Triangulated Categories
Amnon Neeman
Author address:
Department of Mathematics, The Australian National
University, Canberra, ACT 0200, Australia
E-mail address: an3r@virginia.edu
Contents
1. Introduction 6
Chapter 1. Definition and elementary properties of triangulated
categories 33
1.1. Pre-triangulated categories 33
1.2. Corollaries of Proposition 1.1.20 41
1.3. Mapping cones, and the definition of triangulated categories
47
1.4. Elementary properties of triangulated categories 54
1.5. Triangulated subcategories 62
1.6. Direct sums and products, and homotopy limits and
colimits 65
1.7. Some weak "functoriality" for homotopy limits and colimits 70
1.8. History of the results in Chapter 1 72
Chapter 2. Triangulated functors and localizations of triangulated
categories 75
2.1. Verdier localization and thick subcategories 75
2.2. Sets and classes 102
2.3. History of the results in Chapter 2 103
Chapter 3. Perfection of classes 105
3.1. Cardinals 105
3.2. Generated subcategories 105
3.3. Perfect classes 113
3.4. History of the results in Chapter 3 124
Chapter 4. Small objects, and Thomason's localisation theorem 125
4.1. Small objects 125
4.2. Compact objects 130
4.3. Maps factor through ~~fi 132
4.4. Maps in the quotient 137
4.5. A refinement in the countable case 147
4.6. History of the results in Chapter 4 153
3
4 CONTENTS
Chapter 5. The category A(S) 155
5.1. The abelian category A(S) 155
5.2. Subobjects and quotient objects in A(S) 175
5.3. The functoriality of A(S) 181
5.4. History of the results in Chapter 5 186
Chapter 6. The category Ex Sop; Ab 187
6.1. Ex Sop; Ab is an abelian category satisfying [AB3] and
[AB3*] 187
6.2. The case of S = Tff 206
6.3. Ex Sop; Ab satisfies [AB4] and [AB4*], but not [AB5] or
[AB5*] 211
6.4. Projectives and injectives in theicategory ExjSop; Ab 216
6.5. The relation between A(T) and Ex {Tff}op; Ab 220
6.6. History of the results of Chapter 6 225
Chapter 7. Homological properties of Ex Sop; Ab 227
7.1. Ex Sop; Ab as a locally presentable category 227
7.2. Homological objects in Ex Sop; Ab 230
7.3. A technical lemma and some consequences 235
7.4. The derived functors of colimits in Ex Sop; Ab 260
7.5. The adjoint to the inclusion of Ex Sop; Ab 273
Chapter 8. Brown representability 281
8.1. Preliminaries 281
8.2. Brown representability 283
8.3. The first representability theorem 289
8.4. Corollaries of Brown representability 293
8.5. Applications in the presence of injectives 297
8.6. The second representability theorem: Brown representability
for the dual 312
8.7. History of the results in Chapter 8 316
Chapter 9. Bousfield localisation 319
9.1. Basic properties 319
9.2. The six gluing functors 329
9.3. History of the results in Chapter 9 330
Appendix A. Abelian categories 331
A.1. Locally presentable categories 331
A.2. Formal properties of quotients 337
A.3. Derived functors of limits 357
A.4. Derived functors of limits via injectives 366
CONTENTS 5
A.5. A Mittag-Leffler sequence with non-vanishing limn 373
A.6. History of the results of Appendix A 378
Appendix B. Homological functors into [AB5ff] categories 381
B.1. A filtration 381
B.2. Abelian categories satisfying [AB5ff] 390
B.3. History of the results in Appendix B 397
Appendix C. Counterexamples concerning the abelian category
A(T) 399
C.1. The submodules piM 399
C.2. A large R-module 404
C.3. The category A(S) is not well-powered 405
C.4. A category Ex Sop; Ab without a cogenerator 407
C.5. History of the results of Appendix C 418
Appendix D. Where T is the homotopy category of spectra 421
D.1. Localisation with respect to homology 421
D.2. The lack of injectives 434
D.3. History of the results in Appendix D 440
Appendix E. Examples of non-perfectly-generated categories 443
E.1. If T is @0-compactly generated, Top is not even well-
generated 443
E.2. An example of a non @1-perfectly generated T 448
E.3. For T = K(Z), neither T nor Top is well-generated. 453
E.4. History of the results in Appendix E 458
Bibliography 459
Index 461
6 CONTENTS
1. Introduction
Before describing the contents of this book, let me explain its ori-
gins. The book began as a joint project between the author and Vo-
evodsky. The idea was to assemble coherently the facts about triangu-
lated categories, that might be relevant in the applications to motives.
Since the presumed reader would be interested in applications, Voevod-
sky suggested that we keep the theory part of the book free of examples.
The interested reader should have an example in mind, and read the
book to find out what the general theory might have to say about the
example. The theory should be presented cleanly, and the examples
kept separate.
The division of labor was that I should write the theory, Voevodsky
the applications to motives. What then happened was that my part of
this joint project mushroomed out of proportion. This book consists
just of the formal theory of triangulated categories. In a sequel, we
hope to discuss the motivic applications.
