits and colimits which exist in derived categories are finite biproducts.
(ii) More importantly, at the level of chain complexes one can construct functorial
resolutions of diagrams leading to derived limits and derived colimits (see §2.3
and §3.5 for special cases). However, this calculus of derived limits and
derived colimits is not visible at the level of the derived category.
While (i) is a matter of fact one has to live with, there are various ways of trying
to deal with (ii). One way is given by endowing derived categories with additional
structure, thereby turning them into triangulated categories. These triangulations
essentially encode certain shadows of iterated cofiber constructions, hence by §3.5
of iterated derived cokernel constructions.
In this section we review some basics concerning triangulated categories in general and the classical triangulations on derived categories in particular. While here
we only sketch the construction of these classical triangulations most details are
omitted. The point being that these triangulations are special cases of canonical
triangulations in stable derivators which we establish in detail in §9.
Classically, the derived category is obtained by first passing to the homotopy
category of an abelian category and observing that it can be turned into a triangulated category. The derived category is then obtained by a further localization
and, using a theorem of Verdier, this allows one to conclude that derived categories
are triangulated categories. Here we content ourselves by collecting a few of the
key steps.
Despites being a very sucessful theory, triangulated categories suffer certain defects. We conclude this section by a short discussion of such defects. Later in this
book we will see that these defects can be fixed by working with stable derivators
instead.
4.1. The homotopy category of an abelian category. In this subsection we
consider the homotopy category K(A) of an abelian category A. This category
occurs as an intermediate step in the classical construction of the derived cate−1
gory D(A) = Ch(A)[WA
]. Since the chain homotopy relation ' is a congruence
relation (i.e., we have equivalence relations on all homCh(A) (X, Y ) which are compatible 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 chain
homotopy classes of morphisms. The composition is defined by choosing representatives and passing to the chain homotopy class of the corresponding composition.
By definition there is a quotient functor γ 0 : Ch(A) → K(A) which is the universal example of a functor identifying chain homotopic maps. Note that a chain map
Adapt the remainder of
introduction.
42
MORITZ GROTH
f : X → Y is a chain homotopy equivalence if and only if γ 0 (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 0 , we recall the cylinder
construction at the level of chain complexes. This construction is a further instance
of the biadditive pairing described in Digression 3.15. It might be convenient to
briefly recall Examples 3.17.
e cell ([0, 1]+ ) ∈ Chb (FZ ) be the reduced cellular
Examples 4.2.
(i) Let I+ = C
chain complex of [0, 1]+ = [0, 1]t∗ endowed with the CW structure consisting
of 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. The
continuous map [0, 1] → ∗ induces a chain map I+ → S 0 which is easily seen
to be a chain homotopy equivalence. The cylinder cyl(X) of X ∈ Ch(A) is
defined as
cyl(X) = I+ ⊗ X.
It follows from Digression 3.15, Examples 3.17, and Lemma 3.8 that the chain
map I+ → S0 induces a natural chain homotopy equivalence cyl(X) → X.
0
e cell ({0, 1}+ ) ∈ Chb (FZ ) be the reduced cellular chain complex
= C
(ii) Let S+
of {0, 1}+ endowed with the CW structure consisting of three 0-cells only.
0 ∼ 0
Thus, S+
= S ⊕ S 0 and for X ∈ Ch(A) there is a natural isomorphism
0
S+
⊗X ∼
= X ⊕ X.
(Recall that the space {0, 1}+ is the wedge of two copies of {0, 1}.)
0
(iii) The continuous map {0, 1} → [0, 1] yields a map S+
→ I+ in Chb (FZ ) and
0
hence a natural transformation S+ ⊗− → I+ ⊗− of functors Ch(A) → Ch(A).
Under the above isomorphisms, for X ∈ Ch(A) this yields the usual natural
inclusion i : X ⊕ X → cyl(X). We denote the two resulting inclusions by
i0 , 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 chain
homotopies s : f → g and chain maps H : cyl(X) → Y making the following diagram
commute
f
X
i0
i1
$
cyl(X)
:
X
Proof. This proof is left as an exercise.
H
/ Y.
@
g
Thus, as in topology, the cylinder is the natural domain for homotopies. In the
special case of the constant chain homotopy of the identity id : X → X one checks
INTRODUCTION TO THE THEORY OF DERIVATORS
43
that the resulting homotopy
id
X
i0
(4.4)
i1
X
$
HX /X
cyl(X)
@
:
id
is the chain homotopy equivalence cyl(X) → X from Examples 4.2(i). These observations can be reformulated as follows.
Lemma 4.5. Let A be an abelian category and let C be a category. The following
are equivalent for a functor F : Ch(A) → C.
(i) For every pair of chain homotopic maps f, g we have F f = F g.
(ii) The functor F sends chain homotopy equivalences to isomorphisms.
Proof. Clearly, (i) implies (ii). Conversely, (ii) implies that F sends the chain
homotopy equivalences HX in (4.4) to isomorphisms in C. Consequently, the inclusions i0 , i1 : X → cyl(X) have the same images in C and statement (i) is hence an
immediate consequence of Lemma 4.3.
Corollary 4.6. Let A be an abelian category and let HEA be the class of chain
homotopy equivalences in Ch(A). The functor γ 0 : 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 −1 ].
A
Proof. In view of Lemma 4.5 this is simply a reformulation of the universal property
of the quotient functor γ 0 : Ch(A) → K(A).
This corollary shows us that the homotopy category is indeed an intermediate
step in the construction of the derived category. Since K(A) is a quotient category
it 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 additive
categories by additive congruence relations are again additive. We leave the details
to the reader.
Warning 4.8. It is not true, in general, that the homotopy category of an abelian
category is abelian.
The additive category K(A) can be endowed with some additional structure
which can be thought of as shadows of the existence of short exact sequences on
the category Ch(A). First, the suspension functor Σ : Ch(A) → Ch(A) clearly
induces a suspension functor Σ : K(A) → K(A), which is again an equivalence of
categories.
A triangle in K(A) is a diagram of the form X → Y → Z → ΣX and a
morphism of triangles is a commutative diagram
X
f
X0
/Y
g
/ Y0
/Z
h
/ Z0
/ ΣX
Σf
/ ΣX 0 .
44
MORITZ GROTH
Note that we insist that the vertical morphism to the very right is the suspension of
the morphism to the very left. These notions of course make sense in any category
with an endofunctor.
We now single out a particular class of triangles on K(A). Recall from (3.18)
and Definition 3.19 that associated to any chain map f : X → Y there is the cone
Cf which is endowed with a chain map cof(f ) : Y → Cf . There is also a canonical
map Cf → ΣX. To construct it, we note that by the definition of the cone CX of
a chain complex there is a natural short exact sequence
q
i
0 → X → CX → ΣX → 0.
(In fact, by means of Examples 3.17 this is induced by S 0 ⊗ − → I ⊗ − → S 1 ⊗ −.)
The universal property of the defining pushout diagram (3.18) applied to q and the
zero map 0 : Y → ΣX implies that there is a unique map Cf → ΣX making the
following diagram
X
f
i
CX
/Y
cof(f )
/ Cf
0
∃!
" / ΣX
q
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 distinguished if it is isomorphic to a standard triangle.
The additional structure on K(A) consisting of the suspension functor and the
above class of distinguished triangles satisfies certain properties as axiomatized by
the notion of a triangulated structure.
4.2. Triangulated categories. In this subsection we take a glimpse at the theory of triangulated categories. Here we only include what is strictly necessary in
this motivational section. For more details we refer the reader to the established
literature which includes [Nee01, HJR10].
Definition 4.9. Let T be an additive category with a self-equivalence Σ : T → T
and a class of distinguished triangles X → Y → Z → ΣX. The pair consisting of Σ
and the class of distinguished triangles defines a triangulated structure on T if
the following four axioms are satisfied.
id
(T1) For every X ∈ T , the triangle X → X → 0 → ΣX is distinguished. Every
morphism in T occurs as the first morphism in a distinguished triangle and the
class of distinguished triangles is replete, i.e., is closed under isomorphisms.
f
g
h
(T2) If the triangle X → Y → Z → ΣX is distinguished then also the rotated
g
h
−Σf
triangle Y → Z → ΣX → ΣY is distinguished.
INTRODUCTION TO THE THEORY OF DERIVATORS
45
(T3) Given two distinguished triangles and a commutative solid arrow diagram
/Y
X
u
/Z
v
X0
/ ΣX
∃w
/ Y0
Σu
/ Z0
/ ΣX 0
there exists a dashed arrow w : Z → Z 0 as indicated such that the extended
diagram commutes.
f2
f1
(T4) For every pair of composable arrows f3 : X → Y → Z there is a commutative
diagram in which the rows and columns are distinguished triangles:
X
f1
/Y
g1
f2
X
f3
/Z
g3
g2
/ C3
C2
C2
h1
h3
/ ΣX
/ ΣX
Σf1
h2
/ ΣY
Σg1 ◦h2
h2
ΣY
/ C1
Σg1
/ ΣC1
A triangulated category is an additive category together with a triangulated
structure.
A few comments about these axioms are in order.
Remark 4.10.
(i) Axiom (T2) is the rotation axiom, axiom (T3) the (mapping) cone axiom, and axiom (T4) the octahedron axiom. The name
octahedron axiom is motivated by a different way of drawing the diagram in
axiom (T4); see for example [Wei94, p. 375].
(ii) Triangulated categories were invented independently by Puppe [Pup67] (motivated by Algebraic Topology) and Verdier [Ver96] (motivated by Algebraic
Geometry) in his 1967 thesis, the difference being that Puppe triangulations
are only asked to satisfy axioms (T1)-(T3).
(iii) One way to think of distinguished triangles is that they are some shadows of
certain derived cokernel constructions on ‘a model in the background’ (this
will be justified in §4.3). Motivated by this we refer to the third object in a
distinguished triangle as a cone (as already reflected in the name of axiom
(T3)). The axioms (T1)-(T4) capture some compatibility properties satisfied
by such derived cokernel constructions.
(iv) Using this interpretation of the third objects in distinguished triangles, we
can think of the octahedron axiom as a triangulated category version of the
third Noether isomorphism theorem. In fact, while the Noether isomorphism
theorem says that ‘the quotient of two quotients is again a quotient’ the
octahedron 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 to
46
MORITZ GROTH
say that the cone is not necessarily a functorial construction T [1] → T . This
lack of functoriality has important implications and we come back to this
in §4.4.
(vi) There is a stronger version of the rotation axiom, asking that a triangle is
distinguished if and only if the rotated diagram is distinguished. It can be
shown under the assumption of (T1) and (T3) that this stronger version of
the rotation axiom is a consequence of the weaker one.
(vii) Also the octahedron axiom is given in a seemingly weaker form than in
other references. The observation that the stronger version follows from this
weaker one goes back to [KV87] and this form was recently used by Holm
and Jørgenson in their survey article in [HJR10]. For a short discussion of
this fact and related references see [HJR10, Remark 3.2]. The motivation for
us to choose these slightly simpler axioms is that this will later simplify the
proof that stable derivators give rise to canonical triangulations.
We suggest the reader who just saw the definition of a triangulated category for
the first time to provide the proofs of the following statements.
Proposition 4.11. Let T be a triangulated category and let W ∈ T . For every
distinguished 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).
Proof. This proof is left as an exercise.
Using the rotation axiom (TR2) in its symmetric form we see that represented
functors send distinguished triangles to long exact sequences. This is often referred
to 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
/Z
/ ΣX
/Y
X
f
X0
g
/ Y0
h
/ Z0
Σf
/ ΣX 0
be a morphism of distinguished triangles. If two of the maps f, g, h are isomorphisms
then so is the third one.
Proof. This proof is left as an exercise.
Corollary 4.13. Let X → Y be a morphism in a triangulated categories. Any two
distinguished 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 back
to this in §4.4.
Lemma 4.15. A morphism in a triangulated category is an isomorphism if and
only if any of its cones is zero.
Proof. This proof is left as an exercise.
Solutions to all these exercises and much more can be found in [Nee01, §1].
INTRODUCTION TO THE THEORY OF DERIVATORS
47
4.3. Exact morphisms and classical triangulations. We again consider an
abelian category A. Recall from the end of §4.1 that we defined a suspension
functor 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 together
with the equivalence Σ : K(A) → K(A) and the above class of distinguished triangles
is a triangulated category.
Proof. The category K(A) is additive by Lemma 4.7 and Σ : K(A) → K(A) is
an equivalence. Axiom (T1) is immediate while the remaining axioms are more
involved. We refer the reader to [HJR10] or [Wei94, Proposition 10.2.4] for more
details.
In the notation of Corollary 4.6 there clearly is an inclusion HEA ⊆ WA and, by
that corollary, the localization functor γ : Ch(A) → D(A) hence factors uniquely
over γ 0 : Ch(A) → K(A), yielding the canonical functor
γ 00 : K(A) → D(A).
Passing through the homotopy category K(A) as an intermediate step, the derived
category is obtained as the localization D(A) = K(A)[W −1 ]. There are general
techniques which allow us to conclude that such localizations are again triangulated
categories and that the localization functor is exact in the following sense; see for
example [Wei94, §10] and [Kra10].
Definition 4.17. Let T , T 0 be triangulated categories. An exact functor T → T 0
is a pair (F, σ) consisting of
(i) an additive functor F : T → T 0 and
(ii) a natural transformation σ : F Σ → ΣF
h
such that for every distinguished triangle X → Y → Z → ΣX in T the image
σ◦F h
triangle F X → F Y → F Z → ΣF X is distinguished in T 0 .
Remark 4.18. Note that exactness of a functor of triangulated categories is not a
property but that it amounts to specifying an additional structure. If we want to
emphasize this then we refer to σ : ΣF → F Σ as an exact structure on F .
The following theorem is a consequence of the more general Verdier localization
theorem; again we refer the reader to the literature.
Theorem 4.19. Let A be an abelian category. The derived category D(A) can be
turned into a triangulated category and the localization functor γ 00 : K(A) → D(A)
into an exact functor.
4.4. Beyond triangulated categories. The theory of triangulated categories is
a large theory and has been successful in many applications in various areas of
pure mathematics for roughly fifty years by now. For a few sample applications in
various areas we refer the reader to the book [HJR10].
Despites all these successes, from the very beginning on (see already the introduction to [Hel68]) it was also apparent that the axioms of a triangulated category
come with certain defects. We include a few comments along these lines.
derived functors
between derived
categories are often
exact
48
MORITZ GROTH
(i) One crucial observation is that the cone construction at the level of triangulated categories is not functorial. For every morphism in T there is a
distinguished triangle (axiom (T1)), hence we can associate a cone object to
it. There are the following (closely related) drawbacks to this definition of a
cone.
(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 morphism on cones (axiom (T3)), but, again, this morphism is not unique
and hence it lacks functoriality.
And in fact, it turns out that if there is a cone functor T [1] → T then the
triangulated category is semisimple ([Ver96, Proposition 1.2.13]). This is to
be seen in contrast to Example 3.31.
(ii) The axioms of a triangulated category ask for the additional structure of
a suspension functor and of distinguished triangles. This datum does not
satisfy any universal property and is hence non-canonical. This is partially
reflected in the fact that exact morphisms of triangulated categories have
to be defined by means of additional structure as well. But in the typical
examples the fact that we have an exact morphism of triangulated categories
reflects the idea that ‘in the background on the models’ there is a morphism
of ‘stable homotopy theories’ which has the property of preserving certain
finite homotopy (co)limits.
(iii) As we saw in Theorem 3.42 the cone functor C : D(A[1] ) → D(A) is a left
derived cokernel functor, hence satisfies a universal property. There are of
course many further derived (co)limit functors which are important in applications. Given a nice abelian category A and a small category A, often
one can define derived (co)limit functors LcolimA , RlimA : D(AA ) → D(A).
A crucial observation however is that, say, LcolimA does not factor through
the underlying diagram functors diaA : D(AA ) → D(A)A of Proposition 3.47,
i.e., in general, there is not a dashed arrow
D(AA )
Lcolim/
D(A)
:
diaA
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 their
homotopy (co)limits)’. A different but related slogan is that the underlying diagram functor amounts to passing from strict diagrams to homotopy
commutative diagrams, and that such homotopy commutative diagrams do
not carry enough information. (In fact, one would need homotopy coherent
diagrams to construct associated homotopy (co)limits.)
(iv) Still a related observation is as follows. Given a triangulated category T
and a small category A there is no canonical triangulation on the diagram
category T A .
INTRODUCTION TO THE THEORY OF DERIVATORS
49
(v) A further related observation is that the axioms of a triangulated category,
while satisfied in many examples, might miss interesting aspects of homological algebra or homotopy theory. As mentioned above, the axioms encode
some structure which is a shadow of iterated cofiber seqences and certain
compatibilities. The octahedron axiom (T4) goes one step further and considers two composable morphisms together with the various associated distinguished triangles. However, the axioms have nothing to say if one wants
to consider longer strings of composable arrows and all the related distinguished triangles. Pursuing this further leads to higher triangulations; see
for example [BBD82, Remark 1.1.14] and [Mal05].
The aim of the remainder of this course is to show that stable derivators provide
an alternative theory which successfully adresses the above issues. In fact, one can
think of a stable derivator as a minimal, purely categorical extension of a derived
category to a framework which comes with a well-behaved calculus of derived limits
and derived colimits.
5. Kan extensions
In this section we briefly review some basics concerning the theory of Kan extensions from ordinary category theory. Kan extensions are certain universal constructions which can be thought of as ‘relative versions of limits and colimits’. The basic
idea behind a derivator is to axiomatize key formal properties which are common
to
(i) the calculus of limits, colimits, and Kan extensions in complete and cocomplete categories,
(ii) the calculus of derived limits, derived colimits, and derived Kan extensions
in derived categories, and
(iii) the calculus of homotopy limits, homotopy colimits, and homotopy Kan extensions in Quillen model categories or complete and cocomplete ∞-categories.
While here we establish these formal properties for ordinary categories, in §6 they
are essentially turned into the definition of a derivator and then one establishes
the non-trivial results that abelian categories, Quillen model categories, and ∞categories yield derivators (see §6 and Appendix B).
In §5.1 we define Kan extensions and observe that they generalize (co)limits.
In §5.2 we recall a short discussion of final functors. In §5.3 we note that in complete
and cocomplete categories Kan extensions can be constructed pointwisely using the
more well-known (co)limits. In §5.4 we collect a few basic results about ordinary
Kan extensions which we later extend to arbitrary derivators. We also mention
some first examples to illustrate Kan extensions.
5.1. Motivation and definition. To motivate the notion of a Kan extension, we
recall from §3 (see, in particular, Theorem 3.42) that for every abelian category A
the cone functor C : D(A[1] ) → D(A) is the derived cokernel. Given a morphism
f : X0 → X1 in Ch(A), the cone Cf can be constructed as the pushout object in
X0
i
CX0
f
/ X1
cof(f )
/ Cf.
50
MORITZ GROTH
Thus, in all detail, the cone of a chain map is obtained by associating a particular
span to it, passing to the pushout square of the span, and then evaluating the
pushout square on the final vertex.
X0
f
/ X1
(5.1)
/ X1
f
X0
X0
i
f
cof(f )
i
CX0
/ X1
/ Cf
CX0
Cf
Similarly to the case of cokernels, also for cones it is often important to keep track
of the morphism cof(f ) : X1 → Cf instead of the chain complex Cf only. And to
obtain a detailed description of the construction of cof(f ) it suffices to replace the
final step in (5.1) by the restriction to the vertical morphism on the right.
Note that the fact that this construction of the cone is seemingly complicated is
not a consequence of working in the derived setting. In fact, this is already true for
cokernels in ordinary category theory: given a morphism f in a finitely cocomplete
category C admitting a zero object, the cokernel is not the colimit of the morphism
(which would be isomorphic to its target) but it is an instance of the more general
notion of a weighted colimit. Similarly to the derived setting, the cokernel of a
morphism f : X0 → X1 in such a category C can be obtained in the following steps.
X0
f
/ X1
X0
f
/ X1
X0
(5.2)
0
0
f
/ X1
/ cok(k)
cok(k)
The only difference between (5.1) and (5.2) is that in the derived setting we replace
the map X0 → 0 by the inclusion X0 → CX0 in order to fix the problem that, in
general, pushouts are not exact or homotopy-invariant.
As an upshot, a detailed construction of the cokernel of a morphism passes
through intermediate steps, the first two being the extension of the morphism to
a span and then the extension to a pushout square. The purely categorical notion
which is in the background of these two intermediate steps is the notion of a Kan
extension as we define it next.
To abstract from our specific situation, let us consider the following extension
problem. Let u : A → B be a functor between small categories and let X : A → C
be a functor taking values in a not necessarily small category C. The aim is to find
‘extensions of X along u’,
A
(5.3)
u
B.
/C
>
X
∃?
The notion of ‘an extension’ can be made precise in different ways, namely by asking
that the diagram (5.3)
(i) commutes on the nose,
(ii) commutes up to a specified natural isomorphism, or
(iii) commutes up to a specified universal natural transformation.
INTRODUCTION TO THE THEORY OF DERIVATORS
51
The first condition is ‘against the philosophical principle from category theory that
one should not ask that functors are equal but only naturally isomorphic’. While
satisfied in many specific 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 two versions of such universal natural transformations, depending
on whether we want the datum to be initial or final.
Definition 5.4. Let u : A → B be a functor between small categories, let C be a
category, and let X : A → C.
Draw pasting.
(i) A left Kan extension of X along u is a functor LKanu (X) : B → C together
with a natural transformation η : X → LKanu (X) ◦ u satisfying the following
universal property. For every pair (Y : B → C, α : X → Y ◦ u) there is a
unique 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 → C
together with a natural transformation : RKanu (X) ◦ u → X satisfying the
following universal property. For every pair (Y : B → C, α : Y ◦ u → X) there
is a unique β : Y → RKanu (X) such that α = · (βu) : Y ◦ u → X.
These notions are dual to each other and in this section we will mostly focus
on left Kan extensions. For convenience, we included the following diagramatic
description of this definition. A left Kan extension (LKanu (X), η) can be depicted
by a triangle populated by a natural transformation as in
A
X
/C
?
u
B
LKanu (X)
and the universal property of the left Kan extension can be illustrated as follows
To develop some intuition for this notion let us consider the following special
case. Recall that we denote the terminal category by 1.
Example 5.5. Let A be a small category and let X : A → C be given. A left
Kan extension of X along πA : A → 1 consists of a functor LKanπA (X) : 1 → C
and a universal natural transformation η : X → LKanπA (X) ◦ πA . The functor
LKanπA (X) : 1 → C amounts to picking an object c ∈ C and the natural transformation is of the form η : X → ∆A (c),
a
∀f
0
a,
X(a)
X(f )
X(a0 ),
ηa
/= c
ηa0
thereby specifying a cocone on X. Together with the initiality of this datum, we
see that the left Kan extension (c, η) is an initial cocone on X, which is to say that
c = colimA X and that η is the colimiting cocone.
Thus, we just saw that the notion of a left Kan extension along πA : A → 1
reduces to the notion of a colimit. In particular, such Kan extensions exist as soon
as the target category is cocomplete. The goal of §5.3 is to show that this is true
more generally and that those more general Kan extensions can be constructed
Do it!
52
MORITZ GROTH
pointwisely by certain colimits. Before we get to that, in §5.2 we include a short
digression on final functors.
5.2. Final functors. Let C be a cocomplete category and let u : A → B. For
every diagram X : B → C there is a canonical comparison map between colimB X
and colimA (X ◦ u), and we say that u is final if this comparison map is always an
isomorphism.
We begin by defining the canonical comparison map and for this purpose we
consider colimB X ∈ C together with its colimiting cocone η which has components
ηb : Xb → colimB X, b ∈ B. Precomposition along u yields the functor X ◦u : A → C.
Applying the same precomposition to the colimiting cocone η we obtain a cocone α
on X ◦ u given by αa = ηua : (X ◦ u)(a) → colimB X, a ∈ A. In fact, for every
morphism f : a → a0 in A the triangle
X(ua)
/ colimB X
9
ηua
X(uf )
ηua0
X(ua0 )
commutes since η is a cocone. But also the diagram X ◦ u : A → C has a colimit
colimA (X ◦ u) and α hence factors uniquely through the corresponding colimiting
cocone. Thus, there is a canonical morphism
(5.6)
colimA (X ◦ u) → colimB X,
which is compatible with the cocones, i.e., such that the diagram
X(ua)
αa =ηua
colimA (X ◦ u)
/ colimB X
7
∃!
commutes for all a ∈ A. (The unlabelled morphisms belong to the colimiting cocone
of X ◦ u.)
Dually, if C is a complete category then for every X : B → C there is a canonical
morphism
(5.7)
lim X → lim(X ◦ u)
B
A
which is induced from the respective limiting cones.
Definition 5.8. Let u : A → B be a functor between small categories.
(i) The functor u is final if the canonical map (5.6) is an isomorphism for every
cocomplete category C and every diagram X : B → C.
(ii) The functor u is cofinal if the canonical map (5.7) is an isomorphism for
every complete category C and every diagram X : B → C.
Since these notions are dual to each other we mostly focus on final functors.
Proposition 5.9. Right adjoint functors between small categories are final.
INTRODUCTION TO THE THEORY OF DERIVATORS
53
Proof. Let C be a cocomplete category and let (u, v) : A B be an adjunction
between small categories. It is easy to check that the associated precomposition
functors define an adjunction
(v ∗ , u∗ ) : C A C B .