The project was initially intended to be purely expository. We
meant to cover many topics, but had no new results. This was to be
an exposition of the known facts about Brown representability, Bous-
field localisation, t-structures and triangulated categories with tensor
products. The results should be presented in a unified, clear way, with
the exposition accessible to a graduate student wishing to learn the
theory. The catch was that the theory should be developed in the
generality one would need motivically. The motivic examples, unlike
the classical ones, are not compactly generated triangulated categories
(whatever this means). The classical literature basically does not treat
the situation in the generality required.
My job amounted to modifying the classical arguments, to work in
the greater generality. As I started doing this, I quickly came to the
conclusion that both the statements and the proofs given classically
are very unsatisfactory. The proofs in the literature frequently rely on
lifting problems about triangulated categories to problems about more
rigid models. Right at the outset I decided that in this book, I will do
everything to avoid models. Part of the challenge was to see how much
of the theory can be developed without the usual crutch. But there
was a far more serious problem. Many of the statements were known
only in somewhat special cases, decidedly not including the sort that
come up in motives. Thomason once told me that "compact objects
are as necessary to this theory as air to breathe". In his words, I was
trying to develop the theory in the absence of oxygen.
1. INTRODUCTION 7
The book is the result of my work on the subject. It treats a
narrower scope of topics than initially planned; we deal basically only
with Brown representability and Bousfield localisation. But in some
sense we make great progress on the problems. In the process of setting
up the theory in the right generality and without lifting to models,
we end up with some new and surprising theorems. The book was
meant to be an exposition of known results. The way it turned out, it
develops a completely new theory. And this theory gives interesting,
new applications to very old problems. Now it is time to summarise
the mathematical content of the book.
The first two chapters of the book are nothing more than a self-
contained exposition of known results. Chapter 1 is the definitions
and elementary properties of triangulated categories, while Chapter 2
gives Verdier's construction of the quotient of a triangulated category
by a triangulated subcategory. This book was after all intended as a
graduate textbook, and therefore assumes little prior knowledge. We
assume that the reader is familiar with the language of categories and
functors. The reader should know Yoneda's Lemma, the general facts
about adjoint functors between categories, unit and counit of adjunc-
tion, products and coproducts. It is also assumed that the reader has
had the equivalent of an elementary course on homological algebra. We
assume familiarity with abelian categories, exact sequences, the snake
lemma and the 5-lemma. But this is all we assume. In particular, the
reader is not assumed to have ever seen the definition of a triangulated
category. In practice, since we give no examples, the reader might wish
to find one elsewhere, to be able to keep it in mind as an application of
the general theory. One place to find a relatively simple, concrete ex-
position of one example, is the first chapter of Hartshorne's book [19 ].
This first chapter develops the derived category. Note that, since we
wish to study mostly triangulated categories closed under all small co-
products, the derived category of most interest is the unbounded derived
category. In [19 ], this is the derived category that receives probably the
least attention. There is another short account of the derived category
in Chapter 10, pages 369-415 of Weibel's [37 ]. There are, of course,
many other excellent accounts. But they tend to be longer. Anyway,
if the reader is willing to forget the examples, begin with the axioms,
and see what can be proved using them, then this book is relatively
self-contained.
Chapter 1 and 2 are an account of the very classical theory. There
are some expository innovations in these two Chapters, but otherwise
little new. If the reader wants to be able to compare the treatment
given here with the older treatments, at the end of each Chapter there
8 CONTENTS
is a historical summary. In the body of the Chapters, I rarely give
references to older works. The historical surveys at the end of each
Chapter contain references to other expositions. They also try to point
out what, if anything, distinguishes the exposition given here from older
ones.
Starting with Chapter 3, little of what is in the book may be found
in the literature. For the reader who has some familiarity with trian-
gulated categories, it seems only fair that the introduction summarise
what, if anything, he or she can expect to find in this book which they
did not already know. It is inevitable, however, that such an explana-
tion will demand from the reader some prior knowledge of triangulated
categories. The graduate student, who has never before met triangu-
lated categories, is advised to skip the remainder of the introduction
and proceed to Chapter 1. After reading Chapters 1 and 2, the rest of
the introduction will make a lot more sense.
Let me begin with the concrete. Few people have a stomach strong
enough for great generalities. Sweeping theorems about arbitrary 2-
categories tend to leave us cold. We become impressed, only when we
learn that these theorems teach us something new, preferably about an
old, concrete example that we know and love. Before I state the results
in the book in great generality, let me tell the reader what we may
conclude from them, about the special case, where T is the homotopy
category of spectra.
Let T be the homotopy category of spectra. Let E be a spectrum
(ie. an object of T). Following Bousfield, the full subcategory TE T,
whose objects are called the E-acyclic spectra, is defined by
Ob (TE) = {x 2 Ob (T) | x ^ E = 0}:
The full subcategory ? TE T, whose objects are called the E-local
spectra, is defined by
?
Ob TE = {y 2 Ob (T) | 8x 2 Ob (TE ); T(x; y) = 0}:
An old theorem of Bousfield (see [6 ]) asserts that one can localise spec-
tra with respect to any homology theory E. In the notation above,
Bousfield's theorem asserts
Theorem (Bousfield, 1979). Let E be a spectrum, that is E
is an object of T. Let TE T and ?TE T be defined as above. Suppose
x is an object of T. Then there is a triangle in T
xE --- ! x --- ! ?xE -- - ! xE ;
with xE 2 TE , and ?xE 2 ?TE .
1. INTRODUCTION 9
Bousfield's theorem has been known for a long time. What this article
has to add, are surprising structure theorems about the categories TE
and ?TE . We prove the following representability theorems
Theorem (New, this book). Let E be a spectrum. Let TE T
and ?TE T be defined as above. The representable functors
TE (-; h) and ?TE (-; h)
can be characterised as the homological functors H : TopE-! Ab (re-
spectively H : ?TopE-! Ab) taking coproducts in TE (respectively ?TE )
to products in Ab. The representable functors
TE (h; -)
can be characterised as the homological functors H : TE - ! Ab, taking
products to products.
Proof: The characterisation of the functors TE (-; h) and ? TE(-; h)
may be found in Theorem D.1.12. More precisely, the characterisa-
tion of TE (-; h) is D.1.12.1, while the characterisation of ?TE (-; h) is
D.1.12.5. The characterisation of the functors TE (h; -) may be found
in Lemma D.1.14. 2
Representability theorems are central to this subject. What we
have achieved here, is to extend Brown's old representability theorem
of [7 ]. Brown proved that the functors T(-; h) can be characterised
as the homological functors Top -! Ab taking coproducts to products.
We have generalised this to TE , TopEand ? TE , but unfortunately not
to ?TopE. We do not know, whether the the functors ?TE (h; -) can be
characterised as the homological functors taking products to products.
Another amusing fact we learn in this book, is that the categories
TE and ?TE are not equivalent to Top. There are, in fact, many more
amusing facts we prove. Let us give one more. We begin with a defini-
tion.
Definition. Let ff be a regular cardinal. A morphism f : x -! y
in T is called an ff-phantom map if, for any spectrum s with fewer than
ff cells, any composite
f
s --- ! x -- - ! y
vanishes.
With this definition, we are ready to state another fun fact that we
learn in this book.
Theorem (New, this book). Let ff > @0 be a regular cardinal.
There is an object z 2 T, which admits no maximal ff-phantom map
10 CONTENTS
y -! z. That is, given any ff-phantom map y -! z, there is at least
one ff-phantom map x -! z not factoring as
x --- ! y --- ! z:
Proof: The proof of this fact is Proposition D.2.5, coupled with
Lemma 8.5.20. 2
Remark 1.1. It should be noted that the above is surprising. If
ff = @0, the ff-phantom maps are the maps vanishing on all finite
spectra. These are very classical, and have been extensively studied
in the literature. Usually, they go by the name phantom maps; the
reference to ff = @0 is new to this book, where we study the natu-
ral large-cardinal generalisation. From the work of Christensen and
Strickland [9 ], we know that every object z 2 T admits a maximal @0-
phantom map y -! z. There is an @0-phantom map y -! z, so that
all other @0-phantom maps x -! z factor as
x --- ! y --- ! z:
What is quite surprising is that this is very special to ff = @0.
So far, we have given the reader a sampling of facts about the ho-
motopy category of spectra, which follow from the more general results
of this book. I could give more; but it is perhaps more instructive, to
indicate the broad approach.
The idea of this book is to study a certain class of triangulated
categories, the well-generated triangulated categories. And the thrust
is to prove great facts about them. We will show, among many other
things
Theorem 1.2. The following facts are true:
1.2.1. Let T be the homotopy category of spectra. Let E be
an object of T. Then both the category TE and the category ?TE
are well-generated triangulated categories.
1.2.2. Suppose T is a well-generated triangulated category.
The representable functors T(-; h) can be characterised as the
homological functors H : Top -! Ab, taking coproducts in T to
products in Ab.
In other words, we will prove a vast generalisation of Brown's repre-
sentability theorem. Not only does it generalise to TE and ?TE , but to
very many other categories as well. The categories that typically come
up in the study of motives are examples. And now it is probably time
1. INTRODUCTION 11
to tell the reader what a well-generated triangulated category is. It
turns out to be quite a deep fact that this structure even makes sense.
Let T be a triangulated category. We remind the reader: a homo-
logical functor T - ! A is a functor from T to an abelian category A,
taking triangles to long exact sequences. We can consider the collection
of all homological functors T - ! A. An old theorem of Freyd's (see
[13 ]) asserts that
Theorem (Freyd, 1966). Among all the homological functors
T - ! A there is a universal one. There is an abelian category A(T)
and a homological functor T - ! A(T), so that any other homological
functor T -! A factors as
T --- ! A(T) --9!-! A
where the exact functor A(T) - ! A is unique up to canonical equiv-
alence. Any natural tranformation of homological functors T - ! A
factors uniquely through a natural transformation of the (unique) exact
functors A(T) -! A.
This theorem tells us that, associated naturally to every triangulated
category T, there is an abelian category A(T). The association is easily
seen to be functorial. It takes the 2-category of triangulated categories
and triangulated functors to the 2-category of abelian categories and
exact functors, and is a lax functor.