Recall that adjunctions can be composed. In our situation, if we combine the above
adjunction with the adjunction (colimB , ∆B ) : C B C, then we see that colimB ◦v ∗
is left adjoint to u∗ ◦∆B . But since the functor u∗ ◦∆B is equal to ∆A , the uniqueness
of left adjoints implies that there is a canonical isomorphism colimA ∼
= colimB ◦v ∗ .
With a bit more care one checks that this isomorphism agrees with the desired
canonical map (5.6).
Corollary 5.10. Let A be a small category admitting a final object ∗. Every category C has colimits of shape A and for X : A → C there is a natural isomorphism
X(∗) ∼
= colimA X.
Proof. This is immediate from Proposition 5.9 since (πA , ∗) : A 1 is an adjunction.
There is a combinatorial criterion which allows us to characterize final functors.
To state the result we need the following definition. For later reference, we also
make the dual definition.
Definition 5.11. Let u : A → B be a functor between small categories and let
b ∈ B be an object.
(i) The slice category (b/u) has as objects pairs (a ∈ A, f : b → u(a)) and as
morphisms (a, f ) → (a0 , f 0 ) morphisms a → a0 in A such that
b
f
f0
/ u(a)
u(a0 )
commutes.
(ii) The slice category (u/b) has as objects pairs (a ∈ A, f : u(a) → b) and as
morphisms (a, f ) → (a0 , f 0 ) morphisms a → a0 in A such that
u(a)
u(a0 )
f
>/ b
f0
commutes.
Proposition 5.12. Let u : A → B be a functor between small categories.
(i) The functor u is final if and only if the categories (b/u), b ∈ B, are non-empty
and connected.
(ii) The functor u is cofinal if and only if the categories (u/b), b ∈ B, are nonempty and connected.
To be completely specific, u is final if and only if for each b ∈ B the following
two properties are satisfied.
54
MORITZ GROTH
(i) There is an object in (b/u), i.e., an object a ∈ A and a morphism b → u(a)
in B.
(ii) Any two objects in (b/u) can be connected by a finite zigzag of morphisms.
For a proof of Proposition 5.12 we refer the reader, for example, to [KS06, §2.5].
Here we content ourselves by showing how this proposition can be used to reestablish Corollary 5.10 and Proposition 5.9 and by relating the proposition to a
more classical statement about colimits of diagrams defined on partially ordered
sets (aka. posets).
Examples 5.13.
(i) Let B be a poset, let A ⊆ B be a subposet, and let i : A → B
be the corresponding inclusion functor. The inclusion is final if and only if for
every b ∈ B the category (b/i) is non-empty and connected. Thus, we ask for
the existence of a ∈ A such that b ≤ a. Moreover, any a1 , a2 ∈ A such that
b ≤ a1 and b ≤ a2 have to be connected by a finite zigzag of morphisms in
(b/i). A typical sufficient condition is that A is directed, i.e., that any two
elements in A have a common successor. Thus, a directed subposet which is
unbounded in the larger poset yields a final functor.
(ii) Let B be a small category, 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.
This reproduces Corollary 5.10.
(iii) Let A, B be small categories and let v : B → A be a right adjoint functor.
We want to show that for every a ∈ A the slice category (a/v) is non-empty
and connected. The unit of the adjunction (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 explicit description 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 slice category is, in particular, connected. This
recovers Proposition 5.9.
5.3. Pointwise Kan extensions. In this subsection we show that in cocomplete
categories left Kan extensions along functors between small categories always exist.
Moreover, there is an explicit construction of such left Kan extensions using colimits
over slice categories. Given the importance of these formulas to theory of derivators,
we include complete and fairly detailed proofs. And again, duality allows us to
mostly focus on left Kan extensions.
For the remainder of this subsection we assume that C is a cocomplete category
and that u : A → B is a functor between small categories. The defining universal
property of left Kan extensions (Definition 5.4) suggests the following construction
of them.
Let us consider a diagram X : A → C and let (Y, α) be a datum among which
the left Kan extension is supposed to be initial, i.e., we have a functor Y : B → C
and a natural transformation α : X → Y ◦ u. For every morphism g : a → a0 in A
we obtain the naturality square
X(a)
αa
Y u(a)
X(g)
/ X(a0 )
αa0
Y u(g)
/ Y u(a0 ).
INTRODUCTION TO THE THEORY OF DERIVATORS
55
Now, for every object b ∈ B, the value Y (b) has to be compatible with these
naturality squares in the following sense. Let us assume we can find morphisms
f : u(a) → b and f 0 : u(a0 ) → b in B which are compatible with g in the sense that
f = f 0 ◦ u(g). For each such datum the above naturality square extends to the
following commutative diagram
X(a)
X(g)
/ X(a0 )
αa0
αa
Y u(a)
(5.14)
Y (f )
z
Y (b)
Y u(g)
/ Y u(a0 )
Y (f 0 )
in C. If we ignore the line in the middle, then we see that Y (b) is part of a cocone
on a certain diagram. And the universal property of a left Kan extension suggests
that we should form universal such cocones by passing to colimits. It turns out
that this indeed works and we now carry out the details.
Note first that in the above heuristics (5.14) the slice category (u/b) from Definition 5.11 came up.
Lemma 5.15. Let u : A → B be a functor between small categories.
(i) For every b ∈ B there is a projection functor p = pb : (u/b) → A defined on
objects by (a, f : u(a) → b) 7→ a and on morphisms by g 7→ g.
(ii) For every morphism h : b → b0 in B there is a functor (u/h) : (u/b) → (u/b0 )
defined on objects by (a, f : u(a) → b) 7→ (a, h ◦ f : u(a) → b → b0 ) and on
morphisms by g 7→ g.
(iii) For every morphism h : b → b0 the diagram
(u/b)
pb
z
A
(u/h)
/ (u/b0 )
pb 0
commutes.
Proof. This is obvious and is left to the reader.
Thus, for every object b ∈ B we can consider the diagram
X ◦ pb : (u/b) → A → C,
and let us note that in the above heuristics (5.14) we observed that Y (b) is part of
a cocone on precisely this diagram. This suggests that we make the definition
(5.16)
L(X)(b) = colim(u/b) X ◦ pb ,
b ∈ B.
Moreover, as a special instance of the canonical morphisms (5.6), for every morphism h : b → b0 in B we use the relation pb0 ◦ (u/h) = pb from Lemma 5.15 to
obtain a canonical morphism
(5.17)
L(X)(h) : colim(u/b) X ◦ pb → colim(u/b0 ) X ◦ pb0 .
56
MORITZ GROTH
The uniqueness of these canonical morphisms show that they are functorial, thereby
concluding the definition of the functor L(X) : B → C.
Lemma 5.18. Let C be a cocomplete category, let u : A → B be a functor between
small categories, and let X : A → C. The assignments (5.16) and (5.17) define a
functor L(X) : B → C.
Proof. This is immediate from the above discussion.
In order to conclude the construction of a left Kan extension, we need a natural
transformation η : X → L(X) ◦ u. For a ∈ A the pair (a, id : u(a) → u(a)) defines
an object in (u/u(a)) and we let ηa be the corresponding canonical map belonging
to the colimiting cocone of L(X)(u(a)) = colim(u/u(a)) X ◦ pu(a) ,
(5.19)
ηa = ι(a,idu(a) ) : Xa → colim(u/u(a)) X ◦ pu(a) .
Given a morphism g : a → a0 in A, let us consider the following diagram
Xa
ι(a,idu(a) )
X(g)
Xa0
ι0(a,u(g))
ι0(a0 ,id
/ colim(u/u(a)) X ◦ pu(a)
*
/ colim(u/u(a0 )) X ◦ pu(a0 ) ,
)
u(a0 )
in which the respective colimiting cocones are denoted by ι and ι0 . The upper
triangle commutes by definition of (5.17) as a particular instance of the canonical
morphism (5.6), while the lower triangle commutes since ι0 is a cocone. Thus, the
morphisms (5.19) define a natural transformation η : X → L(X) ◦ u.
Proposition 5.20. Let C be a cocomplete category, let u : A → B be a functor
between small categories, and let X : A → C. The pair (L(X), η) consisting of
the functor L(X) : B → C defined by (5.16),(5.17) and the natural transformation
η : X → L(X) ◦ u defined by (5.19) is a left Kan extension of X along u.
Proof. By Definition 5.4 we have to show that for every Y : B → C the assignment
φ : homC B (L(X), Y ) → homC A (X, Y ◦ u) :
β 7→ (βu) ◦ η
is a bijection. It follows from the explicit construcion (5.16) and (5.19) that the
component of φ(β) at an object a ∈ A is given by
φ(β)a = βua ◦ ι(a,id) : X(a) → colim X ◦ pua → (Y ◦ u)(a).
We next construct the following map in the converse direction
ψ : homC A (X, Y ◦ u) → homC B (L(X), Y ).
To this end, let α : X → Y ◦ u and let us fix an object b ∈ B. For every object
(a, f : u(a) → b) ∈ (u/b) we form the map Y f ◦ αa : Xa → (Y ◦ u)(a) → Y (b).
By (5.14) these maps define a cocone on X ◦ pb : (u/b) → C and there is hence a
unique map ψ(α)b : L(X)(b) = colim X ◦ pb → Y (b) such that
colim X
O ◦ pb
(5.21)
∃! ψ(α)b
ι(a,f )
X(a)
/ Y (b)
O
Yf
αa
/ Y (ua)
INTRODUCTION TO THE THEORY OF DERIVATORS
57
commutes. Here, the morphism ι(a,f ) belongs to the colimiting cocone. We leave it
to the reader to check that for every morphism h : b → b0 in B the diagram
colim X ◦ pb
ψ(α)b
L(X)(h)
colim X ◦ pb0
/ Y (b)
h
ψ(α)b0
/ Y (b0 )
commutes, i.e., that ψ(α) : L(X) → Y is a natural transformation.
It remains to check that φ and ψ are inverse to each other. Let β : L(X) → Y
and let α = φ(β) = (βu) ◦ η. For every b ∈ B and (a, f ) ∈ (u/b) we consider the
diagram
colim
O ◦ pb
8 X
ι(a,f )
βb
/ Y (b)
O
L(X)(f )
X(a)
ηa
/ colim X ◦ pua
Yf
βua
/ Y (ua),
in which the square is a naturality square of β. Since ηa = ιa,idua (see (5.19))
and ιa,f belong to the respective colimiting cocones, the definition of L(X)(f )
(see (5.17)) as an instance of the canonical map (5.6) implies that also the triangle
commutes. But the uniqueness of the map ψ(α)b in the definition of ψ (see (5.21))
implies that βb = ψ(φ(β))b .
Conversely, let α : X → Y ◦ u be given and we want to show that (φψ)(α) = α,
i.e., that (ψ(α)u) ◦ η = α. If we consider a ∈ A and the corresponding object
(a, idu(a) ) ∈ (u/a), then as a special case of (5.21) we see that the diagram
colim XO ◦ pua
ψ(α)ua
/ Y (ua)
O
ι(a,idua )
X(a)
=
αa
/ Y (ua)
commutes. Using once more the precise form of η (see (5.19)) we conclude that
φψ(α) = α, concluding the proof.
Definition 5.22. Let C be a category and let u : A → B be a functor between
small categories. The assignments
u∗ : C B → C A :
X 7→ X ◦ u,
β 7→ βu
define a restriction functor or precomposition functor.
For later reference we summarize our findings concerning the existence and the
precise form of left Kan extensions in the following theorem.
Theorem 5.23. Let C be a cocomplete category and let u : A → B be a functor
between small categories.
(i) For every diagram X : A → C a left Kan extension LKanu (X) exists and this
defines a functor LKanu : C A → C B which is left adjoint to u∗ : C B → C A ,
(LKanu , u∗ ) : C A C B .
58
MORITZ GROTH
(ii) For every diagram X : A → C and every object b ∈ B there is a canonical
isomorphism
colim(u/b) X ◦ pb ∼
= LKanu (X)b .
Proof. Associated to u there is the restriction functor u∗ : C B → C A and a left
Kan extension of X along u is precisely an initial object in the category (X/u∗ ).
By Proposition 5.20 we know that such an initial object always exists, and from
this it follows immediately that any choices of such left Kan extensions define a
functor LKanu : C A → C B which is left adjoint to u∗ : C B → C A .
Moreover, any two initial objects are canonically isomorphic, implying that any
left Kan extension LKanu (X) is canonically isomorphic to the one constructed
in (5.16).
The corresponding result for right Kan extensions is dual. Given a functor
u : A → B between small categories and b ∈ B there is a functor
(5.24)
q = qb : (b/u) → A
defined on objects by (a, f : b → u(a)) 7→ a and on morphisms by g 7→ g.
Theorem 5.25. Let C be a complete category and let u : A → B be a functor
between small categories.
(i) For every diagram X : A → C a right Kan extension RKanu (X) : B → C
exists and this defines a functor RKanu : C A → C B which is right adjoint to
u∗ : C B → C A ,
(u∗ , RKanu ) : C B C A .
(ii) For every diagram X : A → C and every object b ∈ B there is a canonical
isomorphism
RKanu (X)b ∼
= lim X ◦ qb .
(b/u)
Proof. This is the dual of Theorem 5.23.
Thus, in sufficiently (co)complete categories, Kan extensions always exist and
can be constructed using pointwise formulas, thereby sort of reducing the more
general notion of a Kan extension to the more well-known notion of a limit or a
colimit.
Ref!
Warning 5.26. There are examples of Kan extensions (necessarily in contexts lacking certain limits or colimits) which cannot be calculated using pointwise formulas.
For a particular example we refer the reader to [?]. Such Kan extensions will not
play a role in this book.
Before we take up more interesting examples in §5.4, we revisit Example 5.5.
Example 5.27. Let C be a complete and cocomplete category, let A be a small
category, and let πA : A → 1 be the unique functor.
∗
(i) Under the isomorphism C 1 ∼
) : C A C 1 gets
= C the adjunction (LKanπA , πA
A
identified with (colimA , ∆A ) : C C.
∗
(ii) Under the isomorphism C 1 ∼
, RKanπA ) : C 1 C A gets
= C the adjunction (πA
A
identified with (∆A , limA ) : C C .
This leads us to think of Kan extensions as ‘relative versions of limits and colimits’, which are obtained by replacing πA : A → 1 by more general functors u : A → B
between small categories.
INTRODUCTION TO THE THEORY OF DERIVATORS
59
5.4. Basic properties and first examples. In this subsection we illustrate the
notion of Kan extensions by collecting a few examples and also establish the useful
property that Kan extensions along fully faithful functors are again fully faithful,
thereby justifying the terminology Kan extensions.
Since Kan extensions will be used a lot in the remainder of this book, let us
introduce the following notation.
Notation 5.28. Let C be a complete and cocomplete category and let u : A → B
be a functor between small categories. We also denote Kan extension functors by
u! = LKanu : C A → C B
and
u∗ = RKanu : C A → C B .
In the special case that u = b : 1 → B classifies the object b ∈ B we thus write
b! = LKanb : C → C B
and
b∗ = RKanb : C → C B .
As first examples we already saw that Kan extensions along functors πA : A → 1
reduce to limits and colimits (Example 5.27). The other extreme case is given by
Kan extensions along functors 1 → B defined on the terminal category, and this
leads to free and cofree diagrams.
Example 5.29. Let B be a small category, let b ∈ B, and let C be a complete and
cocomplete category (actually, the existence of products and coproducts suffices).
We describe the Kan extension functors b! , b∗ : C → C B along b : 1 → B.
(i) The left Kan extension functor b! : C → C B sends an object X ∈ C to the free
diagram b! (X) : B → C generated by X. Given an object b0 ∈ B, the slice
category (b/b0 ) is simply the discrete category on homB (b, b0 ). The pointwise
formula yields the first isomorphism in
a
b! (X)b0 ∼
X.
= colim(b/b0 ) X ◦ pb0 ∼
=
homB (b,b0 )
The second isomorphism is a consequence of (b/b0 ) being a discrete category
so that the colimit reduces to a coproduct. The defining universal property
of the free diagram b! (X) is that there are natural isomorphisms
homC B (b! (X), Y ) ∼
= homC (X, Y (b)), Y ∈ C B ,
i.e., b! is left adjoint to the evaluation functor b∗ : C B → C.
(ii) The right Kan extension functor b∗ : C → C B sends an object X ∈ C to the
cofree diagram b∗ (X) : B → C generated by X. In this case there are
natural isomorphisms
Y
0 ∼
b∗ (X)b0 ∼
X
◦
q
X.
= lim
=
b
0
(b /b)
homB (b0 ,b)
Remark 5.30. Note that this example shows that Kan extensions are, in general,
not fully faithful. In other words, assuming that the respective Kan extensions
exist, for a diagram X : A → C the diagrams
LKanu (X) ◦ u, RKanu (X) ◦ u : A → C
obtained by first extending and then restricting back are, in general, not naturally
isomorphic to X. In fact, for free and cofree diagrams we have isomorphisms
a
Y
b! (X)b ∼
X
and
b∗ (X)b ∼
X.
=
=
homB (b,b)
homB (b,b)
60
MORITZ GROTH
We illustrate the concept of free diagrams by two more specific examples.
Examples 5.31. Let C be a complete and cocomplete category.
(i) Given a quiver Q and a vertex q0 ∈ Q we obtain the free diagram functor
(q0 )! : C → C Q . Specializing to Mod(R) for a ring R we obtain a functor
(q0 )! : Mod(R) → Mod(R)Q . The universal property of free diagrams yields
natural isomorphisms
homMod(R)Q ((q0 )! (R), X) ∼
= homR (R, X(q0 )) ∼
= Xq ,
0
i.e., (q0 )! (R) corepresents the evaluation functor (q0 )∗ : Mod(R)Q → Mod(R).
It follows that (q0 )! (R) is a projective object in the abelian category Mod(R)Q .
Moreover, (q0 )! (R) is an indecomposable object since any non-trivial decomposition would by the Yoneda lemma be induced by a non-trivial idempotent
endomorphism of q0 ∈ Q which is impossible since Q is a free category.
To specialize this a bit, let us consider a field R = k so that (q0 )! (k) is the
usual indecomposable projective representation P (q0 ) corresponding to the
vertex q0 ∈ Q. In the case of the linearly oriented A3 -quiver (1 → 2 → 3)
this yields the indecomposable projective representations
id
id
P (1) = (k → k → k),
id
P (2) = (0 → k → k),
and P (3) = (0 → 0 → k).
(ii) Let G be a discrete group, considered as a groupoid with one object, and let
i : ∗ → G classify the unique object. The free and the cofree diagram functors
i! , i∗ : C → C G , respectively, send an object X ∈ C to the G-objects
Y
a
∼
∼
X
and
i∗ (X) =
X.
i! (X) =
g∈G
g∈G
The G-actions simply permute summands and factors.
Specializing this to C = Mod(R) for a ring R and implicitly using the
equivalence Mod(R)G ' Mod(RG), this recovers the induction and coinduction functors Mod(R) → Mod(RG), i.e., the respective adjoints of the
restriction of scalar functor i∗ : Mod(RG) → Mod(R).
We mention one more example of Kan extension functors.
Example 5.32. Let C be a complete and cocomplete category, let G be a discrete
group, and let H < G be a subgroup. The Kan extensions along the corresponding
functor i : H → G give rise to adjunctions
(i! , i∗ ) : C H C G
and
(i∗ , i∗ ) : C G → C H .
The functors i! and i∗ are, respectively, referred to as induction and coinduction,
and they make perfectly well sense for not necessarily injective group homomorphisms. The reader is invited to use the pointwise formulas to obtain a more precise
description of these functors and to specialize to the case of modules over a ring.
The following property of Kan extensions is important, and this result justifies
the terminology Kan extensions.
Proposition 5.33. Let C be a complete and a cocomplete category and let u : A →
B be a functor between small categories.
(i) The functor u! : C A → C B is fully faithful and induces an equivalence onto the
full subcategory of C B spanned by all Y such that the counit : (u! ◦u∗ )Y → Y
is an isomorphism.
INTRODUCTION TO THE THEORY OF DERIVATORS
61
(ii) The functor u∗ : C A → C B is fully faithful and induces an equivalence onto the
full subcategory of C B spanned by all Y such that the unit η : Y → (u∗ ◦ u∗ )Y
is an isomorphism.
Proof. By duality it is enough to take care of the first statement, and it follows
from Lemma A.4 that u! is fully faithful if and only if the unit η : 1 → u∗ u! is
a natural isomorphism. For X : A → C and a ∈ A it follows from the pointwise
formula (Theorem 5.23) that u! (X)u(a) ∼
= colim(u/u(a)) X ◦ pu(a) . Note that the
fully faithfulness of u implies that (a, idu(a) : u(a) → u(a)) is a terminal object in
(u/u(a)). Hence by Corollary 5.10 the functor (a, idu(a) ) : 1 → (u/u(a)) is final,
and, as a particular instance of (5.6), there is a canonical isomorphism
X(a) = X ◦ pu(a) (a, 1) → colim(u/u(a)) X ◦ pu(a) .
Note that this canonical morphism is simply the map ι(a,1) belonging to the colimiting cocone of colim(u/u(a)) X ◦ pu(a) . But this map agrees with the component ηa
of the adjunction unit η : 1 → u∗ u! by the above explicit construction of left Kan
extensions (see (5.19)). Since this is the case for all a ∈ A, we conclude that η
indeed is a natural isomorphism, thereby establishing that u! is fully faithful. The
description of the essential image of u! is a special case of Lemma A.4.
Given a diagram X : A → C in a cocomplete category, let us recall that the colimit
on X is an initial cocone on X, i.e., it consists of the colimit object colimA X ∈ C
together with the colimiting cocone ιa : Xa → colimA X, a ∈ A. The colimiting
cocone is a natural transformation X → ∆A colimA X, and as noted in Lemma 3.34
the natural domain for natural transformations is the cylinder A × [1]. If the target
of the transformation is a constant diagram, then the tranformation factors over
the cocone in the sense of the following definition, and dually.
Definition 5.34. Let A be a small category.
(i) The cocone AB of A is the small category obtained by freely adjoining a
new terminal object ∞ ∈ AB .
(ii) The cone AC on A is the small category obtained by freely adjoining a new
initial object −∞ ∈ AB .
Let us be more specific about the construction of the cocone AB . The objects
in A are the objects in A and a unique new object ∞ ∈ AB . The set of morphisms
is defined by

 homA (x, y) , x, y ∈ A,
∗
, y = ∞,
homAB (x, y) =

∅
, x = ∞, y ∈ A.
And the composition law in AB is unique determined by the fact that there is a
fully faithful inclusion functor A → AB : x 7→ x. The construction of AC is dual
and there is also a fully faithful functor A → AC .
Examples 5.35. Let C be a complete and cocomplete category and let A be a small
category.
(i) Since the inclusion functor i : A → AB is fully faithful, we obtain by PropoB
sition 5.33 the left Kan extension functor i! : C A → C A is fully faithful. One
can check that the essential image consists precisely of the colimiting cocones.
In particular, for X : A → C, the diagram i! (X) : AB → C encodes colimA X
together with the colimiting cocone.
62
MORITZ GROTH
(ii) Dually, since the inclusion functor i : A → AC is fully faithful, the right
C
Kan extension functor i∗ : C A → C A is fully faithful. The essential image
consists precisely of the limiting cones. Starting with X : A → C, the diagram
i∗ (X) : AC → C encodes limA X together with the limiting cone.
(iii) We denote by = [1] × [1] the commutative square, i.e., the category
/ (1, 0)
(0, 0)
(0, 1)
=
/ (1, 1)
and by ip : p→ and iy : y → the full subcategories obtained by removing
the final object (1, 1) and the initial object (0, 0), respectively. Note that is the cocone on p and the cone on y. Hence, specializing the previous two
points, the Kan extension functors
(ip )! : C p → C and
(iy )∗ : C y → C are fully faithful and the essential images consist precisely of the pushout
squares and the pullback squares, respectively.
To conclude this section, let us briefly come back to the construction of kernels
and cokernels in terms of Kan extensions; see again the discussion of (5.2) and its
dual. With Examples 5.35 in mind it only remains to take care of the passage from
a morphism to the span in (5.2). To this end we consider the functor
j : [1] → p
and
k : [1] →y
classifying the horizontal morphism (0, 0) → (1, 0) and (0, 1) → (1, 1), respectively.
Lemma 5.36. Let C be a category admitting a zero object and let X0 → X1 be a
morphism in C, thereby determing a functor X : [1] → C.
(i) The right Kan extension (i0 )∗ (X) : p→ C is naturally isomorphic to the diagram on the left in the diagram
/ X1
X0
0
0,
X0
/ X1 .
(ii) The left Kan extension (i1 )! (X) : y → C is naturally isomorphic to the diagram on the right in the above diagram.
Proof. By duality it is enough to take care of the first statement. The functor
i0 : [1] → p is fully faithful. Hence also (i0 )∗ : C [1] → C p is fully faithful. In particular, for X : [1] → C, the counit : (i0 )∗ (i0 )∗ X → X is an isomorphism. Thus, it
remains to show that the value of (i0 )∗ (X) at the lower left corner (0, 1) ∈ p is a
zero object. By the pointwise formula for right Kan extensions (Theorem 5.25) we
calculate
(i0 )∗ (X)(0,1) ∼
= lim X ◦ q(0,1)
((0,1)/i0 )
= lim X ◦ q(0,1)
∅
∼
= 0.
INTRODUCTION TO THE THEORY OF DERIVATORS
63
In fact, it is obvious that the slice category ((0, 1)/i0 ) is empty, so that the limit
reduces to a zero object.