One can wonder about the homological algebra of the abelian cat-
egory A(T). Freyd proves also
Proposition (Freyd, 1966). Let T be a triangulated category.
The abelian category A(T) of the previous theorem has enough projec-
tives and enough injectives. In fact, the projectives and injectives in
A(T) are the same. An object a 2 A(T) is projective (equivalently,
injective) if and only if there exists an object b 2 A(T), so that
a b 2 T A(T):
That is, a is a direct summand of an object a b, and a b is in the
image of the universal homological functor T -! A(T). This universal
homological functor happens to be a fully faithful embedding; hence I
allow myself to write T A(T).
It turns out to be easy to deduce the following corollary:
Corollary 1.3. Let F : S - ! T be a triangulated functor. If
F has a right adjoint G : T - ! S, then G is also triagulated, and
A(G) : A(T) -! A(S) is right adjoint to A(F ) : A(S) -! A(T). But
more interesting is the following. If every idempotent in S splits, then
12 CONTENTS
F : S -! T has a right adjoint if and only if A(F ) : A(S) -! A(T) does.
That is, if A(F ) : A(S) -! A(T) has a right adjoint "G: A(T) -! A(S),
then F : S -! T has a right adjoint G : T - ! S, and of course A(G)
is naturally isomorphic to G".
Proof: Lemma 5.3.6 shows that the adjoint of a triangulated functor
is triangulated, Lemma 5.3.8 proves that if G is right adjoint to F then
A(G) is right adjoint to A(F ), while Proposition 5.3.9 establishes that
if A(F ) has a right adjoint G", then F has a right adjoint G. 2
Remark 1.4. It turns out that many of the deepest and most in-
teresting questions about triangulated categories involve the existence
of adjoints. This suggests that Corollary 1.3 should be great. It tells
us that finding adjoints to triangulated functors between triangulated
categories, a difficult problem, is equivalent to finding adjoints to exact
functors between abelian categories. We feel much more comfortable
with abelian categories, so the Corollary should make us very happy.
The problem is that the abelian categories that arise are terrible.
For example, let T be the category D(Z), the derived category of the
category of all abelian groups. Then the abelian group Z can be viewed
as an object of D(Z); it is the complex which is the group Z in dimen-
sion 0, and zero elsewhere. The universal homological functor
D(Z) --- ! A D(Z)
takes Z to an object of A D(Z) . I assert that this object, in the abelian
category A D(Z) , has a proper class of subobjects. The collection of
subobjects of Z 2 A D(Z) is not a set; it is genuinely only a class.
The proof may be found in Appendix C.
In the light of Remark 1.4, it is natural to look for approximations
to the abelian category A(T). It seems reasonable to try to find other
abelian categories A, together with a exact functors A(T) -! A, which
are "reasonable" approximations. It is natural to want the objects of
A to only have sets (not classes) of subobjects. But otherwise it would
be nice if A be as close as possible to the universal abelian category
A(T).
The universal property of A(T) asserts that exact functors A(T) -!
A are in 1-1 correspondence with homological functors T - ! A. We
therefore want to find reasonable homological functors T - ! A, for
suitable A.
Let T be a triangulated category. It is said to satisfy [TR5] if the
coproduct of any small set of objects in T exists in T. If the dual
category Top satisfies [TR5], then T is said to satisfy [TR5*]. Let ff be
1. INTRODUCTION 13
an infinite cardinal. Let T be a triangulated category satisfying [TR5]
and S a triangulated subcategory. We say that S is ff-localising if any
coproduct of fewer than ff objects of S lies in S. We call S T localising
if it is ff-localising for every infinite cardinal ff.
Given a triangulated category T and an ff-localising subcategory
S T, there is a God-given abelian category one can construct out of
S, and a homological functor
T --- ! Ex Sop; Ab :
Now it is time to define these.
The category Ex Sop; Ab is the abelian category of all functors
Sop -! Ab which preserve products of fewer than ff objects. Recall
that S is ff-localising. Given fewer than ff objects in S, their coproduct
exists in T because T satisfies [TR5], and is contained in S because S is
ff-localising. That is, the product exists in the dual Sop, and we look
at functors to abelian groups
Sop --- ! Ab
preserving all such products. The reader can easily check (see Lemma 6.1.4
for details) that Ex Sop; Ab is an abelian category. Futhermore, there
is a homological functor
T --- ! Ex Sop; Ab :
It is the functor that takes an object t 2 T to the representable functor
T(-; t), restricted to S T. We denote this restriction
T (-; t)|S:
This construction depends on the choice of an infinite cardinal ff, and
an ff-localising subcategory S T. In what follows, it is convenient
to assume that the cardinal ff is regular. That is, ff is not the sum of
fewer than ff cardinals, all smaller than ff.
Starting with any regular cardinal ff and any ff-localising subcate-
gory S T, we have constructed a homological functor
T --- ! Ex Sop; Ab :
It factors uniquely through the universal homological functor, to give
an exact functor
A(T) -- ss-!Ex Sop; Ab :
It is very easy to show that the functor ss has a left adjoint F :
Ex Sop; Ab -! A(T). We deduce a unit of adjunction
" : 1 -! ssF:
We prove
14 CONTENTS
Proposition 1.5. Suppose " : 1 -! ssF is the unit of adjunction
above. Then " is an isomorphism if and only if the functor ss : A(T) -!
Ex Sop; Ab preserves coproducts.
Proof: In some sense, all the statements are implicit in Poposition 6.5.3,
although in Proposition 6.5.3 we assert only one of the "if and only if"
implications. But a careful reading of the proof given there will show
that we, in fact, also prove the more precise Proposition 1.5. Let me
elaborate a little.
The proof that the functor ss has a left adjoint F makes up the first
half of the proof of Proposition 6.5.3. Here one uses no hypothesis on
S. In the second part, one shows that ssF is the identity, provided ss
respects coproducts. But the proof makes it clear that the preservation
of coproducts is necessary and sufficient. 2
By Gabriel's theory of localisations of abelian categories (see Ap-
pendix A), the unit of adjunction " : 1 - ! ssF is an isomorphism if
and only if Ex Sop; Ab is a quotient of A(T), with ss being the quotient
map. It is therefore natural to want to study the S T for which this
happens. By Proposition 1.5, this amount to asking when the map
A(T) --ss-!Ex Sop; Ab
preserves coproducts. It turns out that, for a given regular cardinal ff,
this depends on a choice of the ff-localising subcategory S T. We
proceed now to describe the complete answer.
Definition 1.6. Let ff be a regular cardinal. Let T be a triangu-
lated category satisfying [TR5]. An object t 2 T is called ff-small if any
morphism from t to a coproduct
a
t --- ! X
2
factors through a coproduct of fewer than ff objects. There is a subset
0 , 0 of cardinality < ff, and a factorisation
a a
t --- ! X X :
20 2
The full subcategory of all ff-small objects in T is denoted T(ff). Next
we need
Definition 1.7. Let ff be a regular cardinal. Let T be a triangu-
lated category satisfying [TR5]. A class T , containing 0, of objects in
T is called ff-perfect if, for any collection {X ; 2 } of fewer than ff
1. INTRODUCTION 15
objects of T, any object t 2 T , and any map
a
t --- ! X
2
there is a factorisation
a
f
a 2 a
t -- - ! t --- - ! X
2 2
with t in T . Furthermore, if the composite
a
f
a 2 a
t -- - ! t --- - ! X
2 2
vanishes, then each of the maps
f
t --- ! X
factors as
t --- ! u -- - ! X
with u 2 T , so that the composite
a a
t -- - ! t --- ! u
2 2
already vanishes.
With these two definitions, we have a theorem
Theorem 1.8. Let ff be a regular cardinal, T a triangulated cate-
gory satisfying [TR5]. Let S T be an ff-localising subcategory. The
natural functor
A(T) --ss-! Ex Sop; Ab
preserves coproducts if and only if
1.8.1. The objects of S are all ff-small; that is S T(ff).
1.8.2. The class of all objects in S is ff-perfect, as in Defi-
nition 1.7.
Proof: Lemma 6.2.5. Once again, in the statement of Lemma 6.2.5 we
only assert the sufficiency; if S satisfies 1.8.1 and 1.8.2, then ss preserves
coproducts. But the proof immediately also gives the converse. 2
16 CONTENTS
It is therefore of interest to study ff-localising subcategories S
T(ff), whose collection of objects form an ff-perfect class. The remark-
able fact is that there is a biggest one. For any regular cardinal ff, we
can define a canonical, God-given ff-localising subcategory Tff. It is
given by
Definition 1.9. The full subcategory Tff T, of all ff-compact
objects in T, is defined as follows. The class of objects in Ob (Tff) is
the unique maximal ff-perfect class in T(ff). It of course needs to be
shown that such a maximal ff-perfect class exists. It is also relevant to
know that Tff T is an ff-localising triangulated subcategory. All this
is proved in Chapters 3 and 4.
The categories Tffare, in a certain sense, the optimal choices for S.
For each ff, we have an exact functor
i j
A(T) -- ss-!Ex {Tff}op; Ab
preservingicoproductsjand products, having a left adjoint F , and so
that Ex {Tff}op; Ab is the Gabriel quotient of A(T) by the Serre sub-
category of all objects on which the functor ss vanishes. It is natural
to study the subcategories Tff T.
Example 1.10. The previous paragraphs may be a little confus-
ing. But the upshot is the following. Suppose we are given a trian-
gulated category T closed under coproducts, and a regular cardinal ff.
There is some mysterious, canonical way to define an ff-localising tri-
angulated subcategory, denoted Tff T. The reader might naturally
be curious to know, what Tffis, in some simple examples.
If T is the homotopy category of spectra, then Tffis the full subcat-
egory of spectra with fewer than ff cells. If T is the derived category of
an associative ring R, then the objects of Tffturn out to be chain com-
plexes of projective R-modules, whose total rank(=rank of the sum of
all the modules) is < ff. If ff = @0, then Tff= T@0 is the subcategory
of compact objects in T, and its study is very classical. But even for
ff > @0, we are dealing with a fairly natural subcategory.