Thus, the intended passage from a morphism to the span in (5.2) amounts to
forming the right Kan extension along i0 : [1] → p. Combining this with Examples 5.35 we see that for a category C with a zero object the Kan extension functors
(i0 )∗
i
C [1] → C p →! C amount to passing from a morphism f : X0 → X1 to the pushout square
X0
0
f
/ X1
/ cok(f )
containing the cokernel.
While it is arguable how much is gained by this description of the cokernel if
one only cares about ordinary cokernels, the point is that the same description
yields a construction of functorial cones at the level of derived categories of abelian
categories and homotopy categories of Quillen model categories or ∞-categories.
This description only relies on having a suitable framework for a ‘calculus of abstract
Kan extensions’, and precisely such a framework is axiomatized by the notion of
a derivator. In §6 we define derivators and in §§7-10 we illustrate the notion by
indicating a few constructions available in such a derivator.
6. Basics on derivators
In §§6.1-6.2 we define derivators and introduce some basic terminology. To put it
as a slogan, derivators can be thought of as a minimal, purely categorical extension
of a derived category to a framework which comes with a powerful calculus of
derived limits and, more generally, derived Kan extensions. Similarly, the derivator
of spaces is such an extension of the homotopy category of spaces to a framework
with a calculus of homotopy limits and homotopy Kan extensions.
In §6.3 we mention quite a few examples of derivators, but postpone the proofs
that these actually are examples to the appendix. Finally, in §6.4 we emphasize
the difference between categorical limits on the one hand side and derived limits or
homotopy limits on the other side.
6.1. Prederivators. Let A be an abelian category and let D(A) be the derived
category. By Theorem 3.42 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 emphasized
in §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 and
these diagrams allow for many important constructions.
(ii) Diagram categories in derived categories are less well behaved in that their
objects do not carry enough information and many constructions can not be
performed 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 Proposition 3.47 and Warning 3.48. Since
the categories D(AA ) can not be reconstructed from D(A), this suggests that the
TODO: proof-read and
expand from here to
end of file.
64
MORITZ GROTH
passage to 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 (not
necessarily small) categories, functors, and natural transformations.
Definition 6.1. A prederivator D is a strict 2-functor D : Cat op → CAT .
The 2-category Cat op is obtained from Cat by reversing the direction of the functors
while the direction of the natural transformations is unchanged. Thus, given a
prederivator D, for every small category A we obtain a category D(A). Associated
to a functor u : A → B in Cat, there is an induced precomposition functor or
restriction 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, we
obtain an induced natural transformation α∗ : u∗ → v ∗ , as depicted in the diagram
u
A
u∗
'
7 B,
D(B)
v
v∗
+
3 D(A).
This datum is compatible with compositions and identities in a strict sense, i.e., we
have equalities of the respective expressions and not only coherent natural isomorphisms between them.
Recall that 1 denotes the terminal category. Given a prederivator D, we refer to
D(1) as the underlying category of D. As we will see in this book, the structure
of 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 prederivator
y(C) defined by
y(C)(A) := C A , A ∈ Cat.
Given a functor u : A → B in Cat, we define y(C)(u) to be the precomposition
functor u∗ : C B → C A : X 7→ X ◦ u. Finally, given two functors u, v : A → B
and a natural transformation α : u → v the reader easily checks that
∗
αX
= X ◦ α : X ◦ u → X ◦ v,
X ∈ CB ,
defines a natural transformation α∗ : u∗ → v ∗ , concluding the definition of
the 2-functor y(C). The underlying category of y(C) is canonically isomorphic
to 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 be
defined for derivators such that in the represented case they reduce to the
classical ones.
(ii) Let A be an abelian category, let Ch(A) be the category of unbounded
chain complexes, and let W be the class of quasi-isomorphisms in Ch(A).
Given a small category A, we again denote by W A the levelwise quasiisomorphisms, i.e., those natural transformations X → Y : A → Ch(A) such
INTRODUCTION TO THE THEORY OF DERIVATORS
65
that all components are quasi-isomorphisms. The homotopy prederivator
DA : Cat op → CAT is defined by the localizations
DA (A) := Ch(A)A [(W A )−1 ],
∗
A ∈ Cat.
B
Since the restriction functors u : Ch(A) → Ch(A)A clearly preserve levelwise quasi-isomorphisms, there are canonical functors u∗ : DA (B) → DA (A)
compatible with the localization functors (compare to Lemma 3.45). Using
that these localizations are 2-localizations (see Proposition 3.35), the reader
easily concludes the definition of the 2-functor DA : Cat op → CAT . The underlying category of this prederivator is canonically isomorphic to the derived
category D(A). More generally, it follows from Lemma 3.10 that there are
isomorphisms
DA (A) ∼
= D(AA ), A ∈ Cat.
(iii) Let Top denote the category of topological spaces and continuous maps. We
recall that a continuous map f : X → Y is a weak homotopy equivalence if
for 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 W A the levelwise
weak homotopy equivalences in TopA . The (homotopy) prederivator of
topological spaces is defined by
H oTop (A) := TopA [(W A )−1 ],
A ∈ Cat.
As in the previous example, the reader easily checks (using a variant of Proposition 3.35) that this extends to a 2-functor H oTop : Cat op → CAT . The underlying category of H oTop is canonically isomorphic to the usual homotopy
category of spaces,
H oTop (1) ∼
= Ho(Top).
(iv) We conclude by a broad class of examples induced by Quillen model categories; see the original [Qui67], the monographs [Hov99, Hir03] or the introductory [DS95]. Let M be a Quillen model category with weak equivalences W . For every small category A, the category MA comes with the
class W A of levelwise weak equivalences. Although W A does not necessarily
belong to a Quillen model structure on MA , we can define the homotopy
prederivator H oM of M by
H oM (A) := (MA )[(W A )−1 ],
and it can be shown that these categories are locally small. The underlying
category is canonically isomorphic to the usual homotopy category,
H oM (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 one
of Heller [Hel88] and Franke [Fra96]) is based on the idea that we want to
model diagrams (covariant functors) in a fixed abstract homotopy theory.
This results in the use of Cat op 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 Cat op is replaced by Cat coop , obtained
by also reversing the orientations of the natural transformations. Although
the resulting theories are equivalent, many statements differ slightly since the
66
MORITZ GROTH
directions of various natural transformations in each convention are reversed
with respect to the other.
(ii) Often (for example under certain finiteness conditions) it is convenient to have
some flexibility with respect to the domain of definition of (pre)derivators by
restricting the class of ‘admissible shapes’. There is the notion of a category
of diagrams Dia, a full sub-2-category of Cat enjoying certain closure properties, and the related definition of a (pre)derivator of type Dia as a 2-functor
Dia op → CAT . In this introductory book we will not get into this.
We establish some basic terminology related to prederivators D. Motivated by
Examples 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 an
abstract 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 of
shape A as we discuss next.
∼ A1 and hence write
We keep using implicitly the canonical isomorphisms A =
a : 1 → A for the functor classifying a ∈ A. As a special case of a precomposition
functor we obtain an evaluation functor a∗ : D(A) → D(1). Given a morphism
g : X → Y in D(A) we will write ga : Xa → Ya for the induced map in the underlying
category D(1).
Similarly, every morphism f : a → b in A yields a natural transformation of the
corresponding functors 1 → A, and for every prederivator we hence obtain
a
1
b
a∗
'
7 A,
D(A)
b∗
*
4 D(1) .
Evaluated at X ∈ D(A) this defines a map f ∗ : Xa → Xb . We summarize the
functoriality 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 functor
diaA (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 underlying diagram functor
diaA : D(A) → D(1)A .
Proof. The proof is left as an exercise. We suggest the reader to really do this
exercise in order to get used to the concept of 2-functoriality.
This construction is an abstract version of Proposition 3.47.
Examples 6.6. We take up again Examples 6.2 and describe the associated underlying diagram functors.
(i) For represented prederivators y(C) the functors diaA : C A → (C 1 )A reduce to
the canonical isomorphisms C A ∼
= (C 1 )A .
INTRODUCTION TO THE THEORY OF DERIVATORS
67
(ii) In the case of the homotopy prederivator DA of an abelian category, the
functor diaA : DA (A) → DA (1)A corresponds under DA (A) ∼
= D(AA ) to the
A
A
functor diaA : D(A ) → D(A) of Proposition 3.47. In general, this functor
is not 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 : H oM (A) → H oM (1)A gets identified with MA [(W A )−1 ] → Ho(M)A .
If we use the description of Ho(M) as the category of cofibrant and fibrant
objects and homotopy classes of morphisms, then this example makes precise
that 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 an
ordinary diagram which is to say that we draw diaA (X). However, it is important to
note 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 diagrams
and incoherent ones. The main point of the theory of derivators is that many
important constructions which are available at the level of coherent diagrams can
not be performed anymore at the level of underlying incoherent diagrams. To put
it more frankly, this suggests that the functors diaA should not be applied.
Prederivators allow us to describe systems of categories of diagrams and associated restriction functors and similarly for derived categories or homotopy categories
of such diagrams. However, to be able to perform more interesting constructions
we 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 derived categories or homotopy categories together with a well-behaved calculus of
homotopy Kan extensions. For (pre)derivators we follow the established terminology for ∞-categories and simply speak of Kan extensions as opposed to homotopy
Kan extensions (and similarly for related notions). This does not result in a risk
of ambiguity since in the context of an abstract (pre)derivator the concept ‘Kan
extension’ 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 restriction
functor 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 restriction
functor u∗ : D(B) → D(A) has a right adjoint u∗ : D(A) → D(B),
(u∗ , u∗ ) : D(B) D(A).
Motivated by Example 5.27, in the case of left Kan extensions along functors
πA : A → 1 to the terminal category we also speak of colimits of shape A and
write (πA )! = colimA . Similary, right Kan extensions along πA will be referred
to as limits of shape A and will occasionally be denoted by (πA )∗ = limA .
Sometimes we want to emphasize notationally with respect to which prederivator
certain functors are considered. In that case, we will also write
u∗D ,
uD
! ,
colimD
A,
uD
∗ ,
and
limD
A.
68
MORITZ GROTH
(5.24)
In classical category theory, a key formal property of Kan extensions is that in
sufficiently (co)complete categories they can be calculated pointwise; see §??. Let C
be a complete and cocomplete category and let X : A → C be a diagram. For every
u : A → B in Cat the left Kan extension u! (X) : B → C and the right Kan extension
u∗ (X) : B → C exist (??). Moreover, by ?? they can be calculated pointwise in that
for every b ∈ B certain canonical natural transformations
colim(u/b) X ◦ p → u! (X)b
u∗ (X)b → lim(b/u) X ◦ q
and
are isomorphisms. Here, (u/b) and (b/u) are slice categories and p : (u/b) → A and
q : (b/u) → A are the corresponding canonical projections functors.
In the represented case one can use these pointwise (co)limit expressions to
construct a diagram of shape B which will be a Kan extension. This is impossible
for an abstract prederivator D since objects in D(B) are just abstract objects,
in particular, not ordinary diagrams. For a derivator we nevertheless ask that
Kan extensions exist and can be calculated pointwise. To make this precise, let us
consider the slice squares
(6.9)
π(u/b)
/A
p
(u/b)
1
and
u
/B
b
π(b/u)
/A
q
(b/u)
AI
1
b
u
/ B.
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 corner
we obtain u(a) while the other path yields b. Thus, in general the square does not
commute 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.
Proof. This proof is left as an exercise.
Now, let us assume that the prederivator D admits the necessary Kan extensions. Associated to the slice squares (6.9) we obtain the following canonical matetransformations
(6.11)
(6.12)
η
colim(u/b) p∗ = π! p∗ → π! p∗ u∗ u! → π! π ∗ b∗ u! → b∗ u!
η
and
b∗ u∗ → π∗ π ∗ b∗ u∗ → π∗ q ∗ u∗ u∗ → π∗ q ∗ = lim(b/u) q ∗ ,
where the transformations denoted η and are suitable adjunction units and counits,
respectively. The transformation (6.11) is hence defined as the following pasting
D(1) o
Y
id
p∗
D(u/b) o
O
π!
π∗
D(1) o
b∗
D(A) j
O
u∗
D(B) o
id
u!
η
D(A),
INTRODUCTION TO THE THEORY OF DERIVATORS
69
and similarly for (6.12). (We will say a bit more about the passage to canonical
mates in §7.1.) It turns out that the good way to express that Kan extensions in D
are 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.
op
(Der1) D : Cat`
→ CAT takes
Q coproducts to products, i.e., the canonical functor 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 matetransformations (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 disjoint union is canonically determined by its restrictions to the summands.
Axiom (Der2) encodes the idea that a transformation of two coherent diagrams of the same shape is an isomorphism if and only if all components
are. This is an abstraction of the fact that a natural transformation is invertible if and only if each component is. The remaining two axioms encode a
‘(homotopical ) completeness and cocompleteness property’ together with the
pointwise formulas for Kan extensions.
(ii) The formalism behind axiom (Der4) (based on the calculus of canonical
mates) is arguably the key tool in the theory of derivators and will be studied
in more detail in §7. Although a bit heavy on a first view, this formalism
turns 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 isomorphisms’ first of all exist and second turn out to be isomorphisms.
(iii) Note that axioms (Der1)-(Der4) are asking for properties while the only actual structure is the underlying prederivator. This is in contrast to more
classical approaches including triangulated categories in which case some noncanonical 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’ formal
properties of the calculus of Kan extensions. In a way, it is surprising that
these 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 that
there is some flexibility with respect to the allowable shapes in the definition
of a derivator. By the very definition a derivator encodes the idea of having a
complete and cocomplete homotopy theory (enjoying some nice properties).
Under certain finiteness assumptions (like in K-theoretic contexts) one wants
to restrict to finite shapes only and this can be done by means of categories
of diagrams. Related to this, there are also variants of derivators satisfying
only half of (Der3) and half of (Der4), leading to left derivators and right
derivators. We will not get into these variants in this introductory book.
70
MORITZ GROTH
Related to Warning 6.7 we remark the following.
Warning 6.15. Note that we defined a derivator to be a prederivator which is
homotopically cocomplete and homotopically complete in that all restriction functors
u∗ : D(B) → D(A) have left adjoints and right adjoints. It is important to be
aware 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 an
abstract derivator, in general, ordinary categorical Kan extensions and categorical
(co)limits do not exist.
We will expand on this warning in §6.4.
6.3. Examples of derivators. In this subsection we collect a few examples of
derivators (see again Examples 6.2). Detailed proofs that we actually obtain derivators are a bit more involved and we suggest the reader to treat them as black boxes
on a first reading. For convenience, in the appendix we nevertheless include detailed
proofs at least in the case of represented derivators and in the case of homotopy
derivators 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 a
derivator if and only if C is complete and cocomplete.
As already mentioned, this example is convenient when it comes to generalizing
notions 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 to
derivators, i.e., that the typical passages to homotopy categories factor through
derivators. Here we will focus on Quillen model categories. To begin with, let us
recall that a Quillen model category is combinatorial if the model structure is cofibrantly generated and if the underlying category is locally presentable. We do not
include a discussion of locally presentable categories here and instead only mention
that chain complexes over a ring (more generally, chain complexes in Grothendieck
abelian categories) and simplicial sets yield examples of locally presentable categories. 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 homotopy
prederivator
H oM : Cat op → CAT : A 7→ MA [(W A )−1 ]
is a derivator, the homotopy derivator of M.
A proof of this result [Gro13] is included in §B.2.
Remark 6.18. Renaudin
Before we collect more specific examples, let us recall the following definition.
Definition 6.19. Let A be an abelian category.
(i) A generator is an object G ∈ A such that f : X → Y in A is an isomorphism
if 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 filtered
colimits in A are exact, and if it has a generator.
INTRODUCTION TO THE THEORY OF DERIVATORS
71
It can be shown that Grothendieck abelian categories are also complete, see for
example [KS06, §8.3]. To illustrate this notion we include the following examples.
Examples 6.20.
(i) The category of R-modules, R a ring, is a Grothendieck
abelian category.
(ii) The category of quasi-coherent OX -modules on an arbitrary scheme X is a
Grothendieck abelian category.
(iii) For every Grothendieck abelian category A the category Ch(A) of chain complexes in A is again a Grothendieck abelian category.
A proof of the following result can be found in [Hov01].
Theorem 6.21. Let A be a Grothendieck abelian category. The category Ch(A)
admits a combinatorial model structure with the class of quasi-isomorphisms as
class of weak equivalences.
The results stated so far already allow us to obtain interesting examples of derivators arising in homological algebra, algebraic geometry, and topology (see Examples 6.24). To indicate that there are many additional examples of derivators, we
state the following two theorems. The first result is due to Cisinski [Cis03] and it
is a rather deep result.
Theorem 6.22. Let (M, W ) be a model category. The homotopy prederivator
H oM : Cat op → CAT : A 7→ MA [(W A )−1 ]
is a derivator, the homotopy derivator of M.
Similarly, ∞-categories give rise to derivators. A sketch proof of the following
result can be found in [GPS14]. Recall that the nerve N A of a small category A
is the simplicial set which in simplicial degree n is given by the set of strings of n
composable arrows a0 → a1 → . . . → an .
Theorem 6.23. Let C be a complete and cocomplete ∞-category. There is a derivator
H oC : Cat op → CAT : A 7→ Ho(C N A ),
the homotopy derivator of the ∞-category C.
Thus, derivators encode aspects of the calculus of derived Kan extensions in nice
abelian categories and, more generally, of the calculus of homotopy Kan extensions
in model categories and ∞-categories. We now include a few more specific examples.
Examples 6.24.
(i) For every ring R the category of R-modules is Grothendieck
abelian and
DR : Cat op → CAT : A 7→ Ch(R)A [(W A )−1 ]
hence defines a derivator, the derivator of the ring R. In particular, we
have the derivator DZ of the integers and the derivator Dk of a field k. These
derivators enhance the more classical derived categories.
Note that DR encodes a lot of information concerning the ring R as we
illustrate 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. Moreover, as special cases of (derived) Kan extension functors, DR encodes
group cohomology and group homology as well as their versions for chain
complexes (see §2.3).
finitely many objects
72
MORITZ GROTH
(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 any
quiver Q are encoded by DR .
(c) As a further variant, DR also encodes derived categories of incidence
algebras RP of (finite) posets P as well as derived categories of category
algebras RA (of finite categories A).
(ii) Given a scheme X, let Ch(X) be the category of chain complexes of quasicoherent OX -modules on X. For every small category A we denote by W A
the levelwise quasi-isomorphisms in Ch(X)A . The prederivator
DX : Cat op → CAT : A 7→ Ch(X)A [(W A )−1 ]
is a derivator, the derivator of the scheme X. The underlying category
DX (1) is isomorphic to the more classical derived category D(X).
(iii) More generally, for every Grothendieck abelian category A there is the derivator
DA : Cat op → CAT : A 7→ Ch(A)A [(W A )−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 equivalences. Denoting by W A the levelwise weak homotopy equivalences in TopA ,
there is a derivator
H oTop : Cat op → CAT : A 7→ TopA [(W A )−1 ],
the (homotopy) derivator of topological spaces.
(v) Let us recall the notion of a spectrum in the sense of topology. A spectrum X consists of pointed topological spaces Xn , n ≥ 0, and pointed maps
ΣXn → Xn+1 . A morphism f : X → Y of spectra is a collection of pointed
maps fn : Xn → Yn compatible with the structure maps. The category Sp
of spectra can be endowed with the stable model structure, having the property that the homotopy category Ho(Sp) is the classical stable homotopy
category SHC (see [Vog70] and [Ada74, Part III]). We denote the associated
homotopy derivator H oSp by
(6.25)
Sp : Cat op → CAT : A 7→ SpA [(W A )−1 ]
and refer to it as the derivator of spectra.
There are many additional examples and we will mention some of them in our
short 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 concerning
limits. This duality principle is based upon the passage to opposite categories. In
order to obtain a similar principle we include the following example.
Example 6.26. Given a prederivator D, the opposite prederivator D op is defined
by setting
D op (A) = D(Aop )op .
The following result formalizes the self-duality of the axioms of a derivator. It
leads to the important duality principle. In more detail, many statements in the
theory of derivators have dual versions and by the proposition it will be sufficient
to state and prove only one of them.
INTRODUCTION TO THE THEORY OF DERIVATORS
73
Proposition 6.27. A prederivator D is a derivator if and only if the opposite D op
is a derivator.
Proof. The proof is left as an exercise. As a hint we mention that for u : A → B
op
the left Kan extension functor uD
with respect to D op is closely related to the
!
op D
right Kan extension functor (u )∗ with respect to D.
6.4. Limits versus homotopy limits. We expand a bit on Warning 6.15 and
consider a derivator D and a small category A, asking ourselves whether the category D(A) admits categorical (co)limits of shape B. In this subsection only, we
use the expressions categorical (co)limits and categorical Kan extensions in order
to refer to the usual constructions from category theory and the terms homotopy
(co)limits and homotopy Kan extensions to refer to adjoints to restriction functors
u∗ : D(D) → D(C) in derivators D.
By definition of categorical (co)limits these would be adjoint functors to the
diagonal functor ∆B : D(A) → D(A)B which sends X ∈ D(A) to the constant Bshaped diagram with value X. The point of this subsection is that, in general, the
axioms 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 be
∗
: D(A) → D(B × A). By
the projection away from B with restriction functor πB
∗
admits a left adjoint (πB )! : D(B × A) → D(A) and a
definition of a derivator, πB
right adjoint (πB )∗ : D(B × A) → D(A). Note that the two functors
(6.28)
∆B : D(A) → D(A)B
and
∗
πB
: D(A) → D(B × A)
have the same domain but that the targets, in general, are essentially different
categories. In fact, objects in D(B × A) are diagrams which are coherent in the Aand the B-direction, while objects in D(A)B are coherent in the A-direction only.
The two functors in (6.28) are related by the following partial underlying diagram
functors. Given categories A and B, evaluating on objects and morphisms in B
only 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 given
X ∈ D(B × A) we set
(6.29)
diaB,A (X)(b) = (b × 1A )∗ X ∈ D(A).
Lemma 6.30. 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 (6.29).
∗
(ii) The functor diaB,A satisfies ∆B = diaB,A ◦ πB
: D(A) → D(A)B .
Proof. This proof is left as an exercise.
The underlying diagram functor diaB : D(B) → D(1)B of Lemma 6.5 is canonically 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 from
being equivalences.
74
MORITZ GROTH
Thus, given a derivator D and small categories A, B there is the diagram
D(B × A)
O
(6.31)
∗
(πB )! πB
D(A)
diaB,A
/ D(A)B
O
(πB )∗
∆B
/ D(A).
id
If the partial underlying diagram functor diaB,A : D(B × A) → D(A)B is an equivalence, then also ∆B : D(A) → D(A)B has adjoints on both sides, which is to say
that D(A) has categorical (co)limits of shape B.
Lemma 6.32. 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.
Proof. Let S be a set which we consider as a discrete category, i.e., a category
with identity
` morphisms
Q only. As a special case of axiom (Der1), the canonical
functor 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)
Q
diaS,A
/ D(A)S
commutes where the vertical maps are the canonical identifications. Thus, for
discrete categories S the functors diaS,A are equivalences as well. It is immediate
from the above discussion (see (6.31) in the case of B = S) that D(A) hence has
S-fold products and coproducts. Specializing to S = ∅ we deduce the existence of
terminal and initial objects.
Remark 6.33.
(i) One way to refer to this result is by saying that in derivators
categorical (co)products and homotopy (co)products both exist and that they
agree. This is particular to (co)products in that, in general, D(A) does not
admit further categorical (co)limits.
(ii) Using categories of diagrams one can also include examples of derivators such
that 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 formalism
relies on the calculus of mates, a certain calculus which applies to natural transformations. We begin in §7.1 by reviewing this calculus and illustrate it by some
examples.
In §7.2 we introduce the related notion of homotopy exact squares, arguably the
main technical tool in the theory of derivators. While working with derivators, one
often runs into the situation that outputs of certain constructions ‘obviously are
isomorphic’. The formalism of homotopy exact squares often allows one to, first,
construct canonical maps between such gadgets and, second, to deduce that these
INTRODUCTION TO THE THEORY OF DERIVATORS
75
canonical maps actually are isomorphisms. To conclude this section, in §7.3 we
collect a few first applications of homotopy exact squares to the calculus of Kan
extensions in derivators.
7.1. The calculus of mates. Before we define canonical mates, let us quickly
recall the notion of pasting. This is a composition operation which applies to natural
transformations living in ‘larger diagrams’. Here we refrain from giving a precise
definition of such diagrams and instead only mention a few relevant examples. Let
us consider the diagram
v1∗
CO 1 o
}
u∗
1
(7.1)
D1 o
w1∗
v2∗
CO 2 o
u∗
2
D2 o
CO 3
}
u∗
3
D3 ,
w2∗
which consists of categories, functors, and transformations α1 : v1∗ u∗2 → u∗1 w1∗ and
α2 : v2∗ u∗3 → u∗2 w2∗ . We define the natural transformation α1 α2 as
α1 α2 = (α1 w2∗ ) ◦ (v1∗ α2 ) : v1∗ v2∗ u∗3 → v1∗ u∗2 w2∗ → u∗1 w1∗ w2∗ ,
and refer to it as the horizontal pasting of the squares in (7.1). Similarly, we
define the vertical pasting of squares. The pasting operation also applies to more
complicated such diagrams as long as all natural transformations ‘point in the same
direction’. 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 counit
: LR → 1. We recall from §A.1 that the unit and the counit are subject to the
triangular identities,
Lη
L
/ LRL
/ RLR
ηR
R
L
R
+ L,
=
+ R.