Now we return from the examples to the general theory. The first,
trivial property of Tff T is
Lemma 1.11. If ff < fi are regular cardinals, then Tff Tfi.
Proof: Lemma 4.2.3. 2
To get further, we need the notion of generation.
1. INTRODUCTION 17
Definition 1.12. Let ff be an regular cardinal. Let S be a class of
objects of T. Then ~~~~ffstands for the smallest ff-localising subcategory
of T containing S. The symbol ~~~~ stands for the smallest localising
subcategory containing S. That is,
[ ff
~~~~ = ~~~~ :
ff
We say that S generates T if T = ~~~~ . With this definition, we are
ready for Thomason's localisation theorem. The theorem is essentially
the statement that the subcategories Tffbehave well with respect to
quotient maps. We begin with a statement involving only one triangu-
lated category T.
Lemma 1.13. Suppose ff is a regular cardinal, and T is a triangu-
lated category satisfying [TR5]. Suppose further that Tffgenerates T;
that is,
= T:
Then for any regular cardinal fi > ff,
fi = Tfi:
Note that the conclusion of the theorem might be confusing. Since we
are assuming = T, then surely it should be a tautology that
fi = Tfi:
Just replace the by T. But this misses the point that there are two
ways to read the symbol fi. One is to note that, for any triangulated
category S closed under coproducts, there is a canonical way to define
a subcategory Sfi; and then we apply the construction to S = . But
given a collection of objects S 2 T, there is also a canonical way to
define a subcategory ~~~~fi, the fi-localising subcategory generated by
S. And we could do this construction to S = Tff. The assertion of the
Lemma is that under reasonable conditions, these two agree, and the
notation leads to no confusion.
Proof: Lemma 4.4.5. 2
This lemma is actually of great practical use. It says that once we have
computed Tff, then as long as Tffgenerates, we know all Tfifor fi > ff.
They are just the closure of Tffwith respect to coproducts of fewer
than fi objects, and triangles. Call this statement zero of Thomason's
localisation theorem. The rest of the theorem concerns the situation of
a Verdier quotient.
18 CONTENTS
Theorem 1.14. Let S be a triangulated category satisfying [TR5],
R S a localising subcategory. Write T for the Verdier quotient S=R.
Suppose there is a regular cardinal ff, a class of objects S Sffand
another class of objects R R \ Sff, so that
R = and S = ~~~~:
Then for any regular fi ff,
fi= Rfi= R \ Sfi;
~~~~ fi= Sfi:
The natural map
Sfi=Rfi-! T
factors as
Sfi=Rfi-! Tfi T;
and the functor
Sfi=Rfi-! Tfi
is fully faithful. If fi > @0, the functor
Sfi=Rfi-! Tfi
is an equivalence of categories. If fi = @0, then every object of Tfiis a
direct summand of an object in Sfi=Rfi.
Proof: Theorem 4.4.9. 2
Since I have been telling the reader that this theory is quite new,
the reader may well wonder why Thomason's name is attached to it.
Thomason proved the special case where ff = fi = @0, and T is the de-
rived category of the category of coherent sheaves on a quasi-compact,
separated scheme. (Actually, he studies the slightly more general sit-
uation of a semi-separated scheme. A scheme is semi-separated if it
has an open cover by affine open subsets with affine intersections.) For
details, the reader is referred to Thomason's [34 ]. In all fairness to
Thomason, his wonderful observation was that this fact had great ap-
plications in K-theory. In any case, what is really new here is the
generalisation to arbitrary regular cardinals ff.
This raises, of course, the question of why one cares. Thomason
proved the theorem where ff = fi = @0, and T is the derived category of
the category of coherent sheaves on a quasi-compact, separated scheme.
The author gave a simpler proof, which also generalised the result to
1. INTRODUCTION 19
all T, as long as ff = fi = @0. This may be found in [23 ]. The obvious
question is: who cares about the case of large regular cardinals?
The short answer is that everybody should. First of all, it has
already been mentioned that in the applications to motives, the case
ff = fi = @0 does not apply. Only rarely is there a set of objects
T T@0, with = T. But even the people with both their feet firmly
on the ground, the ones who could not care less about motives, should
be interested in the case of large cardinals.
The reason is the following. If T is the homotopy category of spectra,
it has been known for a long time that T@0 generates T. But now let E be
a homology theory. Following Bousfield, let TE T be the subcategory
of E-acyclic spectra, and let ?TE be the subcategory of E-local spectra.
@0
In general, {TE }@0 and ? TE are small and very uninteresting. It
ff
is only for sufficiently large ff that the categories {TE }ffand ? TE
start generating. See Remark D.1.15, for an estimate on how large
ff must be. The moral is very simple. Suppose the main object of
interest is a triangulated category to which Thomason's theorem, or
my old generalisation of it, apply. That is, the main object of study is
a category for which the case ff = fi = @0 is non-trivial. As soon as
we Bousfield localise it, we get a category for which we are naturally
forced into the large cardinal ff generalisation.
So far we have seen that, for each regular cardinal ff, it is possible
to attach to T a canonically defined ff-localising subcategory Tff. We
have also seen Thomason's localisation theorem, which says that the
subcategories Tffbehave well with respect to Verdier quotients. But
to convince the reader that the exercise is worthwhile, I must use the
subcategories Tff T to prove a statement not directly involving them.
First we need a key definition.
Definition 1.15. Let ff be a regular cardinal. Let T be a triangu-
lated category with small Hom-sets satisfying [TR5]. If the subcategory
Tffis essentially small, and if = T, we say that T is ff-compactly
generated. It turns out that if T is ff-compactly generated, then it is
also fi-compactly generated for any fi > ff. A triangulated category T
is said to be well generated if
1.15.1. T has small Hom-sets.
1.15.2. T satisfies [TR5].
1.15.3. For some regular ff, T is ff-compactly generated.
Of course, a well generated triangulated category is in fact fi-compactly
generated for all sufficiently large fi.
20 CONTENTS
Now we begin the main theorems of the article.
Theorem 1.16. (Brown representability). Let T be any well-
generated triangulated category. Let H be a contravariant functor H :
Top -! Ab, where Ab is the category of abelian groups. The functor H
is representable if and only if it is homological, and takes coproducts in
T to products of abelian groups. In other words, the representable func-
tors T(-; t) can be characterised as the cohomological functors taking
coproducts in T to products in Ab.
Proof: Theorem 8.3.3. 2
This theorem has several immediate corollaries. One of them is
Corollary 1.17. Let T be any well-generated triangulated cate-
gory. Then T satisfies [TR5*]; all small products exist in T.
Proof: Given a set {X ; 2 } of objects in T, the following functor
Top -! Ab
Y
H(-) = T(-; X )
2
is homological, and takes coproducts in T to products of abelian groups.
By Theorem 1.16 it is representable. The representing object is nothing
other than the product, in T, of {X ; 2 }. See also Proposition 8.4.6.
2
In other words, we learn that a well-generated triangulated cate-
gory is closed under products. This leads naturally to the question of
whether the dual of Brown's representability theorem holds. We prove
Theorem 1.18. (Brown representability for the dual).
Let T be any well-generated triangulated category. Choose some regular
cardinal ff, for which Tiis ff-compactlyjgenerated. Suppose for that ff,
the abelian category Ex {Tff}op; Ab has enough injectives.
Let H be a covariant functor H : T - ! Ab. Then H will be rep-
resentable if and only if it is homological, and takes products in T to
products of abelian groups. In other words, the representable functors
T(t; -) can be characterised as the homological functors respecting prod-
ucts.
We can formalise this
Definition 1.19. A triangulated category T satisfying [TR5] is
said to satisfy the representability theorem if the (contravariant) rep-
resentable functors T(-; t) are precisely the homological functors H :
Top -! Ab taking coproducts to products.
1. INTRODUCTION 21
The content of Theorems 1.16 and 1.18 is that well-generated cate-
goriesiT satisfyjthe representability theorem, as do their duals, provided
Ex {Tff}op; Ab has enough injectives. An easy proposition states
Proposition 1.20. Let F : S - ! T be a triangulated functor of
triangulated categories. Suppose S satisfies the representability theorem,
as in Definition 1.19. The functor F has a right adjoint G : T -! S if
and only if F respects coproducts.
Proof: Theorem 8.4.4. 2
Remark 1.21. In Remark 1.4 we noted that the deep questions
about triangulated categories involve the existence of adjoints. Let
F : S -! T be a triangulated functor. By Corollary 1.3, F has a right
adjoint if and only if A(F ) : A(S) -! A(T) does. But this amounts to
reducing a difficult problem to an impossible one. The categories A(S)
are terrible, and the author does not know a single example where one
can show directly that A(F ) : A(S) -! A(T) has an adjoint.
By contrast, Proposition 1.20 is practical to apply. If S or Sop is
well-generated, F will have a right adjoint if and only if it preserves
coproducts.
Remark 1.22. It should be noted that Franke has independently
obtained a representability theorem strongly reminiscent of Theorem 1.16.
Franke's theorem also assumes that T can be written as a union of T ff
satisfying suitable hypotheses. But this is where the similarity becomes
confusing. It is not clear whether the Tff's studied here in general sat-
isfy the hypotheses placed on Franke's T ff's. It also is not clear whether
there could be some other choice for Franke's T ff's. In his application,
to the derived category of a Grothendieck abelian category, Franke's
T ffis just the Tffwe have been studying here. See Franke's [11 ].
Franke's method does not generalise to the dual of a well-generated
abelian category.
So far in the Introduction, we have presented the main results of the
book. We ordered them in a way that motivated the definitions. We
began with Freyd's construction of the universal homological functor
T -! A(T). We discussed its properties, and the fact that, in general,
A(T) is terrible. Then we spoke about approximations to A(T), in par-
ticular approximations of the form Ex Sop; Ab , for an ff-localising sub-
category S T. We reasoned that, for every regular cardinal ff, there
is a canonical best choice for S; the largest possible S is Tff. Thoma-
son's localisation theorem is the statement that Tffbehaves reasonably
well with respect to Verdier localisations. Our main theorems give an
22 CONTENTS
application of the Tff's. The first major theorem asserts that the rep-
resentability theorem holds for T whenever Tffis essentially small and
generates T. The second asserts that the representability theoremiholdsj
for Top if T is ff-compactly generated, and furthermore Ex {Tff}op; Ab
has enough injectives.