=
Using the pasting operation we can rewrite the triangular identities concerning the
left adjoint L as
L /
C
D
C
id
η ;C
;C
=
R
id
+C
L
L
/D
D,
and there is a similar picture for R.
As a special case of this pasting operation we now define mates. Let us consider
a natural transformation α : p∗ u∗ → v ∗ q ∗ living in a square of not necessarily small
categories
CO 1 o
(7.3)
v∗
D1 o
p∗
}
q∗
CO 2
u∗
D2 .
76
MORITZ GROTH
Having our applications in mind, in this section functors are often denoted like
precomposition functors u∗ and left adjoints and right adjoints to such functors are
denoted by u! and u∗ , respectively. If the necessary adjoint functors exist, then any
natural transformation α as in (7.3) has two associated canonical mates, namely
η
α
α! : v! p∗ → v! p∗ u∗ u! → v! v ∗ q ∗ u! → q ∗ u!
(7.4)
η
α
and
α∗ : u∗ q∗ → p∗ p∗ u∗ q∗ → p∗ v ∗ q ∗ q∗ → p∗ v ∗ .
(7.5)
Here, η denotes the units of the adjunctions (u! , u∗ ) and (p∗ , p∗ ), and 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
right,
CO 2 k
=
}
p∗
D1U o
}
v!
=
CO 1 o
CO 2 k
p∗
}
v∗
D1 o
q
u∗
∗
CO 1 o
=
D2 o
}
u!
p∗
}
v∗
D1U o
C2 ,
=
q
}
CO 2
u∗
∗
DO 2
q∗
D1 .
Remark 7.6.
(i) Note that the construction of the canonical mates depends on
choosing certain adjoint functors and adjunction (co)units. By Lemma A.11
adjoint functors are unique up to unique isomorphism in a way that the
isomorphism is compatible with adjunction units and counits. This implies
that the canonical mates are well-defined up to some pasting with certain
canonical natural isomorphisms. In particular, the question if a canonical
mate is an isomorphism does not depend on the choices. This motivates us
to 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 functors
have adjoints on both sides. Using ‘conjugations by units and counits from
opposite sides’ the two natural transformations (7.4) and (7.5) are the only
ones which make sense for arbitrary α. Thus, the notation α 7→ α! and
α 7→ α∗ is unambiguous.
Examples 7.7.
(i) Associated to a functor u∗ : D → C there is the identity transformation u∗ → u∗ , which populates the two squares
CO o
u∗
Do
u∗
|
id
=
CO o
DO
=
D,
=
{
=
Co
id
u∗
CO
u∗
D.
If u∗ admits a left adjoint u! : C → D then the canonical mates (7.4) of
the two squares yield the counit : u! u∗ → id and the unit η : id → u∗ u! ,
respectively. Similarly, if u∗ admits a right adjoint u∗ : C → D, then the
canonical mates (7.5) of these two squares are the unit η : id → u∗ u∗ and the
counit : u∗ u∗ → id, respectively.
INTRODUCTION TO THE THEORY OF DERIVATORS
77
(ii) Let us consider functors u∗ , v ∗ : D → C and a natural transformation u∗ → v ∗
which we want to rewrite as two different squares
CO o
=
|
v∗
Do
u∗
id
{
=
Co
D,
=
u∗
CO o
CO
DO
=
id
v∗
D.
populated by this transformation. If u∗ and v ∗ admit left adjoints u! and
v! , 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 the
canonical mate (7.5) of the square on the right is the total mate v∗ → u∗ as
defined in (A.7).
(iii) Let A be a small category and let F : C → D be a functor. We abuse notation
and simply write F : C A → DA for the induced functor. By the very definition
we have the two commutative squares
/ CA
O
F
DO A
!
∆A
D
F
F
DO A
∆A
Ya
∆A
/ C,
D
/ CA
O
∆A
F
/ C.
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 the
right, 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 functor
between small categories, and let b ∈ B. In the background of ?? there were
the slice squares
p
(u/b)
π(u/b)
1
/A
u
b
/B
(b/u)
and
π(b/u)
/A
q
AI
1
u
/ B.
b
Here, the component of the natural transformation in the square on the left
at the object (a, f : u(a) → b) is given by f , and similarly in the square on
the right. Passing to diagram categories we obtain induced natural transformations
p∗
o
C (u/b)
O
∗
π(u/b)
Co
b∗
o
C (b/u)
O
CA
O
u∗
CB
and
q∗
AI
∗
π(b/u)
Co
b∗
CA
O
u∗
CB .
78
MORITZ GROTH
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
b∗ ◦ RKanu → lim(b/u) ◦ q ∗
and
of ??, expressing that Kan extension in sufficiently (co)complete categories
are pointwise.
Lemma 7.8. The passages to canonical mates are compatible with respect to horizontal and vertical pasting as expressed by the formulas
(α1 α2 )! = (α2 )! (α1 )!
(α1 α2 )∗ = (α2 )∗ (α1 )∗ .
and
Proof. Let us consider the horizontal pasting α1 α2 of two natural transformations
α1 : v1∗ u∗2 → u∗1 w1∗ and α2 : v2∗ u∗3 → u∗2 w2∗ ,
CO 1 o
u∗
1
D1 o
v1∗
v2∗
CO 2 o
}
}
u∗
2
w1∗
CO 3
D2 o
u∗
3
w2∗
D3 .
Assuming that the functors u∗1 , u∗2 , and u∗3 have left adjoints, the pasting (α2 )! (α1 )!
is by definition given by
D1U o
(u1 )!
}
=
CO 1 o
u∗
1
D1 o
v1∗
CO 2 k
}
w1∗
=
}
u∗
2
D2U o
(u2 )!
}
CO 2 o
}
u∗
2
=
v2∗
D2 o
w2∗
CO 3 k
u∗
3
D3 o
=
}
(u3 )!
D3 .
Note that the two triangles in the middle cancel out by a triangular identity (see
Example 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
DO 1 o
(7.9)
q∗
D2 o
v!
}
u!
CO 1
p∗
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∗ are
inverse to each other, i.e., we have α = (α! )∗ and α = (α∗ )! .
Proof. This proof is left as an exercise.
Let us now consider a natural transformation (7.3) such that all four functors
have left adjoints. The canonical mate α! can again be written as in (7.9), and by
our assumption it has a further canonical mate (α! )! : q! v! → u! p! .
INTRODUCTION TO THE THEORY OF DERIVATORS
79
Lemma 7.11. Let α be a natural transformation (7.3) such that all functors have
left adjoints. The canonical mate (α! )! : q! v! → u! p! is conjugate to α : p∗ u∗ → v ∗ q ∗ .
Proof. This proof is left as an exercise.
Lemma 7.12. Given a natural transformation α as in (7.3). The canonical mates
α! : v! p∗ → q ∗ u! and α∗ : u∗ q∗ → p∗ v ∗ are conjugate. In particular, α! is an isomorphism if and only if α∗ is an isomorphism.
Proof. The canonical mate α! can again be written as in (7.9). Note that for that
natural transformation all four functors have right adjoints. By Lemma 7.10 we
hence conclude that the canonical mate α∗ : u∗ q∗ → p∗ v ∗ is given by ((α! )∗ )∗ and
the 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 the
calculus of mates to derivators. Let D be a derivator and let us consider a natural
transformation in Cat which lives in a square
(7.13)
v
Since a derivator is a 2-functor Cat
mation
op
/A
p
C
D
|
α
q
u
/ B.
→ CAT we have an induced natural transfor-
D(C) o
O
p∗
v∗
D(D) o
D(A)
O
u∗
α∗
D(B).
q∗
As summarized in §7.1, such a transformation has canonical mates
(7.14)
(7.15)
η
α∗
α! : v! p∗ → v! p∗ u∗ u! → v! v ∗ q ∗ u! → q ∗ u!
η
α
∗
and
α∗ : u∗ q∗ → p∗ p∗ u∗ q∗ → p∗ v ∗ q ∗ q∗ → p∗ v ∗ .
(Again we generically write η for adjunction units and for adjunction counits.)
Thus, α! is a natural transformation of functors D(A) → D(D) and α∗ a natural
transformation of functors D(D) → D(A). It follows from Lemma 7.12 that α! is
an 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 calculated
pointwise and this is made precise by axiom (Der4). Note that the two
relevant natural transformations (6.11) and (6.12) in that axiom are instances
of canonical mates, namely the ones associated to the two types of slice
squares (6.9). Thus, (Der4) is precisely saying that slice squares are homotopy
exact.
80
MORITZ GROTH
(ii) As a special case of the slice squares (6.9) we reconsider the following situation
from Examples 5.35. Let A be a small category and let AB be the cocone
on 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
the square
1
/A
A
πA
1
}
∞
i
/ AB .
In fact, it is easily seen that this square is isomorphic to the slice square
associated to i : A → AB and ∞ ∈ AB , and the square is hence homotopy
exact.
(iii) The class of homotopy exact squares is closed under horizontal and vertical
pasting. In fact, this is immediate from Lemma 7.8.
The point of this definition is that there is a battery of further examples of
homotopy exact squares. These squares allow us to show that in the context of
abstract derivators (like homotopy derivators of model categories or complete and
cocomplete ∞-categories) certain canonical maps exist and are isomorphisms. A bit
more sloppy, these squares hence establish ‘rules which allow us to simultaneously
manipulate categorical Kan extensions, derived Kan extensions, and homotopy Kan
extensions’; see §6.3. For example, one ‘of course’ expects that given a category
A ∈ Cat admitting a terminal object t ∈ A then for every derivator D and diagram
X ∈ D(A) there should be a canonical isomorphism Xt ∼
= colimA X in D(1). This
and further first results along these lines are established in §7.3.
7.3. First applications to Kan extensions. In this section we illustrate the use
of homotopy exact squares by extending a few results about the calculus of Kan extensions from ordinary category theory to the framework of abstract derivators. By
the 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 following
square is homotopy exact
πB
1
/A
v
B
|
1
id
πA
/ 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, ) : A B we obtain an induced adjunction
(v ∗ , u∗ , η ∗ , ∗ ) : D(B) D(A).
Thus, u∗ is a model for the right Kan extension functor v∗ . By definition of a
homotopy exact square we hence have to show that
∗
∗
∗
η ∗ πA
: πA
→ u∗ v ∗ πA
INTRODUCTION TO THE THEORY OF DERIVATORS
81
is an isomorphism for every derivator D. But this natural transformation is given
∗
by η ∗ πA
= (πA η)∗ = 1∗ = 1, and hence clearly is an isomorphism.
In more standard notation, we thus have canonical isomorphisms
X ∈ D(A),
colimB v ∗ X → colimA X,
saying that colimits of coherent diagrams are not affected by restrictions along right
adjoint 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
πA
1
/B
u
A
|
1
=
/1
πB
is homotopy exact.
Examples 7.20.
(i) Identity functors are final as are compositions of final functors.
(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 Proposition 5.12 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 of
Definition 7.19 is much more restrictive. In fact, it turns out that a functor u is
final 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 a
canonical isomorphism
∼
=
Xt → colimA X.
Proof. This is immediate from Lemma 7.18 since t : 1 → A is right adjoint to
πA : A → 1.
Corollary 7.23. Let A ∈ Cat admit a terminal object t. There is a natural iso∗ ∼
morphism πA
= t∗ : D(1) → D(A). Moreover, t∗ is fully faithful and the essential
image consists precisely of those X ∈ D(A) such that all morphisms in diaA (X)
are isomorphisms.
Proof. This proof is left as an exercise.
Example 7.24. The inclusion of the terminal object 1 : 1 → [1] induces a fully
faithful functor i∗ : D(1) → D([1]). An object X ∈ D([1]) lies in the essential image
of 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 of
Kan extensions. Since this is the first proof of this kind in this book, we give a
fairly detailed proof.
82
MORITZ GROTH
Proposition 7.25. If u : A → B is fully faithful, then the square
(7.26)
/A
id
A
|
id
A
u
1
/B
u
is homotopy exact. This is equivalent to saying that for every derivator D the
functors u! , u∗ : D(A) → D(B) are fully faithful.
Proof. The canonical mate (7.14) associated to this square is the adjunction unit
η : 1 → u∗ u! while the canonical mate (7.15) is the counit : u∗ u∗ → 1. Thus, the
square (7.26) is homotopy exact if and only if the Kan extension functors u! , u∗ are
fully faithful.
Let us now show that this square is homotopy exact by showing that the unit
η : 1 → u∗ u! is an isomorphism. Since isomorphisms can be detected pointwise (by
axiom (Der2)) it is enough to show that all components ηa , a ∈ A, are isomorphisms.
For this purpose, let us consider the following pasting
p
(A/a)
(7.27)
π(A/a)
~
1
/A
id
a
/A
/A
id
{
1
u
u
/B
where the square on the left is a slice square (6.9). The functoriality of mates with
respect to pasting implies that the canonical mate of this pasting is
∼
=
ηa
(π(A/a) )! p∗ → a∗ → a∗ u∗ u! ,
where the first map is an isomorphism by (Der4). Thus, ηa is an isomorphism if
and 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)) : (a0 , f : a0 → a) 7→ (a0 , u(f ) : u(a0 ) → u(a))
is an isomorphism of categories. Using this isomorphism we can rewrite the pasting
(7.27) as the pasting
(A/a)
(7.28)
π(A/a)
1
∼
=
/ (A/u(a))
π(A/u(a))
/1
p
u(a)
/A
u
/B
in which the square on the right is a slice square (6.9). Since isomorphisms are final
functors (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, which
concludes 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 only
if the adjunction counit : 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
INTRODUCTION TO THE THEORY OF DERIVATORS
83
object b ∈ B. The point of the following lemma is that it suffices to take care of
the 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 adjunction
counit b : 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. By
Proposition 7.25, u! is fully faithful and X lies in the essential image of u! if and
only if the adjunction counit : u! u∗ (X) → X is an isomorphism. By (Der2) this is
the case if and only if we all components b , b ∈ B, are isomorphisms, establishing
one direction. The converse direction follows from the triangular identity (see (A.3))
1 = u∗ · ηu∗ : u∗ → u∗ u! u∗ → u∗ .
In fact, since 1 and ηu∗ are isomorphisms this is also the case for u∗ , and it is
hence 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 are
stable under shifting.
Example 7.30. Associated to B ∈ Cat there is the 2-functor B × − : Cat → Cat
which sends A to the product B × A. If we are also given a prederivator D, then
we can define the shifted prederivator D B by setting
D B (A) = D(B × A).
More precisely, the shifted prederivator is defined as
B×−
D
D B : Cat op → Cat op → CAT .
In particular, given u : A → A0 in Cat, the induced functor u∗ : D B (A0 ) → D B (A)
is (1B ×u)∗ : D(B ×A0 ) → D(B ×A). The underlying category of D B is canonically
isomorphic to D(B).
Let us illustrate the shifting operation by the following examples. In those
examples we already use the concept of an isomorphism of (pre)derivators – a
notion which will be formally introduced in a later section.
Examples 7.31.
(i) Represented case
(ii) Let DA be the homotopy derivator of a Grothendieck abelian category A
and let B, A ∈ Cat. A combination of Lemma 3.10 with the categorical
exponential law yields canonical isomorphisms
B
DA
(A) = DA (B × A)
B×A −1
= Ch(A)B×A [(WA
) ]
∼
= Ch(AB )A [(W AB )−1 ]
A
=D
AB
(A).
B ∼
Thus, there is an isomorphism of derivators DA
= DAB .
(iii) We now specialize to the derivator DR of a ring R (see also Examples 6.24(i)).
G ∼
(a) For every discrete group G there is an isomorphism DR
= DRG , i.e., the
shifting operation encodes the passage to group algebras.
Comments about dias
in dias.
84
MORITZ GROTH
Q ∼
(b) For every (finite) quiver Q there is an isomorphism DR
= DRQ , i.e., the
shifting operation encodes the passage to path algebras.
P ∼
(c) For every (finite) poset P there is an isomorphism DR
= DRP , i.e., the
shifting operation encodes the passage to incidence algebras.
(iv) Combinatorial model categories
In the above examples we saw instances of derivators D such that the associated
shifted prederivators D B are again derivators. This is fact always the case as we
show next.
Proposition 7.32. For every derivator D and every B ∈ Cat the prederivator
D B is again a derivator, the shifted derivator or the derivator of (coherent)
B-shaped diagrams in D. The Kan extension functors of D B are calculated as
uD
= (1B × u)D
!
!
B
and
uD
= (1B × u)D
∗
∗ .
B
Proof. The verification of the axioms (Der1)-(Der3) for D B is left to the reader. It
remains to check the pointwise formulas encoded by axiom (Der4). Thus, given a
functor u : A → A0 and a0 ∈ A0 let us consider the square
1×p
B × (u/a0 )
(7.33)
1×π
B×1
1×a0
/ B×A
1×u
/ B × A0
which is obtained from the slice square (6.9) by forming the product with B. We
have to show that the canonical mate (1 × π)! (1 × p)∗ → (1 × a0 )∗ (1 × u)! is an
isomorphism of functors D(B × A) → D(B × 1). Since isomorphisms are detected
pointwise by (Der2), it is enough to show that all components of the mate are
isomorphisms. For b ∈ B we consider the following pasting diagram
φ
∼
=
((1 × u)/(b, a0 ))
(7.34)
π
1
=
1
p0
/ ((1 × π)/b)
π
/1
b
/ B × (u/a0 )
1×π
/ B×1
1×p
1×a0
/ B×A
1×u
/ B × A0
in which the square in the middle is the slice square (6.9) associated to the functor
1 × π and b ∈ B ∼
= B × 1. The square on the left is induced from the isomorphisms
of categories
(7.35)
((1 × π)/b) ∼
= (B/b) × (u/a0 ) ∼
= ((1 × u)/(b, a0 ))
with (b, a0 ) : 1 → B × A0 . The functoriality of mates with pasting (Lemma 7.8)
implies that the canonical mate associated to (7.34) factors as
∼
=
∼
=
π! φ∗ (p0 )∗ (1 × p)∗ → π! (p0 )∗ (1 × p)∗ → b∗ (1 × π)! (1 × p)∗ → b∗ (1 × a0 )∗ (1 × u)! .
Here, the second arrow is an isomorphism by (Der4) applied to D and the corresponding slice square. The first arrow is also an isomorphism because the isomorphism (8.33) is final; see Examples 7.20. In order to conclude the proof it suffices
to 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 × A0 and the
object (b, a0 ) ∈ B × A0 , this follows by a further application of (Der4).
INTRODUCTION TO THE THEORY OF DERIVATORS
85
8. Pointed derivators
In this section we introduce pointed derivators as derivators admitting zero objects. Typical examples are homotopy derivators of abelian categories or homotopy
derivators of pointed model categories, like the homotopy derivator of pointed topological spaces.
We show in §8.1 that certain Kan extension functors along (co)sieves amount
to extending diagrams by zero objects. This yields a convenient tool allowing us
to ‘add zero objects where desired’. In §8.2 we illustrate this tool and construct
suspension, loop, cofiber, and fiber functors in pointed derivators, thereby generalizing these classical constructions from homological algebra and homotopy theory. In
§8.3 we include a short discussion of parametrized Kan extensions or Kan extensions
in shifted derivators, while in §8.4 we extend a few well-known results concerning
pushout squares in ordinary categories to cocartesian squares in derivators. These
techniques are illustrated in our discussion of iterated cofiber constructions in §8.5.
8.1. Basics on pointed derivators. Let us recall from Lemma 6.32 that, given a
derivator D and a small 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 a
zero object.
Thus, we ask axiomatically that the unique map ∅ → ∗ in D(1) is an isomorphism. Following standard conventions, any zero object will be denoted by
0 ∈ 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 representing 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 special
cases, the homotopy derivator H oTop∗ of pointed topological spaces and the
derivator Sp of spectra are pointed.
Lemma 8.3.
(i) A derivator D is pointed if and only if D op is pointed.
(ii) If D is a pointed derivator, then D(A), A ∈ Cat, are pointed categories, i.e.,
the shifted derivators D A 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.
Proof. This proof is left as an exercise.
We know from Proposition 7.25 that Kan extensions along fully faithful functors
u : A → B are fully faithful and, by Lemma 7.29, that the essential images can
be 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 essential
images 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 follows
that b lies in the image of u.
86
MORITZ GROTH
(ii) Dually, u is a sieve if for every morphism b → u(a) in B it follows that b lies
in the image of u.
The following proposition will be of constant use in later sections, for example in
the construction of suspensions, cofibers, and cofiber sequences in pointed derivators; see §8.2 and §8.5. To indicate its natural generality, we formulate the result
for 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) lies
in 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 that
u! : D(A) → D(B) is fully faithful and by Lemma 7.29 a diagram X ∈ D(B) lies in
the essential image of u! if and only if the counit b : 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 → B
is a cosieve. By (Der1) the category D(∅) is equivalent to the terminal category
and any object therein is hence a zero object. Since left adjoint functors preserve
initial 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 b : 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 images
agree. We say that X ∈ D(B) vanishes at an object b ∈ B if Xb ∼
= 0, and we
similarly 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 an
equivalence onto the full subcategory of D(B) spanned by all diagrams which
vanish on B − u(A).
(ii) If u is a sieve, then u∗ : D(A) → D(B) is fully faithful and induces an equivalence onto the full subcategory of D(B) spanned by all diagrams which vanish
on B − u(A).
In the situation of the corollary, we refer to u! as left extension by zero and
to u∗ as right extension by zero. These results already allow us to carry out
some interesting constructions in arbitrary pointed derivators, as we illustrate in
the §8.2.
8.2. Suspensions, loops, cofibers, and fibers. In this section we define suspensions, loops, cofibers, and fibers for arbitrary pointed derivators, thereby generalizing the classical constructions from homological algebra and homotopy theory.
INTRODUCTION TO THE THEORY OF DERIVATORS
87
The following definition is motivated by Examples 5.35(iii). Let us recall that we
denote by = [1] × [1] the commutative square, i.e., the category
/ (1, 0)
(0, 0)
(0, 1)
=
/ (1, 1)
and by ip : p→ and iy : y → the full subcategories obtained by removing the
final 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 D
is 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 the
category of cocartesian squares are equivalent categories, and dually for cospans
and 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 a
pushout square (Examples 5.35(iii)). In homotopy derivators of abelian categories
or model categories the notion of cocartesian squares reduces to derived pushout
squares and homotopy pushout squares.
We now reconsider the rather detailed description of the cone construction of
a morphism of chain complexes as given at the beginning of §5.1; see, in particular, (5.1). Starting from a morphism of chain complexes f : X0 → X1 , we first construct 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 step
in the context of abstract pointed derivators. As for the first step, let us recall that
cones of chain complexes are contractible and are hence sent to zero objects in derived categories. As observed in Corollary 8.6, right Kan extensions along inclusions
of sieves are right extensions by zero. This suggests the following generalization of
the 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 inclusions ip : p→ and iy : y → we obtain the fully faithful functors
(8.11)
i0 = ip ◦ i : [1] → p→ and
k 0 = 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
i
(ip )!
(k0 )∗
∗
cof : D([1]) →
D(p) → D() → D([1]).
A further evaluation at 1 ∈ [1] yields C = 1∗ ◦ cof : D([1]) → D(1).
88
MORITZ GROTH
(ii) The fiber functor fib : D([1]) → D([1]) is defined as
(i0 )∗
(iy )∗
k
fib : D([1]) →! D(y) → D() → 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 of
Corollary 8.6 it follows that i∗ is right extension by zero and k! left extension by zero.
Given f ∈ D([1]) with underlying diagram X0 → X1 there are thus a cocartesian
square and a cartesian square in D with respective underlying diagrams
X0
f
/ X1
Ff
fib(f )
cof(f )
f
/ Cf,
0
/ X0
/ X1 .
0
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]) o
i∗
i∗
/ D(p) o
(ip )!
(ip )∗
/ D() o
(iy )∗
(iy )∗
/ D(y) o
k∗
k∗
/ D([1]);
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 at
the lower left corner (0, 1). As a special case of Corollary 8.6 we obtain adjoint
equivalences 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 functors
cof and fib respectively factor as
D([1]) o
i∗
i∗
/ D(p)ex o
(ip )!
(ip )∗
/ D()ex o
(iy )∗
(iy )∗
/ D(y)ex o
k∗
k∗
/ D([1]).
Thus, (cof, fib) is obtained by composing four adjunctions (two of which actually
are adjoint equivalences), concluding the proof.
Remark 8.14. Given a pointed derivator D we denote by D()cof ⊆ D() the
full subcategory spanned by the cofiber squares, i.e., the cocartesian squares
which vanish on the lower left corner. Similarly, let D()fib ⊆ D() be the full
subcategory given by the fiber squares, i.e., the cartesian squares which vanish
on the lower left corner. The proof of Proposition 8.13 together with Remark 8.8
imply that there are equivalences of categories
D([1]) ' D()cof
and
D([1]) ' D()fib .
INTRODUCTION TO THE THEORY OF DERIVATORS
89
Thus, using coherent formulations, a morphism is simply ‘as good as a (co)fiber
square’. We will come back to this in §8.5; see Proposition 8.37.
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 Kan
extension functors are extensions by zero (Corollary 8.6).
For the convenience of the reader, we include a more detailed construction of
these 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 the
functors ip : p→ and iy : y → yields the fully faithful functors
(8.16)
i0 = ip ◦ i : 1 → p→ and
k 0 = 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
(k0 )∗
(ip )!
i
∗
Σ : D(1) →
D(p) → D() → D(1).
(ii) The loop functor Ω : D(1) → D(1) is defined as
(i0 )∗
(iy )∗
k
Ω : D(1) →! D(y) → D() → D(1).
We note again that i : 1 → p is a sieve while k : 1 →y is a cosieve so that i∗ is
right extension by zero and k! left extension by zero (Corollary 8.6). As a summary
of these constructions, for X ∈ D(1) there is a cocartesian square and a cartesian
square in D with respective underlying diagrams
X
/0
ΩX
/0
0
/ ΣX,
0
/ X.
Remark 8.18. Only interesting in derived contexts. C as homotopy cokernel and
F as homotopy kernel. One-categorical these are boring on objects X → 0 and
0 → X, respectively.
Proposition 8.19. 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 the
functors Σ and Ω respectively factor as
D(1) o
i∗
i∗
/ D(p) o
(ip )!
(ip )∗
/ D() o
(iy )∗
(iy )∗
/ D(y) o
k∗
k∗
/ D(1).
Blabla.
90
MORITZ GROTH
Again, there are four adjunctions of which only the outer ones ‘point in the bad
direction’. We 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 at
the lower left corner (0, 1) and the upper right corner (1, 0). As a special case of
Corollary 8.6 we obtain adjoint equivalences of categories
(i∗ , i∗ ) : D(1) ' D(p)ex
(k ∗ , k! ) : D(y)ex ' D(1).
and
Since the functors (ip )! , (ip )∗ and (iy )∗ , (iy )∗ preserve the vanishing conditions at
the corners (0, 1), (1, 0), the functors Σ and Ω respectively factor as
D(1) o
i∗
i∗
/ D(p)ex o
(ip )!
(ip )∗
/ D()ex o
(iy )∗
(iy )∗
/ D(y)ex o
k∗
k∗
/ D(1).
As a composition of two adjoint equivalences and two adjunctions also (Σ, Ω) is an
adjunction.
The remaining main goal in this section is to study iterated constructions of
cofibers which, in stable derivators, will lead to canonical triangulations. For this
purpose however we need some basic results concerning (co)cartesian squares which
we collect in §8.4. Some of these results in turn rely on a basic understanding of
parametrized Kan extensions which we study next.
8.3. Parametrized Kan extensions. The goal of this subsection is to establish a
result making precise that restriction functors and Kan extensions in unrelated variables commute (Proposition 8.20) and to deduce some interesting consequences. In
particular, this allows us to understand (co)cartesian squares in shifted derivators.
To begin with, let us recall the following observation from ordinary category
theory. If C is a cocomplete category and B ∈ Cat, then the diagram category
C B is also cocomplete and colimits in C B are constructed coordinatewise. More
precisely, for every small category A ∈ Cat and diagram X : A → C B the colimit
colima∈A X(a) ∈ C B exists and for every b ∈ B there is an isomorphism
colima∈A X(a) (b) ∼
= colima∈A X(a)(b) .
Actually, a typical proof of the cocompleteness is given by showing that the right
hand side of the above isomorphisms defines a diagram B → C and that it, together
with a suitably defined cocone, yields a colimit colimA X ∈ C B . We sometimes refer
to the colimits in C B as colimits with parameters or parametrized colimits.
It can be shown that every evaluation morphism b∗ : C B → C preserves colimits,
i.e., that the corresponding canonical map is an isomorphism.
More generally, as a cocomplete category C B admits left Kan extensions along
functors u : A → A0 (??) which we also refer to as left Kan extensions with
parameters or parametrized left Kan extensions. It turns out that evaluation
functors also commute with (parametrized) Kan extensions, i.e., that for every
b ∈ B the following square
(C B )A
b∗
∼
=
@H
CA
/ (C B )A0
u!
u!
b∗
/ C A0
INTRODUCTION TO THE THEORY OF DERIVATORS
91
commutes up to a certain canonical isomorphism.
As a final generalization, we can pass from evaluation functors to general restriction functors. In fact, it turns out that, given a functor v : B → B 0 , then the induced
0
restriction functor v ∗ : C B → C B preserves colimits and left Kan extensions. These
results from ordinary category theory can be obtained by specializing the following
proposition to represented derivators. At the same time, that proposition yields
similar results for arbitrary derivators, hence, for example, for homotopy derivators
of abelian categories or model categories.
Proposition 8.20. For functors v : B → B 0 and u : A → A0 between small categories the square
B×A
(8.21)
v×1
1×u
B × A0
1
v×1
/ B0 × A
1×u
/ B 0 × A0
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 this
purpose, we consider the following two pastings
B × (u/a0 )
π
B
1×p
1×a0
/ B×A
v×1
1×u
/ B × A0
1
v×1
/ B0 × A
1×u
/ B 0 × A0
B × (u/a0 )
=
v×1
/ B 0 × (u/a0 )
/ B0
π
B
π
1
v
1×p
/ B0 × A
1×u
1×a0
/ B 0 × A0
in which the square to the very left and to the very right are obtained from slice
squares be forming the product with a fixed category. We recall from the proof of
Proposition 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 to
conclude that (8.21) is homotopy exact it suffices to show that the pasting on the
left is homotopy exact. Since these two pastings agree, a further application of the
functoriality of mates with pasting implies that it suffices to show that the second
square from the right is homotopy exact. This completes the reduction to the case
of u = πA : A → 1.
As an additional reduction we show that is is enough to consider evaluation
functors instead of more general restriction functors. To this end, let us consider
the pasting on the left in
A
(8.22)
π
1
/ (B/b) × A
t×1
1
1
π
/1
p×1
/ B×A
π
b
/B
v×1
v
1
/ B0 × A
π
/ B0
A
=
π
1
vb×1

vb
/ B0 × A
1
π
/ B0.
We have to show that in that pasting the right square is homotopy exact. As a slice
square, we know that the square in the middle is homotopy exact. Moreover, the
square on the left is induced from the functor t : 1 → (B/b) classifying the terminal
92
MORITZ GROTH
object (b, 1 : b → b) ∈ (B/b). Since right adjoint funtors are final, the left square is
homotopy exact as well (Lemma 7.18). By the functoriality of mates with pasting
and (Der2) it hence suffices to show that the pasting is homotopy exact. Since the
pasting is easily seen to agree with the commutative square on the right in (8.22),
this concludes the reduction to evaluation functors.
Finally, let us consider an object b0 ∈ B 0 . Our remaining task is to show that
the commutative square to the left in
A
b0 ×1

π
1
/ B0 × A
1
b0
π
/ B0
A
=
π
1
/ (B 0 /b0 ) × A
t×1
1
=
p×1
π
/1
/ B0 × A
π
/ B0
b0
is homotopy exact. In the pasting to the right, the right square is a slice square
and hence homotopy exact. The remaining square is induced by t : 1 → (B 0 /b0 )
pointing at the terminal object (b0 , 1 : b0 → b0 ). Since the pasting on the right is
easily 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 to
pasting (Example 7.17) conclude the proof.
For Kan extensions along fully faithful functors there is the following consequence.
Corollary 8.23. Let D be a derivator, let u : A → A0 be fully faithful, and let
B ∈ Cat. A coherent diagram X ∈ D(B × A0 ) lies in the essential image of (1 × u)!
if and only if each Xb ∈ D(A0 ), b ∈ B, lies in the essential image of u! .
Proof. Let us consider the following pasting of commutative squares in which the
square to the very left is homotopy exact as an instance of (8.21),
A
b×1
u
A0
/ B×A
1×u
=
1×u
b×1
/ B × A0
/ B × A0
=
/ B × A0
A
=
u
b×1
=
u
A0
/ A0
=
/ A0
/ B × A0
=
b×1
/ B × A0 .
Since 1 × u is fully faithful, X ∈ D(B × A0 ) lies in the essential image of (1 × u)!
if and only if the adjunction counit (1 × u)! (1 × u)∗ → 1 is an isomorphism on
X (Proposition 7.25). We note that this counit is the canonical mate associated
to the second square from the left. Thus, (Der2), the homotopy exactness of the
square 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 ∈ B
the canonical mate of the pasting on the left is an isomorphism. Since the above
two pastings agree, this is the case if and only if for every b ∈ B the canonical
mate of the pasting on the right is an isomorphism on X. To conclude the proof it
suffices to observe that these canonical mates are the counits u! u∗ → 1 applied to
Xb ∈ D(A0 ), b ∈ B.
Note that we can read X ∈ D(B × A0 ) as an object in D B (A0 ) while the objects
Xb , b ∈ B, all live in D(A0 ). In this sense, the essential image of a fully faithful
Kan extension functor in a shifted derivator D B can be detected by evaluating at
INTRODUCTION TO THE THEORY OF DERIVATORS
93
all b ∈ B (see Proposition 7.32 for the construction of Kan extensions in shifted
derivators).
Corollary 8.24. Let D be a derivator and let B ∈ Cat. A square X ∈ D B () is
cocartesian if and only if Xb ∈ D() is cocartesian for all b ∈ B.
Proof. This is an immediate application of Corollary 8.23 to the fully faithful functor ip : p→ .
We recall that an object X ∈ D([1]) is called an isomorphism if the underlying
diagram X0 → X1 is an isomorphism.
Corollary 8.25. Let D be a derivator and let B ∈ Cat. The following are equivalent for a coherent morphism X ∈ D B ([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! : D B (1) → D B ([1]).
Proof. This follows from Example 7.24 and (Der2). Alternatively, we can also
invoke Example 7.24 and Corollary 8.23.
8.4. Cartesian and cocartesian squares. In this subsection we extend a few
well-known results concerning pushouts and pullbacks in ordinary categories to
cartesian and cocartesian squares in arbitrary derivators. Recall the definition of
these squares given in Definition 8.7.
We begin by establishing some terminology. Let D be a derivator and let us
consider 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) and
X(1,0) → X(1,1) both are isomorphisms. There is a similar notion of horizontally
constant 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 8.26. Let D be a derivator.
(i) The functor (id[1] × 0)! : D([1]) → D() is fully faithful and induces an equivalence onto the full subcategory of D() spanned by the vertically constant
squares.
(ii) The functor (0, 0)! : D(1) → D() is fully faithful and induces an equivalence
onto the full subcategory of D() spanned by the constant squares.
Proof. The first statement is immediate from Corollary 8.25 and the second statement 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 8.27. Let D be a derivator. The functor i! : D([1]) → D(p) is fully faithful
and induces an equivalence onto the full subcategory of D(p) spanned by those X
such that X(0,0) → X(0,1) is an isomorphism.
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MORITZ GROTH
Proof. By Proposition 7.25 the functor i! is fully faithful with essential image those
X such that : i! i∗ (X) → X is an isomorphism at (0, 1). To reformulate this let us
consider the pasting on the left
1
∼
=
1
1
/ [1]
/ (i/(0, 1))
1
/1
(0,1)
i
/p
1
1
i
/p
1
/p
=
1
1
(0,0)
z
(0,1)
/p
1
/ p,
in which the square in the middle is a slice square and the square to the left is given
by an isomorphism. Since these two squares are homotopy exact, the functoriality of
canonical mates with pasting implies that (0,1) is an isomorphism on X if and only
if the canonical mate associated to this pasting is an isomorphism at X. Since this
pasting 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 the
proof.
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 of
isomorphisms are isomorphisms.
Proposition 8.28. Let D be a derivator and let X be a square in D such that
X(0,0) → X(0,1) is an isomorphism. The square X is cocartesian if and only if
X(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 consequence
of Corollary 8.26, vertically constant squares are thus cocartesian. Conversely, if
X is a cocartesian square such that X(1,0) → X(1,1) is an isomorphism, then, by
Lemma 8.27, X lies in the essential image of (ip )! ◦ i! ∼
= (id[1] × 0)! and is by
Corollary 8.26 vertically constant.
Let [2] be the poset (0 < 1 < 2) considered as a category. For 0 ≤ i ≤ j ≤ 2
there is the functor ιi,j : [1] → [2] which sends 0 to i and 1 to j. The product
category [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∗01 X ∈ D(), the right square ι∗12 X ∈ D(),
and the composite square ι∗02 X ∈ D(). (Here we abused notation and simply
wrote ιi,j instead of ιi,j × id[1] .)
Cocartesian squares enjoy a composition and cancelation property as made precise by Proposition 8.30. The following lemma is a first step towards that proposition. 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
i
j
k = j ◦ i : A → B → .
INTRODUCTION TO THE THEORY OF DERIVATORS
95
Lemma 8.29. Let D be a derivator.
(i) The functor i! : D(A) → D(B) is fully faithful and induces an equivalence
onto 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 equivalence
onto 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 equivalence
onto the full subcategory spanned by all diagrams X ∈ D() such that both
squares ι∗01 X, ι∗12 X are cocartesian.
Proof. We begin by statement (i). Since i : A → B is fully faithful, i! is fully faithful
and X ∈ D(B) lies in the essential image if and only if the counit : i! i∗ (X) → X
is an isomorphism at (1, 1) (Proposition 7.25). Using the homotopy exactness of
slice squares and the functoriality of mates with respect to pasting, this is the case
if and only if the canonical mate associated to the pasting on the left in
p
π
1
ι01
{
(1,1)
/A
i
/B
1
i
/B
1
/B
1
p
=
π
1
{
/p
ip
ip
(1,1)
/
ι01
1
1
/
1
/
/B
ι01
/B
is an isomorphism on X. In the above pasting on the right, the square to the left is a
slice square. Since the above two pastings coincide, using again the functoriality of
mates 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
∼
=
1,1
colimp i∗p ι∗01 (X) → (1, 1)∗ (ip )! i∗p ι∗01 (X) → (1, 1)∗ ι∗01 (X)
and is hence an isomorphism if and only if the counit 1,1 is an isomorphism on
ι∗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 : j! j ∗ (X) → X is
an isomorphism at (2, 1) (Proposition 7.25). To re-express this differently let us
consider the pasting on the left in
p
ι12
π
/B
π
1
1
/1
1
{
(2,1)
/B
j
j
/
/
1
1
/
1
p
=
π
1
{
(1,1)
/p
ip
ip
/
/
ι12
1
1
/
/
1
ι12
/ ,
and for now let us focus on the two squares to the right. Proposition 7.25, the
homotopy exactness of slice squares, and the compatibility of mates with pasting
imply that X lies in the essential image of j! if and only if the canonical mate of
the pasting of those two squares is an isomorphism on X. We next observe that the
functor ι01 : p→ B is a right adjoint. In fact, the functor s0 : [2] → [1] determined
by 0, 1 7→ 0 and 2 7→ 1 induces a left adjoint to ι01 : p→ B. Hence, by Lemma 7.18
the corresponding square is homotopy and, again by the functoriality of mates
96
MORITZ GROTH
with pasting, we conclude that X lies in the essential image of j! if and only if
the canonical mate of the pasting to the left is an isomorphism on X. Since that
pasting can be rewritten as the above pasting on the right, we can now conclude as
in 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 property is as follows.
Proposition 8.30. Let D be a derivator and let X ∈ D() such that ι∗01 X is
cocartesian. Then ι∗12 X is cocartesian if and only if ι∗02 X is.
Proof. We consider X ∈ D() such that the square ι∗01 (X) is cocartesian. By
Lemma 8.29 and Proposition 7.25 for such a diagram X also the square ι∗12 (X) is
cocartesian if and only if X lies in the essential image of k! : D(A) → D() if and
only if the counit : k! k ∗ (X) → X is an isomorphism at (2, 1). In more detail, by
Proposition 7.25 the essential image of k! is characterized by : k! k ∗ → 1 being
an isomorphism at (1, 1) and (2, 1). Under our assumption the component 1,1 is
always an isomorphism as the counit factors as
∼ j! i! i∗ j ∗ (X) →
j! j ∗ (X) → X
k! k ∗ (X) =
and the component at (1, 1) of the first morphism is an isomorpism by our assumption on X (use Proposition 7.25 and Lemma 8.29) while the corresponding
component of the second morphism is always an isomorphism (Proposition 7.25).
To reformulate this we consider the pasting on the left in
p
ι02
π
/A
π
1
1
/1
1
{
(2,1)
/A
k
k
/
/
1
1
/
1
p
=
π
1
{
(1,1)
/p
ip
ip
/
/
ι02
1
1
/
/
1
ι02
/ .
Here, the functor ι02 : p→ A is a right adjoint so that the square to the very left is
homotopy exact by Lemma 7.18. In fact, a left adjoint is induced by the functor
s1 : [2] → [1] determined by 0 7→ 0 and 1, 2 7→ 1. Together with the homotopy
exactness of slice squares and the functoriality of mates with pasting this shows
that for our diagram X ∈ D() the square ι∗12 (X) is cocartesian if and only if the
canonical mate of the pasting on the left is an isomorphism on X. In the pasting on
the right, the square to the very left is a slice square and one notes that the above
two pasting coincide. Using standard arguments we thus conclude that ι∗12 (X) is
cocartesian if and only if ι∗02 (X) is cocartesian.
8.5. Iterated cofiber constructions. In this subsection we apply our above results on cocartesian squares and briefly discuss cofiber sequences. We will also show
that if we iterate the cofiber construction three times then we obtain the suspension
morphism up to natural isomorphism.
But let us begin with the cofiber sequences themselves.
Definition 8.31. Let D be a pointed derivator. A cofiber sequence in D is a
diagram 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.
INTRODUCTION TO THE THEORY OF DERIVATORS
97
We denote by D()cof ⊆ D() the full subcategory spanned by all cofiber sequences.
Thus, given a cofiber sequence X, we obtain an underlying incoherent diagram
dia (X) : → D(1) looking like
f
x
(8.32)
/y
/0
g
0
/ z
h
/ w.
By Proposition 8.30 also the composite square ι∗02 (X) is cocartesian and it has an
underlying diagram
/0
x
0
/ w.
Let us recall from Definition 8.17 that such squares were used in the construction
of the suspension functor Σ : D(1) → D(1). In fact, let D()ex ⊆ D() be the full
subcategory 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 categories
D(1) ' D()ex and we just saw that ι∗02 (X) ∈ D()ex . It follows from this
observation that for every cofiber sequence X with underlying diagram (8.32) there
is a canonical isomorphism
φ: w ∼
= Σx.
(8.33)
Let [3] be the poset (0 < 1 < 2 < 3) and let k : [3] → be the ‘diagonal
embedding‘ pointing at the objects (0, 0) < (1, 0) < (1, 1) < (2, 1). Given a cofiber
sequence 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 diagram
with the canonical isomorphism (8.33) we obtain, as an upshot, for every cofiber
sequence X as in (8.32) an underlying incoherent cofiber sequence in D(1)
(8.34)
f
g
φ◦h
x → y → z → Σx.
In the context of stable derivators, these incoherent cofiber sequences lead to canonical triangulations as we will see in §9.6.
Remark 8.35. In the construction of incoherent cofiber sequences we applied the
underlying diagram functor dia[3] : D([3]) → D(1)[3] and this step results in a loss
of information. These incoherent cofiber sequences are some non-canonical shadows of certain universal constructions which were applied to a coherent morphism.
In particular, given an incoherent cofiber sequence (8.34) it is true that the compositions x → z and y → Σx are zero morphisms. But at that level we do not
remember the reason why these compositions are zero, namely that they belong to
certain cofiber squares (Remark 8.14) in the background (see (8.32)).
As a related remark, in general, it is not possible to canonically reconstruct a coherent version of the morphism f : x → y starting with the given incoherent cofiber
sequence. This is to be seen in contrast to the following result (Proposition 8.37).
98
MORITZ GROTH
We now show that every coherent morphism can be extended to a cofiber sequence. For this purpose, let A ⊆ again denote the full subcategory obtained by
removing 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 every
derivator D there are related Kan extension functors
(8.36)
k
i
∗
D(A) →! D().
D([1]) →
Proposition 8.37. For every pointed derivator D the functors (8.36) induce an
equivalence of categories
D([1]) ' D()cof .
Proof. The functors i : [1] → A and k : A → are both fully faithful and hence the
same 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 we
deal with both cases individually.
We note first that the functor i is a sieve and that i∗ hence is right extension
by zero (Corollary 8.6). Thus, the essential image of i∗ consists of precisely those
X ∈ D(A) which vanish at (2, 0) and (0, 1). As for k! : D(A) → D() we already
know by Lemma 8.29 that the essential image of k! consists precisely of those
X ∈ D() such that both squares ι∗01 (X), ι∗12 (X) are cocartesian. Since k! is a fully
faithful 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 full
subcategory D(A)ex of D(A) spanned by all diagrams which vanish at these two
objects on the one-hand-side and the category D()cof of cofiber sequences on the
other-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 pointed
derivator is simply as good as a cofiber sequence. Compare this observation to
Remark 8.14.
We now add one more iteration of the construction of the cofiber. To make this
precise, let us consider the full subcategory B ⊆ [2] × [2] obtained by removing
(0, 2),
(8.38)
(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 morphism X ∈ D([1]) to a diagram of the above shape by first adding certain zero
objects 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
INTRODUCTION TO THE THEORY OF DERIVATORS
99
classifying the horizontal morphism (0, 0) → (1, 0) and the obvious fully faithful
inclusion j : A → B. At the level of coherent diagrams in a derivator D we obtain
the corresponding Kan extension functors
(8.39)
j!
i
∗
D(A) → D(B).
D([1]) →
Lemma 8.40. Let D be a pointed derivator and let D(B)cof ⊆ D(B) be the full
subcategory 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 (8.39) induce an equivalence of categories
D([1]) ' D(B)cof .
Proof. Since the proof is similar to the one of Proposition 8.37 we allow ourselves
to be a bit sketchy. Both functors i : [1] → A and j : A → B are fully faithful, hence
the 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 subcategory
D(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 as
the composition of fully faithful inclusions
j1
j2
j3
j : A → A1 → A2 → B
where the individual steps add the objects (1, 1), (2, 1), and (2, 2), respectively. The
left Kan extension functor j! : D(A) → D(B) is accordingly naturally isomorphic
to the composition
(j1 )!
(j2 )!
(j3 )!
D(A) → D(A1 ) → D(A2 ) → D(B).
Similarly to Lemma 8.29, one now observes that each of these three Kan extensions
precisely amounts to adding a new cocartesian square in the obvious sense.
As an upshot, the functor j! hence amounts to adding three cocartesian squares
and 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 a
coherent diagram looking like
x
(8.41)
01
f
/y
/ 02
g
/ z
h
/ x0
f0
03
/ y0
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 next
show that the vertical morphism (f 0 : x0 → y 0 ) is naturally isomorphic to Σf .
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MORITZ GROTH
To make this precise, let us recall from Lemma 8.3 that pointed derivators are
stable under shifting. In particular, if D is pointed then so is D [1] , and the pointed
derivator D [1] comes with a suspension functor Σ : D([1]) → D([1]).
Proposition 8.42. For every pointed derivator D there is a natural isomorphism
cof 3 ∼
= Σ : D([1]) → D([1]).
Proof. Let B be the category (8.38) and let us consider the functors k : [1] → B
and l : [1] → B classifying the morphisms
(0, 0) → (1, 0)
(2, 1) → (2, 2),
and
respectively. The main step consists of showing that for every Y ∈ D(B)ex there
is a natural isomorphism Σ(k ∗ Y ) ∼
= l∗ (Y ). To this end, let Y ∈ D(B)ex with
underlying diagram (8.41). There is a unique functor q : [1] × → B such that
q ∗ (Y ) ∈ D([1] × ) has underlying diagram looking like
/ 02
x
f
(8.43)
/ 02
y
/ x0
01
03
f0
/ y0 .
In this diagram, the [1]-coordinate is drawn diagonally while the -coordinate is
as before and the decoration of the objects implies that there is at most one such
functor (the categories under consideration are posets). We leave it to the reader
that 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 squares
Z0 , Z1 ∈ D() are cocartesian. In fact, Z0 is cocartesian as a horizontal pasting
of two cocartesian squares while Z1 is a vertical pasting of two cocartesian squares
(Proposition 8.30). Thus, by Corollary 8.24, the square Z ∈ D [1] () is cocartesian
when considered as an object of D [1] . Morever, this cocartesian square vanishes
at (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 see
that Z0,0 = (0, 0)∗ q ∗ Y = k ∗ Y , yielding the desired canonical isomorphism
∼ l∗ (Y ).
Σ(k ∗ Y ) =
We leave it to the reader to apply Lemma 8.40 to conclude the proof of this
proposition.
We will see in §9.6 that stable derivators canonically take values in triangulated
categories. Ignoring signs for the moment, this proposition is a coherent version of
the following observation. Recall the rotation axiom (T2) for triangulated categories
(Definition 4.9) by which a distinguished triangle can be rotated backwards. At the
level of coherent morphisms this amounts to passing to the cofiber of the coherent
INTRODUCTION TO THE THEORY OF DERIVATORS
101
morphism. If we take a distinguished triangle and rotate it three times then we
obtain a distinguished triangle which is the (negative of the) suspension of the
original triangle. The corresponding statement at the level of coherent morphisms
is Proposition 8.42.
9. Stable derivators
Stable derivators are obtained from pointed derivators by imposing the linearity
condition that a square is cocartesian if and only if it is cartesian. This linearity
axiom allows us to establish derivator versions of well-known results from homological algebra and stable homotopy theory, i.e., the study of spectra in the sense
of topology. (In the next section we will see less well-known such results.) Stable
derivators canonically take values in triangulated categories (based on the incoherent cofiber sequences introduced in §8), and stable derivators hence belong to the
family of different enhancements of triangulated categories.
We begin in §9.1 with the definition of stable derivators and some first consequences of these axioms. In §9.2 we show that the values of stable derivators are
preadditive categories and in §9.3 we sketch ideas involved in a proof that these
categories actually are additive. In §9.4 we include a short digression on morphisms
of derivators and natural transformations between such morphisms. We also briefly
talk about adjunctions and equivalences of derivators and related notions. Finally,
in §9.6 we show that (strong) stable derivators take values in triangulated categories
which are compatible with restriction functors and Kan extension functors.
9.1. Basics on stable derivators. Similarly to the theory of model categories
and ∞-categories, we obtain stable derivators from pointed ones by imposing the
following linearity condition.
Definition 9.1. A pointed derivator D is stable if a square in D is cartesian if
and only if it is cocartesian. A bicartesian square is square which is both cartesian
and cocartesian.
Examples 9.2.
(i) The homotopy derivator of a Grothendieck abelian category
is 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 the
universal example of a stable derivator, namely the free stable derivator generated by the sphere spectrum.
(iii) The homotopy derivator of a stable model category or stable ∞-category is
stable, giving rise to a plethora of stable derivators. We refer the reader to
[GŠ14c, 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 D op 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 derivator
D B is stable.
Proof. By Lemma 8.3 the shifted derivator D B is pointed, thus it remains to show
that a square in D B is cocartesian if and only if it is cartesian. This follows from
102
MORITZ GROTH
our discussion of parametrized Kan extensions in §8.3. In fact, by Corollary 8.24
and its dual we can use the stability of D to see that X ∈ D B () = D(B × ) is
cocartesian if and only if all Xb ∈ D(), b ∈ B, are cocartesian if and only if all
Xb ∈ D(), b ∈ B, are cartesian if and only if X ∈ D B () = D(B ×) is cartesian.
Thus, D B is also stable.
We begin by a few sanity checks, indicating that Definition 9.1 captures some of
the expected phenomenons.
Proposition 9.5. Let D be a stable derivator. A morphism f ∈ D([1]) is an
isomorphism if and only if Cf ∼
= 0 if and only if F f ∼
= 0.
Proof. It follows from the construction of C : D([1]) → D(1) that for every morphism f ∈ D([1]) there is a cocartesian square
x
0
f
/y
/ Cf.
If f is an isomorphism then the same is true for 0 → Cf by Proposition 8.28, establishing one direction. Conversely, let us assume that Cf ∼
= 0. Thus, the bottom
horizontal morphism in the above cocartesian square is an isomorphism. Since D is
stable, the square is also cartesian and it follows from the dual of Proposition 8.28
that 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 the
squares ι∗01 (X), ι∗12 (X), and ι∗02 (X) are bicartesian then so is the third one.
Proof. This is immediate from Proposition 8.30 and its dual.
We now establish a derivator version of the 5-lemma. In that result we consider
two cofiber squares X, X 0 ∈ D()cof looking like
x
/y
x0
/ y0
0
/ z,
0
/ z0
and a morphism X → X 0 with components f : x → x0 , g : y → y 0 , and h : z → z 0 .
Proposition 9.7. Let D be a stable derivator and let F : X → X 0 be a morphism
of cofiber squares with components f : x → x0 , g : y → y 0 , and h : z → z 0 . If two of
the 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 )! implies
that 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 fiber
squares).
We attack the remaining case and assume that f and h are isomorphisms. A
right Kan extension by zero followed by a left Kan extension allows us to extend
INTRODUCTION TO THE THEORY OF DERIVATORS
103
X and X 0 to cofiber sequences X̃ and X̃ 0 ,
x
/y
/0
x0
/ y0
/0
0
/ z
/ w,
0
/ z0
/ w0 ,
and to obtain an induced morphism of cofiber sequences F̃ : X̃ → X̃ 0 . The restriction ι∗0,2 (F̃ ) to the cocartesian composite squares is an isomorphism. In fact, it is
an isomorphism if we restrict it further along ip : p→ since f is an isomorphism
so that axiom (Der2) allows us to conclude. By the first part of this proof it follows
that ι∗0,2 (F̃ ) is an isomorphism. Thus, the restriction ι∗1,2 (F̃ ) to the squares on the
right 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 hence
conclude 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 indicated
in
D([1]) o
i∗
i
∗
/ D(p) o
(ip )!
(ip )
∗
/ D() o
(iy )∗
(iy )∗
/ D(y) o
k∗
k∗
/ D([1]).
These four adjunctions restrict to equivalences as follows. 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 satisfying the
following exactness conditions. In the first two cases we impose the vanishing
condition at (0, 1) while in the third case we consider only the squares which are
bicartesian and satisfy this vanishing condition. It follows from Corollary 8.6 and
Proposition 7.25 that the functors cof and fib respectively factor as
D([1]) o
i∗
i
∗
/ D(p)ex o
(ip )!
(ip )
∗
/ D()ex o
(iy )∗
(iy )∗
/ D(y)ex o
k∗
k∗
/ D([1]).
and that each individual step is an equivalence.
As already mentioned, we will see in §9.6 that stable derivators D canonically
take values in triangulated categories. In particular, the underlying category D(1)
can be endowed with a triangulated structure. The proposition above establishes
that we can rotate triangles back and forth without a loss of information. And the
following 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 being
that we impose an additional vanishing condition at the upper right corner (1, 0).
The details are left as an exercise to the reader.
104
MORITZ GROTH
We conclude by mentioning the following theorem, offering different characterizations 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 that
the values D(A), A ∈ Cat, of a stable derivator D are preadditive categories. As
a preparation we generalize the following result from ordinary category theory in
which we consider a pushout square
∅
/Y
X
/W
in a cocomplete category such that the upper left corner is populated by an initial
object. In this situation the object W is isomorphic to the coproduct X t Y , more
precisely, 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)) : 1 t 1 → which factors as compositions of fully faithful functors
(9.11)
i
i
p
1 t 1 → p→
and
j
i
y
1 t 1 →y →
.
Definition 9.12. Let D be a derivator. A coherent cospan X ∈ D(y) is a coproduct 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 the
full subcategory D()copr ⊆ D() spanned by the cocartesian squares X such that
X(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 and
the lower left corner is fully faithful. By Proposition 7.25 the same is true for
k! : D(1 t 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 to
obtain a different description of the essential image. Using the natural isomorphism
k! ∼
= (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 the
essential 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 preadditivity
of stable derivators.
Proposition 9.14. Let D be a stable derivator and A ∈ Cat. The category D(A)
is preadditive.
INTRODUCTION TO THE THEORY OF DERIVATORS
105
Proof. Let D be a stable derivator and A ∈ Cat. The derivator D A is again stable
(Lemma 9.4) and there is an isomorphism of categories D A (1) ∼
= D(A). Thus, it is
enough to show that the underlying category D(1) is preaddive. By Lemma 6.32,
the category D(1) admits small (co)products. Hence, in order to conclude it is
enough to show that for X, Y ∈ D(1) the canonical map X t Y → X × Y is an
isomorphism. To this end, we begin by observing that (Der1) yields a canonical
equivalence of categories D(1 t 1) ' D(1) × D(1). For X, Y ∈ D(1) we denote
by (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
/ (2, 0)
/ (1, 0)
(0, 0)
(0, 1)
/ (1, 1)
/ (2, 1)
(0, 2)
/ (1, 2)
/ (2, 2).
Related to [2] × [2] there are categories and fully faithful functors
i
j
k
1 t 1 → A → B → [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 define
the 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 adding
the object (2, 2) and the functor j : A → B is the obvious fully faithful inclusion. 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 fully
faithful Kan extension functors
(9.15)
i
j∗
k
D(1 t 1) →! D(A) → D(B) →∗ D([2] × [2]).
Moreover, since i is a cosieve and j a sieve it follows from Corollary 8.6 that i! and
j∗ are left and right extension by zero, respectively. To understand the remaining
functor k∗ we note that k : B → [2] × [2] factors as a composition of four fully
faithful inclusions,
k
k
k
k
k : B →1 B1 →2 B2 →3 B3 →4 [2] × [2],
which are obtained by adding the objects (1, 1), (1, 0), (0, 1), and (0, 0) one by one
in turn. Using arguments similar to the proof of Lemma 8.29, the reader checks
that each of the corresponding right Kan extension functors precisely amounts to
adding 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 diagrams
Q ∈ D([2] × [2]) satisfying the following exactness properties.
(i) The diagram Q vanishes at the three corners (0, 2), (2, 0), and (2, 2).
106
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 coherent
diagram. The diagram Q looks like
/ X0
Z
(9.16)
Y0
0
1
3
/B
/X
/0
2
4
/Y
/ 0.
Considering the squares 2 , 4 only, Proposition 9.6 and (the dual of) Proposition 8.28 imply that the composite square is cartesian and that the structure map
X 0 → X is an isomorphism. Similarly, we obtain that the structure map Y 0 → Y is
an 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 and
Lemma 9.13 hence implies that B is the coproduct of X 0 , Y 0 . A final combination
of these observations with the isomorphisms X 0 ∼
= X and Y 0 ∼
= Y concludes the
proof.
9.3. The additivity of stable derivators. In this subsection we mention ingredients 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 that
the abelian monoid structures on sets of morphisms in the preadditive categories
D(A), A ∈ Cat, actually are group structures.
To this end, let D be a pointed derivator and let X ∈ D(1). Mimicking a classical
construction from topology, one can show that the loop object ΩX ∈ D(1) can be
turned 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 topological
space. We do not include the construction of these concatenation maps here
and instead refer the reader to [Gro13]. Nevertheless we collect the following result
making precise that loop objects are group objects.
Theorem 9.18. Let D be a pointed derivator and let X ∈ D(1). The concatenation
map (9.17) endows ΩX with the structure of a group object.
While the precise definition of the concatenation (9.17) is not relevant here, we
will later need some hint where the inversion map comes from. This hint is obtained
from the following heuristics in the context of pointed topological spaces.
Let (X, x0 ) be a pointed topological space. Then a typical model for the loop
space is given by the space of maps [0, 1] → X which send the boundary points
to x0 (the topology is the compact-open topology). Here it is more convenient to
use a different description of the same homotopy type. To begin with, let P X be
the space of paths [0, 1] → X starting at x0 and let ev1 : P X → X be the map
which evaluates a path at its target 1 ∈ [0, 1]. We define the loop space ΩX to be
INTRODUCTION TO THE THEORY OF DERIVATORS
107
the pullback
p2
ΩX
p1
(9.19)
/ PX
ev1
PX
ev1
/ X.
Thus, up to homeomorphism, ΩX is the space of maps [−1, 1] → X which send
the boundary points −1, 1 to the base point x0 . The base point of this space is the
constant map [−1, 1] → X with value x0 .
In this model for the loop space, the inversion of loops ι : ΩX → ΩX is given by a
reparametrization via the reflection at the origin 0 ∈ [−1, 1] and this reparametrization can be obtained by interchanging the two copies of P X in the pullback square
(9.19). More formally, the outer commutative square in the diagram
ΩX
p1
"
p2
ΩX
p2
p1
PX
)/
PX
ev1
ev1
/X
induces by the universal property of the pullback square a canonical dashed morphism making everything commute. The reader easily checks that this morpism is
the above reparametrization ι.
To mimick this in arbitrary pointed derivators, let
σ : y →y
be the swap symmetry which interchanges the objects (1, 0) and (0, 1). Given
a pointed derivator D and X ∈ D(1) it can be shown that this swap symmetry
induces a loop inversion ι : ΩX → ΩX, yielding the inverses for the group structure
of Theorem 9.18. In more detail, ι is essentially the canonical mate limy σ ∗ → limy
associated to
σ /
y
y
C
;
1
π
π
1 = /1
evaluated on the coherent cospan (1, 1)! (X) ∈ D(y). And this induces the loop
inversion map.
Theorem 9.20. 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) is
additive. By Proposition 9.14 the category D(1) is preadditive and we have to
show that the abelian monoid structure on homD(1) (Y, Y ), Y ∈ D(1), is a group
structure. The stability of D implies by Proposition 9.9 that there is an equivalence
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 can show
Adapt, simplify, less
detailed.
108
MORITZ GROTH
that the loop inversion map (??) is an additive inverse to id : ΩX → ΩX, concluding
the proof by Lemma 2.4. We refer the reader to [Gro13] for more details.
We conclude this subsection by a remark sketching an alternative proof of the
additivity of stable derivators.
Remark 9.21.
(i) Abstracting a classical result concerning the homotopy category Ho(Top∗ ) of pointed topological spaces, one can show that in arbitrary
pointed derivators 2-fold loop objects are abelian group objects. More precisely, let D be a pointed derivator and let X ∈ D(1). The object Ω2 X can
be endowed with two structures of a group object, respectively coming with
multiplication morphisms
∗1 : Ω2 X × Ω2 X → Ω2 X
and ∗2 : Ω2 X × Ω2 X → Ω2 X.
These morphisms are abstractions of the loop concatenations with respect to
different sphere coordinates in topology. One can show that these two group
structures agree and actually are abelian group structures.
(ii) The previous point yields a different proof of the additivity of stable derivators
(Theorem 9.20). Again, it suffices to show that the preadditive category D(1)
is additive. The preadditivity implies that every object X ∈ D(1) can be
essentially uniquely endowed with the structure of a commutative monoid
object, and by Lemma 2.4 it remains to show that this unique commutative
monoid structure is a group structure. Since D is stable there is a natural
isomorphism X ∼
= Ω2 Σ2 X and the concatenation morphism on 2-fold loop
objects turns X into an abelian group object. Thus, the unique abelian
monoid structures are abelian group structures, concluding the proof.
9.4. Morphisms and natural transformations. In this subsection we define
morphisms of derivators and natural transformations between such morphisms.
These definitions are special cases of the more general concepts of pseudo-natural
transformations and modifications.
Definition 9.22. A morphism of prederivators is a pseudo-natural transformation.
Let us unravel this definition. If D and E are prederivators, then a morphism
F : 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.
u
v
(b) For A → B → C in Cat the equation γvu = γu v ∗ · u∗ γv holds,
D(C)
FC
E (C)
(vu)∗
∼
=
?G
(vu)∗
/ D(A)
FA
/ E (A)
D(C)
=
FC
E (C)
v∗
∼
=
@H
v∗
/ D(B)
FB
/ E (B)
u∗
∼
=
@H
u∗
/ D(A)
FA
/ E (A).
(c) For functors u1 , u2 : A → B and a natural transformation α : u1 → u2 we
have the equality γu2 · (α∗ FB ) = (FA α∗ ) · γu1 .
INTRODUCTION TO THE THEORY OF DERIVATORS
109
Thus, a morphism of prederivators is given by a collection of functors which are
compatible with restrictions up to specified, coherent isomorphisms. A morphism
of derivators is simply a morphism of underlying prederivators. We say that a
morphism is strict if all γu are identity transformations (which is to say that we
have a 2-natural transformation).
Examples 9.23.
(i) Let F : C → D be a functor between ordinary categories.
Associativity of composition of functors implies that the induced functors
C A → 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 0 be in Cat. The restriction
functors (v × 1A )∗ : D(B 0 × A) → D(B × A), A ∈ Cat, assemble to a strict
0
restriction morphism v ∗ : D B → D B . In particular, there are evaluation
∗
B
morphisms b : D → D, b ∈ B.
(iii) Left Quillen functors between combinatorial model categories have associated
left derived morphisms between homotopy derivators, and similarly for right
Quillen functors; see §B.2 for more details.
Definition 9.24. Let F, G : D → E be morphisms of prederivators. A natural
transformation φ : F → G is a modification.
Thus, such a φ : F → G consists of natural transformations φA : FA → GA which
for each u : A → B make the following diagram commute
u∗ FB
γF,u
u∗ φB
u∗ GB
/ FA u∗
φA u∗
γG,u
/ GA u∗ .
The same definition also applies to derivators. The reader easily defines composition
laws for morphisms and natural transformations of (pre)derivators and checks that
these yield 2-categories.
Notation 9.25. 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 sub2-category of PDER. Often we simplify notation and drop some of the decorations
of the components of morphisms and natural transformations of (pre)derivators.
Examples 9.26.
(i) Let F, F 0 : C → D be functors between ordinary categories
and let φ : F → F 0 be a natural transformation. For A ∈ Cat there is the
2-functor (−)A : CAT → CAT and φ hence yields natural transformations
φA : F A → F 0A of functors C A → DA , defining a natural transformation
y(φ) : y(F ) → y(F 0 ). In fact, the passage to represented prederivators defines
an embedding
y : CAT → PDER.
By restriction, there is a similar embedding of the 2-category of complete and
cocomplete categories into the 2-category DER of derivators.
110
MORITZ GROTH
(ii) Let D be a prederivator, let u, v : A → B be in Cat, and let φ : u → v be a
natural transformation. The transformations φ × 1 : u × 1C → v × 1C induce
a natural transformation φ∗ : u∗ → v ∗ of strict morphisms D B → D A . In
fact, the following stronger result is true: the passage to shifted prederivators
defines a 2-functor
(−)(−) : Cat op × PDER → PDER : (A, D) 7→ D A .
In particular, for every morphism F : D → E and B ∈ Cat there is an induced
shifted morphism
F A : DA → E A .
(9.27)
Recall that a morphism of derivators is a pseudo-natural transformation coming
with structure isomorphisms. We can use the isomorphisms γu−1 : FA u∗ → u∗ FB
and γu : u∗ FB → FA u∗ ,
D(B)
u∗
FB
E (B)
∼
=
u∗
/ D(A)
u∗
D(B)
∼
=
FB
FA
/ E (A),
E (B)
/ D(A)
@H
u∗
FA
/ E (A),
in order to talk about morphisms of derivators which preserve certain Kan extensions.
Definition 9.28.
(i) A morphism of derivators F : D → E preserves left Kan
extensions along u : A → B if the canonical mate transformation
(9.29)
η
γ −1
u
u! FA → u! FA u∗ u! →
u! u∗ FB u! → FB u!
is an isomorphism. A morphism of derivators is cocontinuous if it preserves
arbitrary left Kan extensions.
(ii) A morphism of derivators F : D → E preserves right Kan extensions
along u : A → B if the canonical mate transformation
η
γu
FB u∗ → u∗ u∗ FB u∗ → u∗ FA u∗ u∗ → u∗ FA
is an isomorphism. A morphism of derivators is continuous if it preserves
arbitrary right Kan extensions.
Specializing to Kan extensions along πA : A → 1, there are obvious variants of
morphisms preserving limits of a fixed shape or arbitrary limits, and dually.
Example 9.30. Let F : C → D be a functor between complete and cocomplete
categories and let A ∈ Cat. If we consider the induced morphism of derivators
y(F ) : y(C) → y(D) then the canonical mate (9.29) associated to u = πA : A → 1
can 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 9.31. A morphism of derivators is cocontinuous if and only if it preserves
colimits.
Proof. This proof is left as an exercise.
As a consequence one can deduce the following.
INTRODUCTION TO THE THEORY OF DERIVATORS
111
Remark 9.32. Let C and D be complete and cocomplete categories. A functor
F : C → D is cocontinuous in the usual sense of Definition A.21 if and only if the
strict morphism y(F ) : y(C) → y(D) is cocontinous in the sense of Definition 9.28.
Our discussion of parametrized Kan extensions in §8.3 immediately establishes
the following result.
Proposition 9.33. Let D be a derivator and let v : B → B 0 be a functor. The
0
restriction morphism v ∗ : D B → D B 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 shifted
derivators. From this it follows that, given a functor u : A → A0 , we have to show
that a certain canonical map
(1 × u)! (v × 1)∗ → (v × 1)∗ (1 × u)!
is an isomorphism. But unraveling definitions the reader checks that this is precisely
the canonical mate expressing that the square (8.21) is homotopy exact and we
hence conclude by Proposition 8.20.
As in ordinary category theory, examples of continuous or cocontinuous morphisms come from adjunctions. To this end we recall from §A.1 that an adjunction
can be specified by two functors and two natural transformations satisfying the
triangular identities (A.3). This description allows us to extend to concept of an
adjunction to arbitrary 2-categories.
Definition 9.34. An adjunction of derivators is an adjunction internal to the
2-category DER.
Thus, an adjunction consists of morphisms L : D1 → D2 and R : D2 → D1
together with natural transformations η : 1 → RL and : LR → 1 satisfying the
triangular identities
1 = L · Lη : L → L
and
1 = R · ηR : R → R.
This general notion can be re-expressed in more elementary terms.
Proposition 9.35.
(i) A morphism of derivators is a left adjoint if and only if
each 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 a
right adjoint and the morphism is continuous.
Proof. We refer the reader to [Gro13] for a proof.
The proof of this result shows the following. Let L : D1 → D2 be a cocontinuous 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 be
assembled to a right adjoint morphism of derivators R : D2 → D1 .
We specialize this to the following situation. Let D be a derivator and let us
0
consider a restriction morphism v ∗ : D B → D B which is cocontinuous by Proposition 9.33. By (Der3) each component of v ∗ has a left adjoint (v × 1)! which hence
0
assemble to a left Kan extension morphism of derivators v! : D B → D B .
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MORITZ GROTH
Corollary 9.36. Let D be a derivator and let u : A → B be a functor. There are
adjunctions of derivators
(u! , u∗ ) : D A D B
and
(u∗ , u∗ ) : D B D A .
In particular, the left Kan extension morphism u! : D A → D B is cocontinuous and
the right Kan extension morphism u∗ : D A → D B is continuous.
Definition 9.37. An equivalence of derivators is an equivalence internal to the
2-category DER.
Thus, an equivalence consists of morphisms F : D1 → D2 and G : D2 → D1
together with natural isomorphisms η : 1 ∼
= GF and : F G ∼
= 1. In that case we
also say that the morphism F or G alone is an equivalence. Similarly to the case
of adjunctions there is the following description of equivalences in more elementary
terms.
Proposition 9.38. A morphism of derivators is an equivalence if and only if each
component is an equivalence.
We conclude this subsection with the following remark.
Remark 9.39. In earlier sections of this book we established results about Kan
extension functors in arbitrary derivators or suitable classes of derivators. If we
take the philosophy of derivators more serious, then we should not content ourselves
with such results concerning categories of coherent diagrams. In fact, based on
the shifting operation of derivators (Proposition 7.32), we should instead aim for
similar such results about derivators of coherent diagrams. More specifically, in the
context of homotopy derivators of model categories we would obtain statements
about homotopy theories of diagrams which, in general, are stricter than mere
statements about homotopy categories of diagrams.
It turns out that with a bit more work one can show that basically all our above
results extend to statements about Kan extension morphisms of derivators. For
example, given a derivator D and a fully faithful functor u : A → B, then the
Kan extension morphisms u! , u∗ : D A → D B are fully faithful and hence induce
equivalences onto their essential images.
These refined versions are central to our applications in abstract representation
theory in the final section, and in that section we freely use these derivator versions
of our earlier results. (In a more systematic treatment one would hence right away
establish these results in their derivator version while for pedagogic reasons we
refrained from doing so here.)
Todo
9.5. Strongness of derivators. Discuss definition and variants. Refer to appendix for proof in case of combinatorial model categories. Discuss the functors
D(k) → D([1])[1] → D(k) for a field k, AR-quivers
9.6. Canonical triangulations in stable derivators. In this subsection we construct canonical triangulations on strong stable derivators. Recall from Lemma 6.30
that for every prederivator D and categories A, B ∈ Cat there is the partial underlying diagram functor
diaB,A : D(B × A) → D(A)B ,
making a coherent diagram incoherent in the B-direction.
INTRODUCTION TO THE THEORY OF DERIVATORS
113
Definition 9.40. A derivator is strong if dia[1],A : D([1] × A) → D(A)[1] is full
and essentially surjective for every A ∈ Cat.
Note that we do not ask that the functor is faithful, so the partial underlying
diagram functors dia[1],A are not asked to be equivalences.
Examples 9.41.
(i) Represented derivators are strong. In fact, in that case all
partial 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 6.24) are strong.
(iii) A derivator D is strong if and only if D op is strong.
(iv) If D is a strong derivator and if B ∈ Cat, then the shifted derivator D B is
also strong.
Remark 9.42. The precise form of Definition 9.40 depends on the author. For
example, Heller [Hel88] asks that the functors diaF,A : D(F × A) → D(A)F are
full and essentially surjective for all finite free categories F and all A ∈ Cat. It
can be shown that the derivators mentioned in Examples 9.41 all satisfy this more
restrictive axiom.
The property of being strong is important when one wants to relate properties
of stable derivators to structure on its values. This is illustrated in the proof of the
following theorem.
As a preparation recall from §8.5 the construction of (coherent) cofiber sequences
in a pointed derivator D. By Proposition 8.37 there is an equivalence of categories
D([1]) ' D()cof sending a morphism to its cofiber sequence. Every coherent
cofiber sequence has an underlying incoherent cofiber sequence, yielding the functor
D()cof → D(1)[3] defined in (8.34). In the case of a stable derivator we denote
the following composition by
(9.43)
tria : D([1]) ' D()cof → D(1)[3]
and refer to tria(X), X ∈ D([1]), as the standard triangle associated to X. A
triangle x → y → z → Σx in D(1) is distinguished if it is naturally isomorphic
to a standard triangle.
Passing to shifted derivators this also defines a class of distinguished triangles on
D(A), A ∈ Cat. In fact, with D also the shifted derivator D A is stable (Lemma 9.4)
and the underlying category of D A is canonically isomorphic to D(A).
Theorem 9.44. Let D be a strong, stable derivator and A ∈ Cat. The suspension
functor Σ : D(A) → D(A) together with the above class of distinguished triangles
turn D(A) into a triangulated category.
Proof. Passing to shifted derivators we may assume that A = 1. By Theorem 9.20
the category D(1) is additive and the suspension functor Σ : D(1) → D(1) is an
equivalence by Proposition 9.9. It remains to show that the above class of distinguished triangles satisfies the axioms (T1)-(T4) of Definition 4.9.
(T1): Let x ∈ D(1) and let us consider the corresponding identity morphism
id : x → x as an object in D(1)[1] . Since D is strong, the underlying diagram functor
D([1]) → D(1)[1] is essentially surjective and we can find a coherent morphism
X ∈ D([1]) and an isomorphism dia[1] (X) ∼
= id (one can also avoid using the
strength here by restricting along π : [1] → 1 or by invoking Example 7.24). To
114
MORITZ GROTH
see that x → x → 0 → Σx is distinguished the reader easily checks that it suffices
to show that the third object in tria(X) is a zero object. And this claim follows
immediately from Proposition 9.5 since the third object in tria(X) is isomorphic to
CX which is trivial as X is an isomorphism.
By definition the class of distinguished triangles is closed under isomorphisms
and for axiom (T1) it hence remains to show that every morphism f : x → y
in D(1) extends to a distinguished triangle. By assumption the diagram functor
D([1]) → D(1)[1] is essentially surjective and we can hence find X ∈ D([1]) and
a natural isomorphism dia[1] (X) ∼
= f . The reader checks that this isomorphism
together with the distinguished triangle tria(X) yields the desired distinguished
triangle extending f (using once more that the class of distinguished triangles is
closed under isomorphisms).
(T2): We leave it to the reader to verify that it suffices to establish the following
claim: if we rotate backwards a standard triangle then we obtain a distinguished
triangle. Since our distinguished triangles are obtained from two iterations of the
cofiber construction, here it is convenient to consider coherent diagrams encoding
three iterations of cofiber constructions as it was already done in §8.5 (we suggest
the reader to recall the statement of Lemma 8.40 as well as Proposition 8.42 together
with its proof).
By this lemma, every coherent morphism X = (f : x → y) ∈ D([1]) can be
extended to a coherent diagram looking like (8.41),
f
x
(9.45)
/ 02
/y
g
01
/ z
h
/ x0
f0
/ y0 ,
03
vanishing as indicated and making all squares cocartesian (we can ignore the subscripts in the objects 01 , 02 , 03 for now). Note that the cofiber sequence consisting
of the two top squares in (9.45) is used to construct the standard triangle tria(f ).
In more detail, one considers the composition of these two squares, looking like
(9.46)
x
/ 02
01
/ x0 ,
and the associated identification φ : x0 ∼
= Σx in order to obtain the standard triangle
(9.47)
f
g
φ◦h
x → y → z → Σx.
We next collect a detailed description of the standard triangle tria(g). To begin
with we note that if we reflect the two squares on the right in (9.45) in order to
INTRODUCTION TO THE THEORY OF DERIVATORS
115
draw them horizontally, then we obtain the cofiber sequence on the left in
y
(9.48)
g
/z
h
/ x0
02
f0
/ 03
y
/ 03
/ y0 ,
02
/ y0 ,
which can hence be used to calculate tria(g). In fact, the cocartesian square on the
right in (9.48) obtained by composing the cocartesian squares on the left yields an
identification ψ : y 0 ∼
= Σy and tria(g) hence looks like
g
ψ◦f 0
h
y → z → x0 → Σy.
(9.49)
It remains to show that this triangle is isomorphic to the rotation of (9.47). To
this end, we recall that f 0 is naturally isomorphic to Σf . Since we need some details
about this identification, we repete from the proof of Proposition 8.42 that there is
a coherent cube (8.43),
/ 02
x
f
/ 02
y
/ x0
01
03
f0
/ y0 ,
making the back face and the front face cocartesian. And the natural isomorphism
f0 ∼
= Σf is obtained from this cube. Thus, in more detail, we use the identifications
x0 ∼
= Σx and y 0 ∼
= Σy induced from the cocartesian squares
(9.50)
x
/ 02
y
/ 02
01
/ x0 ,
03
/ y0 .
Since the square on the left is precisely the square (9.46), the first identification
is again the morphism φ : x0 ∼
= Σx showing up in (9.47). However, if we compare
the square on the right in (9.50) to the square on the right in (9.48), then we see
that they differ by a restriction along the automorphism swapping (1, 0) ↔ (0, 1).
It follows from the sketch proof of Theorem 9.20 that the second identification
induced by (9.50) hence is −ψ : y 0 ∼
= Σy, where ψ is as in (9.49). To summarize,
the natural isomorphism f 0 ∼
= Σf has components
x0
f0
φ ∼
=
Σx
/ y0
∼
= −ψ
Σf
/ Σy.
116
MORITZ GROTH
This finally gives us the desired isomorphism of triangles
/z
g
y
h
φ
=
=
y
/ z
g
ψ◦f 0
/ x0
φ◦h
/ Σy
=
/ Σx
−Σf
/ Σy
from the standard triangle tria(g) (see (9.49)) to the triangle which is obtained
from tria(f ) (see (9.47)) by rotation.
(T3): We leave it to the reader to show that this axiom follows rather directly
from the assumption that D is strong.
(T4): It remains to establish the octahedron axiom, saying that for every pair
of composable morphisms in D(1) there is an associated octahedron diagram. We
first show that every coherent pair of composable morphisms X ∈ D([2]) with
f1
f2
underlying diagram f3 = f2 ◦ f1 : x → y → z gives rise to such a diagram. Choosing
a slightly different notation than in Definition 4.9, we have to show that there is a
commutative diagram
/y
f1
x
=
f2
x
/ z
f3
g2
(9.51)
w
h2
Σy
g1
0
g3
2
=
4
Σg1
/u
h1
1
f4
/ v
h3
3
g4
/ w
h2
/ Σx
=
/ Σx
Σf1
/ Σy
h4
/ Σu
such that the rows and colums are distinguished triangles, i.e., such that
f1
g1
h
f2
g2
h
f3
g3
h
f4
g4
(i) x → y → u →1 Σx,
(ii) y → z → w →2 Σy,
(iii) x → z → v →3 Σx, and
h
(iv) u → v → w →4 Σu
are distinguished triangles. Thus, these four distinguished triangles have to be
constructed such that the squares 0 − 4 in (9.51) all commute.
This can be achieved by considering coherent diagrams looking like
x
(9.52)
0
f1
/y
/z
/0
/ u
/ v
/ x0
/0
0
/ w
/ y0
/ u0 .
f2
INTRODUCTION TO THE THEORY OF DERIVATORS
117
In more detail, let B ⊆ [4] × [2] be the full subcategory obtained by removing the
objects (4, 0), (0, 2). The functor [2] → B classifying the composable morphisms
j
i
(0, 0) → (1, 0) → (2, 0) factors as a composition [2] → A → B of fully faithful
functors where A ⊆ B contains the objects (0, 0), (1, 0), (2, 0) and the four objects
populated by zeros in (9.52). Similarly to Lemma 8.40 one checks that the fully
j!
i
∗
D(A) → D(B) induce an equivalence
faithful functors D([2]) →
D([2]) ' D(B)cof ,
(9.53)
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, this
shows that every X ∈ D([2]) with underlying diagram f3 = f2 ◦ f1 : x → y → z
gives rise to a coherent diagram Q ∈ D(B)cof with underlying diagram
x
(9.54)
0
f1
/y
f2
/z
/0
g3
g1
/ u
0
f4
/ v
h3
/0
Σf1
g4
/ w
/ Σx
h2
/ Σy
Σg1
/ Σu.
(Of course the notation is slightly abusive in that the respective squares in the
underlying diagram are no pushout squares anymore; the decoration is only meant
to indicate that they come from coherent cocartesian squares.) A ‘diagram chase’
in (9.52) combined with the identifications made in (9.54) yields the desired four
distinguished triangles satisfying the above five relations 0 − 4 . To begin with, if
one considers the top three cocartesian squares and glues the two to the right, then
one obtains the first distinguished triangle together with the relation h3 f4 = h1 ,
settling equation 1 . Let us next consider the copy of [2] × [2] in the middle of
the diagram and compose vertically. This gives the second distinguished triangle
together with the relation g2 = g4 g3 , hence taking care of 2 . If we again consider
the top three cocartesian squares but this time glue the two squares to the left,
then we construct the third distinguished triangle. For the remaining distinguished
triangle 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 suffices
to consider the middle square in the top and bottom row of (9.52), respectively.
This completes the construction of an octahedron diagram for a coherent pair of
composable morphisms.
Let us now consider a pair of composable morphisms x → y → z in the underlying
category D(1) and show that it has an associated octahedron diagram. The reader
easily checks that it suffices to construct a coherent diagram X ∈ D([2]) such that
the underlying diagram is naturally isomorphic to x → y → z. If we use the
stronger variant of strength (see Remark 9.42) then this is immediate since [2] is a
finite, free category. The reader who does not want to use this stronger axiom is
118
MORITZ GROTH
invited to come up with an independent construction of such a natural isomorphism
dia(X) ∼
= (x → y → z), using Definition 9.40 only.
The triangulations of the theorem are referred to as canonical triangulations.
As a special case the theorem yields canonical triangulations on underlying categories of strong, stable derivators. We illustrate this by a few examples.
Examples 9.55.
(i) If A is an abelian category, such that the homotopy derivator
DA exists (for example, if A is a Grothendieck abelian category; see Definition 6.19), then the underlying category DA (1) ∼
= D(A) is a triangulated
category. Taking up again Examples 6.20, Theorem 9.44 thus reproduces the
classical triangulations on derived categories of fields, rings, and schemes.
(ii) If we apply Theorem 9.44 to the stable derivator of spectra (6.25), then we
obtain the classical triangulation on the stable homotopy category SHC.
(iii) More generally, homotopy derivators of stable model categories and stable ∞-categories are strong, stable derivators. Applied to such examples,
Theorem 9.44 reproduces the triangulations on homotopy categories of stable model categories ([Hov99, §7]) and homotopy categories of stable ∞categories ([Lur11, §1]).
We conclude this subsection by a short remark on the octahedron axiom (related
to this see also Remark 8.35).
Remark 9.56. Let us recall the main steps of the proof of the octahedron axiom
(T4) for the canonical triangulations from Theorem 9.44. Let D be a strong, stable
derivator and let X ∈ D([2]) be a coherent pair of composable morphisms. Associated to X we obtain a coherent diagram looking like (9.52) satisfying certain
exactness and vanishing properties. More precisely, there is an equivalence of categories (9.53), and one might refer to the target category D(B)cof as the category
of coherent octahedron diagrams. Thus, as long as we stay at the level of
coherent diagrams, having a pair of composable morphisms is equivalent to having
an octahedron diagram.
Now, given a coherent octahedron diagram X ∈ D(B)cof , making some identifications as indicated in (9.54) we obtain an incoherent octahedron diagram looking
like (9.51), and this step amounts to a loss of information since we pass to underlying diagrams. If we denote the shape of an incoherent octahedron diagram
(9.51) by O, then these two steps hence amount to the functorial construction of
octahedron diagrams
octa : D([2]) ' D(B)cof → D(1)O .
If we start with an incoherent pair of composable morphisms instead, then we first
have 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 construction
of cones and distinguished triangles at the level of triangulated categories; see the
beginning of §4.4. Also in these cases we can obtain functorial such constructions
if we stay at the level of coherent diagrams as made precise by Proposition 8.37
and the functor tria : D([1]) → D(1)[3] defined in (9.43). Again, if we instead
begin with incoherent morphisms, then we first have to lift against the functor
dia[1] : D([1]) → D(1)[1] , resulting in the loss of functoriality. These two similar
INTRODUCTION TO THE THEORY OF DERIVATORS
119
observations can be summarized by the following two diagrams
D([1])
dia[1]
D(1)[1]
'
tria
/ D()cof
% D(1)[3] ,
'
/ D(B)cof
octa
[2]
D(1)
% D(1)O ,
D([2])
dia[2]
in which the vertical functors result in a loss of information.
9.7. Exact morphisms of stable derivators. Our next goal is to show that
the triangulations D(A), A ∈ Cat, of Theorem 9.44 are compatible with restriction
functors 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 morphisms
of (strong) stable derivators.
Recall from Definition 9.28 what it means that a morphism of derivators preserves
certain Kan extensions. Specializing this to Kan extensions along πA : A → 1 for
suitable A we obtain the following definition.
Definition 9.57.
(i) A morphism of derivators is left exact if it preserves terminal objects and cartesian squares.
(ii) A morphism of derivators is right exact if it preserves initial objects and
cocartesian squares.
(iii) A morphism of derivators is exact if it is left exact and right exact.
The following is immediate from Remark 9.32 and Proposition A.22.
Example 9.58. A functor F : C → D between complete and cocomplete categories
is left exact in the sense of Definition A.21 if and only if the induced morphism of
derivators y(F ) : y(C) → y(D) is left exact in the sense of Definition 9.57.
Lemma 9.59.
(i) A morphism of stable derivators is left exact if and only if it
is right exact if and only if it is exact.
(ii) Let (F, G) : D E be an adjunction between stable derivators. The morphisms F, G both are exact.
Lemma 9.60. Let F : D → E be a morphism of derivators, let u : A → A0 , and
let B ∈ Cat. If F preserves left Kan extensions along u, then F also preserves left
Kan extensions along 1 × u : B × A → B × A0 .
Proof. TODO.
Let us recall from Examples 9.26 that the shifting operation also applies to
morphisms of derivators (see (9.27)). A different way of phrasing Lemma 9.60 is
by saying that if F : D → E preserves left Kan extensions along u then so does the
shifted morphism F B : D B → E B .
Corollary 9.61. Let F : D → E be an exact morphism of stable derivators and let
B ∈ Cat. The induced morphism F : D B → E B is again exact.
Proof. This is immediate from Lemma 9.60 and its dual.
Recall from Definition 4.17 that an exact functor between triangulated categories
is an additive functor endowed with exact structure.
120
MORITZ GROTH
Proposition 9.62. Let F : D → E be an exact morphism of strong, stable derivators and let A ∈ Cat. The functor FA : D(A) → E (A) can be turned into an exact
functor with respect to canonical triangulations.
Proof. By Corollary 9.61 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 9.63. 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) can
be turned into exact functors with respect to canonical triangulations.
Proof. By Corollary 9.36 there are adjunctions of strong, stable derivators
(u! , u∗ ) : D A D B
and
(u∗ , u∗ ) : D B D A ,
given by restriction and Kan extension morphisms. It follows from Lemma 9.59
that these three morphisms are exact and the underlying functors can hence be
turned into exact functors (Proposition 9.62). We note that these underlying
functors respectively are the functors u∗ : D(B) → D(A), u! : D(A) → D(B), and
u∗ : D(A) → D(B), concluding the proof.
Remark 9.64. Given a strong, stable derivator D, by Theorem 9.44 there are canonical triangulations on D(A), A ∈ Cat, and by Proposition 9.62 the restriction functors u∗ : D(B) → D(A) can be turned into exact functors. There is also such an
observation for natural transformations.
To make this precise, we recall the following definition. Let T , T 0 be triangulated categories and let (F, σ), (F 0 , σ 0 ) be exact functors T → T 0 . A natural
transformation α : F → F 0 is exact if the diagram
FΣ
αΣ
F 0Σ
σ
∼
=
/ ΣF
∼
=
/ ΣF 0
Σα
σ0
commutes. Given a strong, stable derivator D, functors u, v : A → B, and a transformation α : u → v, one can show that the natural transformation α∗ : u∗ → v ∗ is
exact in this sense.
A concise way of summarizing this is as follows. Triangulated categories, exact
functors, and exact natural transformations are assembled in a (again very large)
2-category T riaCAT of triangulated categories. Every strong, stable derivator
D : Cat op → CAT admits a lift against the forgetful 2-functor T riaCAT → CAT ,
T riaCAT
8
∃D
Cat op
D
/ CAT .
10. Towards abstract representation theory
10.1. Morita equivalences.
10.2. Derived equivalences.
INTRODUCTION TO THE THEORY OF DERIVATORS
121
10.3. Strong stable equivalences. Given a ring R we denote by Mod(R) the
abelian category of R-modules. Attributing credit to [?] there is the following
definition.
Definition 10.1. Two rings R, S are Morita equivalent if there is an equivalence
of categories Mod(R) ' Mod(S).
It can be shown that such an equivalence is necessarily given by the tensor product with an (R, S)-bimodule. In representation theory people are often interested
in derived categories of rings and algebras; see for example [Hap88, AHHK07]. The
derived analogue of Definition 10.1 is 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 finite
quivers. Let Q be a finite quiver and let k be a field. We recall that the path
algebra kQ is constructed as follows. The underlying k-vector space has as basis
the set of all paths in Q. More precisely, if Qn , n ≥ 0, denotes the set of all paths
of length precisely n, then we set
M
kQ =
kQn .
n≥0
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 do
not match appropriately). A key feature about this k-algebra is that there is an
equivalence of categories
(10.3)
Mod(kQ) ' Mod(k)Q .
Here, Q also denotes the free category associated to the oriented graph Q, and
representations can hence equivalently be specified by Q-shaped diagrams of vector
spaces. Of course, instead of working over a field k we can consider a ring R and
obtain a similarly defined path algebra RQ and a corresponding equivalence (10.3).
Definition 10.4. Let Q, Q0 be finite quivers and let R be a ring. The quivers Q, Q0
are derived equivalent over R if there is an exact equivalence
∆
D(RQ) ' D(RQ0 ).
Note that in principle the property of being derived equivalent might depend on
the 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. A
quiver is a tree if it admits no non-trivial, unoriented loops.
A reorientation of a quiver Q is a quiver Q0 obtained from Q by changing the
orientation 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 direct
argument showing that the corresponding path algebras over fields are not Morita
equivalent. Nevertheless, it turns out that these quivers are derived equivalent over
arbitrary fields. In fact, this is only a special case of the following more general
result of Happel.
122
MORITZ GROTH
Theorem 10.6. Let T be a finite tree and let T 0 be a reorientation of T . The
quivers T, T 0 are derived equivalent over arbitrary fields.
By a purely combinatorial argument the task is reduced to the following situation. Let T be a finite tree and let q0 ∈ T be a source, i.e., a vertex q0 such that
all edges adjacent to it start at q0 . The reflected tree T 0 of T at the source q0 is
obtained by changing the orientation of all edges starting at q0 and thereby turning
the source q0 into a sink. Now, in order to establish Theorem 10.6 it suffices to show
that finite trees and their reflections at sources and sinks are derived equivalent over
arbitrary fields.
It turns out that such derived equivalences even exist more generally. The notion
of 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 Q0 be the reflection of Q at
a source or a sink. The quivers Q, Q0 are derived equivalent over arbitrary fields.
In fact, such derived equivalences can be explicitly constructed as derived reflection functors in the sense of Bernšteı̆n, Gel0 fand, and Ponomarev (see [BGP73] and
[Hap87]). Using the formalism of derivators, one can show that such equivalences
exist in much broader generality. To this end we begin by a minor reformulation
of Definition 10.4. As a special case of Lemma 3.10 we observe that for a finite
quiver Q there is an isomorphism of categories
∼ Ch(R)Q
Ch(Mod(R)Q ) =
which sends quasi-isomorphisms to levelwise quasi-isomorphisms. If we combine
this with a degreewise application of (10.3) then we obtain an equivalence of categories
Ch(RQ) ' Ch(R)Q ,
again sending quasi-isomorphisms to levelwise quasi-isomorphisms. As an upshot,
if two quivers Q, Q0 are derived equivalent over a ring R, then there is the top
horizontal and the two vertical equivalences
(10.8)
/ D(RQ0 )
'
D(RQ)
'
'
Ho(Ch(R)Q )
'
/ Ho(Ch(R)Q0 ),
hence inducing the bottom equivalence. Thus, to put it in words, the fact that two
quivers are derived equivalent over R means that if we pass to the corresponding
homotopy theories of diagrams in Ch(R) then the associated homotopy categories
are equivalent. In some cases one can obtain similar results for homotopy theories
of representations in arbitrary stable homotopy theories in the sense of the following
definition (see [GŠ14c]).
Definition 10.9. Two quivers Q, Q0 are strongly stably equivalent if for every
stable derivator D there is an equivalence of derivators
0
DQ ' DQ
which is natural with respect to exact morphisms of derivators.
We include a few comments about the definition.
INTRODUCTION TO THE THEORY OF DERIVATORS
123
Remark 10.10.
(i) This definition of course makes perfectly well sense for small
categories A, A0 as well, but here we focus on quivers.
(ii) If two finite quivers Q, Q0 are strongly stably equivalent then they are also
derived equivalent over arbitrary rings. In fact, considering the special case
of the stable derivator DR of a ring (Examples 9.2), we obtain an equivalence
Q
Q0
of derivators DR
' DR
. Passing to underlying categories this yields an
Q
Q0
equivalence of categories DR
(1) ' DR
(1) which is exact with respect to
the canonical triangulations of Theorem 9.44. Combining this with (10.8) we
deduce that Q, Q0 are derived equivalent over R. (One can also invoke the
Q
equivalence of derivators DR
' DRQ ; see Examples 7.31.)
(iii) However, the notion of being strongly stably equivalent is a priori much more
restrictive for the following three reasons.
(a) First, simply by choosing other examples of stable derivators (Examples 9.2) and passing to underlying categories we obtain further exact
equivalences of triangulated categories of representations. For example,
considering the stable derivator DX of a scheme X we obtain equivalent derived categories of representations in quasi-coherent OX -modules.
Similarly, if we pass to the stable derivator DA of a differential-graded
algebra A, then we get exact equivalences of triangulated categories of
differential-graded representations. Moreover, there are further such results for spectral representations, simply by passing to the stable derivator of spectra or the stable derivator DE of a —say— symmetric ring
spectrum E. Still more generally, we can consider homotopy derivators
associated to arbitrary stable model categories or stable ∞-categories,
concluding that there are exact equivalences of triangulated categories of
abstract representations of the quivers under consideration. (We again
refer the reader to [GŠ14c, Examples 5.5] for a fairly long list of additional examples.)
(b) Second, by the very definition we ask for equivalences of derivators as
opposed to merely asking for exact equivalences of the underlying triangulated categories. Recall that, in general, it is a stronger statement to
have an equivalence of homotopy theories and not only an equivalence of
homotopy categories. Thus, there are equivalences of homotopy theories
of abstract representations of the quivers under consideration.
(c) Third, these equivalences are compatible with respect to exact morphisms. In fact, by the very definition suitably restricted corepresented
0
copresheaves (−)Q , (−)Q are asked to be equivalent. In particular, the
various equivalences listed above are compatible with (derived) restrictions of scalars, inductions and coinductions of scalars, (derived) tensor
and hom functors, as well as localization and colocalization functors.
With these remarks in mind, let us mention the following generalization of Theorem 10.6; see [GŠ14b].
Theorem 10.11. Let T be a finite tree and let T 0 be a reorientation of T . The
quivers T, T 0 are strongly stably equivalent.
In §10.4 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 branching
point, i.e., vertex of valences at least three. With more refined techniques one can
124
MORITZ GROTH
mimic the classical construction of reflection functors in the sense of Bernšteı̆n,
Gel0 fand, and Ponomarev [BGP73] in order to obtain the strong stable equivalences
of Theorem 10.11; see [GŠ14b]. Theorem 10.7 can be refined to yield the following
result.
Theorem 10.12. Let Q be a finite, acylic quiver and let Q0 be the reflection of Q
at a source or a sink. The quivers Q, Q0 are derived equivalent over arbitrary fields.
In fact, in [GŠ15] there will be the construction of abstract reflection functors
establishing this result. The techniques of that paper apply even more generally,
yielding additional examples of strong stable equivalences. A further specializing
to derivators of rings and fields gives rise to new examples of derived equivalences
of category algebras, generalizing Theorem 10.7.
10.4. Abstract tilting theory of An -quivers. In this subsection we give a sketch
proof that different Dynkin quivers of type A of the same length are strongly stably
equivalent. These observations are closely related to the construction of canonical
higher triangulations on (strong) stable derivators.
To begin with let us recall that a Dynkin quiver of type A is simply a finite
zig-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 obtain
the An -quivers. Thus, the goal is to see that all An -quivers are strongly stably
equivalent. 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 again
take a look at Remark 9.39.
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 derivators
of representations in D are naturally equivalent. To begin with, since the two
categories (1 → 2 → 3) and (1 ← 2 ← 3) are isomorphic these quivers are strongly
stably equivalent.
We next show that the quivers p= (1 ← 2 → 3) and y = (1 → 2 ← 3) are strongly
stably equivalent. To this end, we consider the functors ip : p→ and iy : y → and the associated fully faithful Kan extension morphisms (ip )! : D p → D and
(iy )∗ : D y → D . The essential image of (ip )! consists precisely of those X ∈ D (A)
such that every Xa ∈ D(), a ∈ A, is cocartesian (Corollary 8.23) and similarly for
the essential image of (iy )∗ . Since D is stable these two essential images coincide
and will be denoted by D ,ex ⊆ D . As an upshot we obtain equivalences of
derivators
D p ' D ,ex ' D y ,
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 abstract
representation of [2]. Then such an equivalence is obtained by passing to the cofiber
INTRODUCTION TO THE THEORY OF DERIVATORS
125
of (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)
(0, 1)
/ (1, 1).
/ (2, 0)
The functor [2] → B classifying (0, 0) → (1, 0) → (2, 0) factors as a composition of
fully faithful functors
j1
i1
A1 → B,
[2] →
where A1 is obtained from B by also removing the object (1, 1). Associated to
these functors there are fully faithful Kan extension morphisms (i1 )∗ : D [2] → D A1
and (j1 )! : D A1 → D B . 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 )! is
easily checked to consist of precisely those coherent diagrams such that the obvious
square is cocartesian. As an upshot, (j1 )! (i1 )∗ : D [2] → D B induces an equivalence
onto the full subderivator of D B spanned by all Z satisfying the following exactness
properties.
(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). This
functor factors as a composition of fully faithful functors
i
j2
2
p→
A2 → B,
where A2 is obtained from B by also removing the object (0, 0). Using similar arguments as above and an additional cofinality argument, one checks that
(j2 )∗ (i2 )! : D p → D B induces an equivalence onto the full subderivator of D B
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 (Corollary 8.23), and let us denote it by D B,ex ⊆ D B . As an upshot, the above constructions yield equivalences of derivators
D [2] ' D B,ex ' D p ,
showing that the A3 -quivers [2] and p are strongly stably equivalent. Since being
strongly 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 ≥ 1
are strongly stably equivalent. It will pay off to approach these constructions a bit
more systematically. To motivate the construction, let us emphasize that in the case
of A3 -quivers, once the dust settles, everything essentially boiled down to expanding
our representations by passing to (co)fibers and adding certain bicartesian squares.
In fact, similar arguments were already used in the proof of Theorem 9.44, namely
in the proof of the octahedron axiom by means of the construction of coherent
octahedron diagrams; see (9.54) and Remark 9.56.
It turns out that one convenient strategy consists of expanding coherent diagrams
looking like (9.54) doubly infinitely by adding more zeros and bicartesian squares
126
MORITZ GROTH
~ n = [n − 1] for the linearly oriented
as we explain next. From now on, let us write A
~
An -quiver. Given an abstract representation X = (x → y → z) ∈ D A3 , suitable
combinations of Kan extensions yield a coherent diagram looking like
<0
···
(10.14)
···
!
#
=0
Az
Ωw
<
=Ωv
@u
;x
"
!
0
0
!
<0
Σx
>
Av
>y
"
···
@0
0
<Σu
"
< Σy
=w
<· · ·
"
"
0
"
"
<· · ·
< Σz
"
···
The defining exactness properties of such coherent diagrams are that they vanish
on the boundary stripes and that all squares are bicartesian. We denote the corresponding shape by M3 ∈ Cat and note that there is the fully faithful inclusion
~ 3 → M3 pointing at the objects populated by (x → y → z). More generfunctor A
ally, for n ≥ 1 there is a similar category Mn ∈ Cat and a fully faithful inclusion
~ n → Mn . Note that this inclusion functor factors as a composition of
functor A
fully faithful functors
i1
i2
i3
i4
~n →
A
B1 →
B2 →
B3 →
Mn ,
where the respective full subcategories of Mn are obtained as follows.
~ n → Mn by adjoining all
(i) The category B1 is obtained from the image of A
objects on the boundary stripes sitting below this image.
(ii) The category B2 is obtained from B1 by adding the remaining objects below
this image.
(iii) The category B3 is obtained from B2 by adjoining the remaining objects on
the boundary stripes.
Given a derivator D, associated to these inclusions there are the following fully
faithful Kan extension morphisms,
(10.15)
~
(i1 )∗
(i2 )!
(i3 )!
(i4 )∗
D An → D B1 → D B2 → D B3 → D Mn .
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 ' D Mn ,ex
Mn ,ex
Mn
where D
⊆ D
is the full subderivator spanned by all coherent diagrams
Z ∈ D Mn 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 right
extension by zero, the second one amounts to adding cocartesian squares in one
direction, the third morphism is left extension by zero, and the fourth morphism
adds cartesian squares in the other direction. Thus, in stable derivators the essential
image is as claimed. For a detailed proof see [GŠ14a, §4].
One reason for us to consider these shapes Mn is the following observation.
~ n → Mn there are many additional embeddings Q → Mn
Besided the inclusions A
INTRODUCTION TO THE THEORY OF DERIVATORS
127
for arbitrary An -quivers Q of the same length. And related to these embedding one
constructs, for every stable derivator D, similar equivalences of derivators
D Q ' D Mn ,ex ,
which again are natural with respect to exact morphisms of stable derivators. Note
that in this case as well as in the equivalence of Theorem 10.16 an inverse equivalence 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 constructions
as in (10.15) induce natural equivalences D p ' D M3 ,ex . If we combine this with
the equivalence of Theorem 10.16, then we obtain natural equivalences
~
D A3 ' D Mn ,ex ' D p ,
~ 3 and p are strongly stably equivalent. The reader easily checks that
showing that A
this reproduces the strong stable equivalence constructed in the proof of Proposition 10.13.
Theorem 10.17. For every fixed n ≥ 1 all An -quivers are strongly stably equivalent.
Proof. Let Q, Q0 be two An -quivers of the same length and let D be a stable derivator. Then we can choose embeddings iQ : Q → Mn and iQ0 : Q0 → Mn as above.
Related to these embedding, the abovementioned more general version of Theorem 10.16 yields natural equivalences
0
D Q ' D Mn ,ex ' D Q ,
showing that Q, Q0 are strongly stably equivalent.
An additional reason for us to consider the shapes Mn ∈ Cat is that they show
up 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 A
consists of a self-equivalence Σ : A ' A and chosen classes of distinguished ntriangles. Here, an n-triangle in A is a diagram Mn → A together with suitably 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 compatibility
axioms, including relations between distinguished n-triangles for varying n.
Thus, 2-triangles look like a more refined version of classical triangles and 3triangles like refined octahedron diagrams; see Remark 9.56. The distinguished
n-triangles for n ≥ 4 can be thought of as ‘higher octahedron diagrams’. A precise
formulation and a proof of the following result can be found in [GŠ14a, §13].
Theorem 10.18. Let D be a strong stable derivator and let A ∈ Cat. The category
D(A) admits a canonical strong triangulation.
As in the case of triangulations (see Proposition 9.62 and Remark 9.64) one can
show that these strong triangulations are natural with respect to exact morphisms
and that a strong stable derivator D : Cat op → CAT lifts against the forgetful
functor from a certain 2-category of strong triangulated categories, exact functors,
and exact transformation. We again refer the reader to [GŠ14a, §13] for more
details.
128
MORITZ GROTH
10.5. Universal tilting modules. Will be added soon.
Topics to be briefly mentioned:
(i) Universality of spaces and spectra
(ii) implications for (co)limits: cofinality, characterization of htpy exact squares:
depends only on spaces, ends and coends, (co)simplicial bar constructions
(iii) bicategory of profunctors
(iv) induced canceling action, universal tilting modules
We include a short remark which is intended for the reader who is familiar with
simplicial sets. It can be safely skipped without any loss of continuity.
Remark 10.19. The characterization of final functors given in the above proposition
sounds like a geometric statement. To make it precise, let us recall that there is
the nerve functor associating a simplicial set N A to every small category A. The
above statement can be reformulated by saying that u : A → B is final if and only
if 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-shaped
and B-shaped homotopy colimits are the same. In that case, the combinatorial
characterization of such functors is that all nerves N (b/u) are weakly contractible
simplicial sets.
Appendix A. Some category theory
In this section we review some basics from category theory and refer the reader
to [ML98, Bor94a, Bor94b] for more details.
A.1. Adjunctions. Let us begin by recalling that an adjunction (L, R) : C D
between two categories C, D consists of two functors L : C → D and R : D → C
together 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 encoding
such 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 we
obtain a natural transformation η : 1 → RL, the (adjunction) unit. Dually, applying φ−1 to identity morphisms 1 : Rd → Rd we obtain a natural transformation
: LR → 1, the (adjunction) counit.
These two transformations are not unrelated since they are constructed using the
isomorphisms (A.1) and their inverses. First, let us note that we can reconstruct φ
from R and η. In fact, the naturality of φ implies that for every f : Lc → d there
is a commutative diagram
homD (Lc, Lc)
φ
f∗
homD (Lc, d)
/ homC (c, RLc)
(Rf )∗
φ
/ homC (c, Rd).
Tracing 1 : Lc → Lc through the diagram and using the definition of η we see
that φ(f ) = (Rf )∗ (ηc ) = Rf ◦ ηc . Similarly, the inverse isomorphism φ−1 can be
INTRODUCTION TO THE THEORY OF DERIVATORS
129
reconstructed from L and , and we have the commutative diagram
homC (RLc, Rd)
6
η∗
R
/
φ
homD (Lc, d) o
h
(A.2)
φ−1
∗
v
homD (Lc, LRd).
(
homC (c, Rd)
L
In particular, starting with any morphism Lc → d or c → Rd an application of
the four maps in the clockwise direction gives us back the same morphism. If we
specialize this to the identity morphisms 1 : Lc → Lc and 1 : Rd → Rd, then we
deduce the following two relations between η and ,
Lη
L
(A.3)
/ LRL
ηR
R
L
=
+ L,
/ RLR
R
=
+ R,
called the triangular identities. It can be shown that an adjunction is determined
by the datum of two functors L : C → D, R : D → C and two natural transformations
η : 1 → RL, : LR → 1 such that the triangular identities are satisfied.
In this book we denote adjunction units generically by η and adjunction counits
by . If we want to emphasize that there are different adjunctions in a given context,
then we use obvious notational variations like η 0 , 0 . We recall the following fact
about 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 a
natural isomorphism. An object d ∈ D lies in the essential image of L if and
only if the counit d : LRd → d is an isomorphism.
(ii) The right adjoint R is fully faithful if and only if the counit : LR → 1 is a
natural isomorphism. An object c ∈ C lies in the essential image of R if and
only 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 are
fully faithful if and only if η and are natural isomorphisms.
It is immediate from (A.1) that adjunctions can be composed. Given adjunctions (L0 , R0 ) : C D and (L00 , R00 ) : D E then composing the respective adjunction isomorphisms we obtain an adjunction (L, R) : C E with L = L00 L0 and
R = R0 R00 . The adjunction unit and counit of this composite adjunction can be
calculated as
η0
η : 1 → R0 L0
R0 η 00 L0
→
R0 R00 L00 L0
and : L00 L0 R0 R00
L00 0 R00
→
00
L00 R00 → 1.
Natural transformations between left adjoint functors are closely related to natural transformations between associated right adjoint functors.
Lemma A.5. Let (L, R), (L0 , R0 ) : C D be adjunctions and let α : L → L0 be a
natural transformation. There is a unique natural transformation α0 : R0 → R such
130
MORITZ GROTH
that the following diagram commutes
homD (Lc, d)
O
(A.6)
∼
=
φ
(α0 )∗
α∗
homD (L0 c, d)
/ homC (c, Rd)
O
φ0
∼
=
/ homC (c, R0 d).
This defines a bijection between natural transformations α : L → L0 and natural
transformations α0 : R0 → R.
Proof. Let us choose c = R0 d and trace the counit 0d : L0 R0 d → d through the
diagram. This implies αd0 = φ(0d ◦ αR0 d ), showing that there is at most one such
natural transformation. Note that using the description of the adjunction isomorphism φ in terms of R and η as in (A.2) we obtain α0 = R0 ◦ RαR0 ◦ ηR0 and this
shows that we actually constructed a natural transformation α0 .
Natural transformations α and α0 making (A.6) commute are conjugate or total
mates. The proof and its converse show that conjugate natural transformations
α, α0 determine each other by means of the relations
(A.7)
α0 = R0 ◦ RαR0 ◦ ηR0 :
R0 → RLR0 → RL0 R0 → R
α = L0 ◦ Lα0 L0 ◦ Lη 0 :
L → LR0 L0 → LRL0 → L0 .
and
(A.8)
The uniqueness of conjugate transformations implies that the construction is compatible with compositions and identity transformations.
Corollary A.9. If α : L → L0 and α0 : R0 → R are conjugate natural transformations, then α is a natural isomorphism if and only if α0 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 in
LAdj(C, D) is a left adjoint L : C → D and morphisms are natural transformations
between left adjoints. Dually, we define the category RAdj(C, D).
Corollary A.10. Let C and D be categories. There is an equivalence of categories
LAdj(C, D) ' RAdj(D, C)op .
Proof. We describe a functor LAdj(C, D) → RAdj(D, C)op . For each left adjoint
functor L : C → D we choose a right adjoint R : D → C. Given these choices, for
every pair of left adjoint functors L, L0 the passage to conjugate transformations
defines a bijection between natural transformations L → L0 and R0 → R, which, as
already observed, is compatible with compositions and identities. Thus, we obtain
a fully faithful functor LAdj(C, D) → RAdj(D, C)op . Clearly this functor is also
essentially surjective and hence an equivalence.
Thus, the specification of a left adjoint is essentially equivalent to the specification of a right adjoint. We will later need the following more precise statement
along these lines.
INTRODUCTION TO THE THEORY OF DERIVATORS
131
Lemma A.11. Let (L, R, η, ) : C D and (L0 , R, η 0 , 0 ) : C D be adjunctions
with the same right adjoint R. There is a unique natural isomorphism α : L → L0
making the following diagrams commute
1
η
/ RL
LR
∼
= Rα
η0
/1
J
αR ∼
=
) 0
RL ,
L0 R.
0
Proof. The existence of a unique natural isomorphism α making the diagram on
the 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 show
that the two compositions have the same image under the natural isomorphism
φ : homD (LRd, d) ∼
= homC (Rd, Rd). By definition of the adjunction counit we
have to show that φ(0 ◦ αR) = 1. By (A.2) we know that φ(0 ◦ αR) is given by
ηR
R0
RαR
R → RLR → RL0 R → R.
Since we already showed that Rα ◦ η = η 0 , we see that φ(0 ◦ αR) = R0 ◦ η 0 R which
is the identity 1 : R → R by a triangular identity (A.3).
A.2. Limits and colimits. We assume the reader to have some basic familiarity with limits and colimits, including a discussion of terminal objects, products,
pullbacks and the dual notions of initial objects, coproducts, and pushouts. Nevertheless we recall some key definitions and results about (co)limits, mainly to
establish notation and to prepare the ground for homotopical generalizations as
discussed 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 → C
be a functor. A cone on X is a pair (l, α) consisting of an object l ∈ C and
morphisms αa : l → Xa, a ∈ A, such that for every morphism f : a → a0 in A the
diagram
/ Xa
αa
l
f∗
αa0
Xa0
commutes. A morphism of cones (l, α) → (l0 , α0 ) is a morphism l → l0 in C such
that
l
αa
l0
α0a
/ Xa
commutes for every a ∈ A.
With the obvious composition law and identity morphisms this defines the category 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 sense
132
MORITZ GROTH
that
Xa
βa
/> c
f∗
βa0
Xa0
commutes for every morphism f : a → a0 . Defining morphisms of cocones in the
obvious 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 → C
be a functor.
(i) A limit of X is a terminal object limA X of cone(X).
(ii) A colimit of X is an initial object colimA X of cocone(X).
Thus, to emphasize, such a limit is a pair consisting of an underlying object,
also denoted by limA X, together with a universal cone which we also refer to as a
limiting cone. Similarly, we abuse notation and write colimA X for the underlying
object 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 up
to unique isomorphisms and this uniqueness implies functoriality. In the following
proposition we write ∆ = ∆A : C → C A for the diagonal functor which associates
to 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.
(i) If every X : A → C has a limit, then the assignment X 7→ limA X extends to
a limit functor limA : C A → C which is right adjoint to ∆A ,
(∆A , limA ) : C C A .
(ii) If every X : A → C has a colimit, then the assignment X 7→ colimA X extends
to a colimit functor colimA : C A → C which is left adjoint to ∆A ,
(colimA , ∆A ) : C A C.
For particular choices of small categories A, there are special names for the
resulting 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 in
the following precise sense. A category is complete or cocomplete if it admits
limits or colimits of all small diagrams, respectively.
Proposition A.15.
(i) A category is complete if and only if it admits equalizers
and (small) products.
(ii) A category is cocomplete if and only if it admits coequalizers and (small)
coproducts.
INTRODUCTION TO THE THEORY OF DERIVATORS
133
Proof. By duality it is enough to take care of the first statement. The task is to
show 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 small
products and equalizers. We consider the following pair of parallel morphisms
Y
Y
Xa1
d0 , d1 :
Xa →
f : a0 →a1
a∈A
Q
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 the
projections onto the factor Xa1 associated to an arbitrary f : a0 → a1 . In the case
of d0 this composition is chosen to be
Y
f∗
Xa → Xa0 → Xa1
a∈A
while in the case of d1 we simply take the projection
Y
Xa → Xa1 .
a∈A
The underlying object of the limit limA X is now defined to be the equalizer of
these two morphisms,
Y
Y
(A.16)
lim X = eq
Xa ⇒
Xa1 .
A
f : a0 →a1
a∈A
Using the universal cone belonging to this equalizer, we can define the universal
cone of limA X to have components
Y
lim X →
Xa → Xa0 , a0 ∈ A,
A
a∈A
where the second morphism is the projection onto the a0 -component. We leave it
to 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
a
(A.17)
colimA X = coeq
Xa0 ⇒
Xa
f : a0 →a1
of two similarly defined morphisms d0 , d1 :
are left to the reader.
`
f : a0 →a1
a∈A
Xb →
`
a∈A
Xa. The details
This proposition admits a few variants, one of which we include here. A category
is finitely complete if it admits finite limits, i.e., limits of diagrams defined on
finite categories (recall that a category is finite if it has finitely many objects and
morphisms 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 is
suffices to show that terminal objects and pullbacks generate finite products and
equalizers. Since equalizers are special cases of pullbacks it remains to construct
finite products. The reader easily checks that products of two objects are obtained
134
MORITZ GROTH
by forming pullbacks over terminal objects, and the case of finite products hence
follows by induction.
We conclude this section by a short discussion of the preservation of (certain) limits by functors. Let us consider a functor X : A → C admitting a colimit colimA X
and let F : C → D be a functor such that also F ◦ X : A → D admits a colimit
colimA F ◦ X. If we apply F to the colimiting cocone of X, then for every f : a → a0
in A there is a commutative diagram
F (Xa)
/ F (colimA X)
8
F (Xa0 )
in D, which is to say that we obtain a cocone on F ◦ X. The universality of the
colimiting cocone of F ◦ X implies that there is a unique morphism
(A.19)
colimA (F X) → F (colimA X)
compatible with these two cocones. We refer to this morphism and a dually defined
morphism
(A.20)
F (limA X) → limA (F X)
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 canonical
morphism colimA (F X) → F (colimA X) is an isomorphism.
Definition A.21.
(i) A functor between complete categories is continuous if
it preserves all limits.
(ii) A functor between cocomplete categories is cocontinuous if it preserves all
colimits.
(iii) A functor between finitely complete categories is left exact if it preserves
finite limits.
(iv) A functor between finitely cocomplete categories is right exact if it preserves
finite colimits.
(v) A functor between finitely complete and finitely cocomplete categories is exact 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 if
and only if it preserves equalizers and products.
(ii) A functor between finitely complete categories is left exact if and only if it
preserves equalizers and finite products if and only if it preserves pullbacks
and 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 continuous and hence left exact.
(ii) A left adjoint functor between cocomplete categories is cocontinuous and hence
right exact.
A.3. Basic 2-categorical terminology. Will be added soon.
INTRODUCTION TO THE THEORY OF DERIVATORS
135
Appendix B. Examples of derivators
B.1. Represented derivators. Will be added soon.
B.2. Homotopy derivators of model categories. In this subsection we include
a detailed proof that combinatorial model categories have underlying homotopy
derivators. The proof relies on the non-trivial results that diagram categories in
combinatorial model categories can be endowed with both the projective and the
injective model structures.
Let M be a Quillen model category and let A ∈ Cat. In general, it is not true
that we can endow the diagram category MA with a model structure such that the
weak equivalences are precisely the levelwise weak equivalences, i.e., those natural
transformations 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 the
one which is adapted to the study of homotopy colimits and homotopy left Kan
extensions.
Definition B.1. Let M 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 on MA , then we refer to it as the
projective model structure. For the notion of a cofibrantly generated model
structure we refer the reader to [Hov99]. Let us only mention that most model
structure showing up in nature (like homotopy theory, homological algebra, and
higher category theory) enjoy this property, so that the following result often applies.
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 denoted
by MA
proj . Recall that the category of simplicial sets can be endowed with the
cofibrantly generated Kan–Quillen model structure. In this case, the corresponding
projective model structures on diagram categories were first established in [].
Projective model structures have good functorial properties. For the study of
homotopy left Kan extensions we observe the following.
Proposition B.3. Let M be a cofibrantly generated model category and let u : A →
B
B be in Cat. The adjunction (u! , u∗ ) : MA
proj Mproj is a Quillen adjunction. In
A
particular, (colimA , ∆A ) : Mproj M is a Quillen adjunction.
Proof. We have to show that u∗ : MB → MA preserves projective fibrations and
acyclic projective fibrations which is immediate since both classes are defined levelwise.
Passing to left derived functors, this proposition yields homotopy left Kan extensions and homotopy colimits. Dualizing Definition B.1 we obtain the following.
136
MORITZ GROTH
Definition B.4. Let M 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 cofibration.
(ii) A morphism f : X → Y is an injective weak equivalence if it is a levelwise
weak equivalence.
(iii) A morphism f : X → Y is an injective fibration if it has the RLP with
respect to acyclic injective cofibrations.
As in the projective case, if the above three classes define a model structure on
MA , then we refer to it as the injective model structure and denote the resulting
model category by MA
inj . One of the first examples was established by Heller in
[Hel88] where he shows that diagram categories in simplicial sets admit the injective
model structure. In general, as of this writing, we only know that injective model
structures exist on diagram categories in combinatorial model categories. Let us
recall 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 model
categories and additional references. Here, we only mention that many model
categories arising in nature are combinatorial.
Theorem B.5. Let M be a combinatorial model category and let A ∈ Cat. The
injective 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 → B
A
be in Cat. The adjunction (u∗ , u∗ ) : MB
inj Minj is a Quillen adjunction. In
particular, (∆A , limA ) : M MA
inj is a Quillen adjunction.
Proof. The functor u∗ : MB → MA preserves injective cofibrations and acyclic
injective cofibrations since both classes are defined levelwise.
Thus, for combinatorial model categories we can use the projective model structures of Theorem B.2 to establish the existence of homotopy left Kan extensions
and the injective model structures of Theorem B.5 to obtain homotopy right Kan
extensions. To relate the domains of the respective functor we make the following
observation.
Proposition B.7. Let M be a combinatorial model category and let A ∈ Cat. The
A
identity adjunction (id, id) : MA
proj Minj is a Quillen equivalence.
Proof.
The main goal of this subsection is to show that combinatorial model categories
have homotopy derivators. The proof that homotopy Kan extensions can be calculated pointwise needs some additional preparation. For this purpose, let us consider
u : A → B, b ∈ B, and the associated slice squares
p
(u/b)
π(u/b)
1
/A
u
b
/B
(b/u)
and
π(b/u)
/A
q
AI
1
b
u
/ B.
INTRODUCTION TO THE THEORY OF DERIVATORS
137
Proposition B.8. Let M be a model category, let u : A → B in Cat, and let b ∈ B.
(i) If M is a cofibrantly generated model category, then the restriction functors
(u/b)
∗
B
p∗ : MA
proj → Mproj and b : M → Mproj are left Quillen functors.
(b/u)
(ii) If M is a combinatorial model category, then the functors q ∗ : MA
inj → Minj
and b∗ : M → MB
are
right
Quillen
functors.
inj
Proof. By Theorem B.2 and Theorem B.5 the respective model structures exist and
it suffices by duality to take care of (i). We begin by showing that b∗ : MB
proj → M is
a left Quillen functor. Since b∗ preserves weak equivalences, it suffices to show that
b∗ preserves projective cofibrations. Considering the adjunction (b∗ , b∗ ) : MB M this is equivalent to b∗ preserving acyclic projective fibrations, i.e., levelwise
acyclic fibrations. Let us recall from the explicit construction of cofree diagrams in
Example 5.29 that there are natural isomorphisms
Y
X,
X ∈ M, b0 ∈ B.
b∗ (X)b0 ∼
=
homB (b0 ,b)
Since products of acyclic fibrations are again acyclic fibration, b∗ sends acyclic
fibrations to acyclic projective fibrations.
(u/b)
We now show that p∗ : MA
proj → Mproj also is a left Quillen functor. TODO Proposition B.9. Let M, N be cofibrantly generated model categories, and let
(L, R) : M N be a Quillen adjunction.
(i) The induced adjunction (F A , GA ) : MA N A is a Quillen adjunction.
(ii) If (F, G) : M N is a Quillen equivalence then so is (F A , GA ) : MA N A .
Proposition B.10. Let F : M → N be a left Quillen functor beween combinatorial model categories and let y(F ) : y(M) → y(N ) be the induced strict morphism
of derivators. Passing to left derived functors of the components of y(F ), we obtain an induced morphism of derivators LF : H o(M) → H o(N ). Similarly, a
right Quillen functor G : N → M between combinatorial model categories induces
a morphism RG : H o(N ) → H o(M).
Note that, in general, the morphisms LF and RG are not strict, since Quillen
functors do not necessarily preserve weak equivalences. This fact is part of the motivation behind defining morphisms of derivators as pseudo-natural transformations
(as opposed to as 2-natural transformations).
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140
MORITZ GROTH
MPIM, Vivatsgasse 7, 53111 Bonn, Germany
E-mail address: [email protected]