Now it is time to explain the way the exposition of these facts is
organised in the book, and to discuss some of the less major theorems
that we prove on the way, or as consequences. The order in which the
results are presented in the book is the Bourbaki order. It is the logical
order, not the order that would motivate the constructions. Chapters 1
and 2 give the elementary properties of triangulated categories. Chap-
ters 3 and 4 give the definitions of the categories Tff, and their formal
properties. This culminates in Thomason's localisation theorem, which
asserts Tffpasses to Verdier quotients. This is quite unmotivated. We
define the categories Tff, and study their formal properties, before we
have any indication that they might be of some use.
Only in Chapter 5 do we treat Freyd's classical theorem, con-
cerning the universal homologicalifunctor. InjChapter 6 we finally
come around to the categories Ex {Tff}op; Ab . We develop the ele-
mentary properties of the categories Ex Sop; Ab , and of the functor
A(T) -! Ex Sop; Ab . Chapter 6 should help clarify, somewhat belat-
edly, the point of studying the categories Tff.
In Theorem 1.18, we saw that Brown representability sometimes
holds for the dual of T; in particular, it holds if Ex Sop; Ab has enough
injectives. It becomes interesting to study whether there are enough
injectives.
The general answer is no. Counterexamples may be found in Sec-
tions C.4 and D.2. Nevertheless, it is possible that Brown representabil-
ity for the dual could be proved with less than the existence of injec-
tives. The homological algebra of the categories Ex Sop; Ab is inter-
esting, and its careful study might yield great results. In Chapter 7, I
assemble assorted facts I know. These do not really lead anywhere yet,
but I thought they might be useful to future researchers. The Chapter
may safely be skipped by all but the truly committed.
The category Ex Sop; Ab does not satisfy [AB5]. We remind the
reader: this means direct limits of exact sequences need not be exact.
It follows that Ex Sop; Ab is not a Grothendieck abelian category, and
hence the classical proofs of the existence of injectives break down.
Of course, the proofs must break down, since we know that there are
not, in general, enough injectives. But it is interesting to analyse just
where the breakdown occurs. We will analyse this for the argument
1. INTRODUCTION 23
that appears in Grothendieck's T^ohoku paper; see Theoreme 1.10.1, on
page 135 of [18 ]. I do not know the origin of the argument; Grothendieck
said that it had been well-known, and he was merely sketching it. The
argument is based on adding cells. We should perhaps remind the
reader.
Let A be an abelian category, x an object of A. We wish to embed
x in an injective I. This means that given any extension in Ext 1(z; x),
that is any exact sequence
0 -- - ! x --- ! y --- ! z -- - ! 0;
the map x -! I should kill it. In other words, the map x -! I should
factor as
x --- ! y --- ! I:
This suggests a natural way to try to construct I. If x is not injective,
it has an extension
0 -- - ! x --- ! y --- ! z -- - ! 0:
If y is not injective, we can repeat the process. We can construct a
sequence of monomorphisms
x = x0 -- - ! x1 --- ! x2 --- ! . . .
and hope that the colimit of the xi will be injective. This is the process
of adding cells, and the proof in [18 ] was based on a slightly refined
version of this construction.
The first and most serious problem with this construction is that,
for an abelian category not satisfying [AB5], it is not clear that x injects
into the colim xi. It might well be that colim xi = 0. We say that an
-! -!
abelian category satisfies [AB4:5] if, for any (transfinite) sequence of
monomorphisms as above, the map x0 -! colim xi is injective.
-!
Actually, for the purpose of the proofs given in the article, it is
convenient to give an equivalent statement in terms of the derived
functors of the colimit. An abelian category satisfies [AB4:5] if, for any
(transfinite) sequence of monomorphisms as above, and for any n 1,
the nth derived functor of the colimit vanishes. That is,
colimn xi = 0:
-!
We do not prove the equivalence of the two statements; we use the
second as a definition, and we use the fact that it implies the first.
The converse is true, but of no importance to us. One can easily
show (Lemma A.3.15) that if an abelian category satisfies [AB3] (has
coproducts) and has enough injectives, then it satisfies [AB4:5].
24 CONTENTS
It is instructive to know that, if S is sufficiently ridiculous, the cat-
egory Ex Sop; Ab need not satisfy [AB4:5]. For this reason, in Chap-
ter 6 we begin by defining Ex Sop; Ab for fairly arbitrary S. For the
S's for which we define it, Ex Sop; Ab always satisfies [AB4]; coprod-
ucts are exact. See Lemma 6.3.2. However, if S is the category of
normed, non-archimedean, complete topological abelian groups, then
Ex Sop; Ab does not satisfy [AB4:5]. See Proposition A.5.12. But in
this book, we are mostly interested in the case where the category S is
triangulated. I have no example of a triangulated category S, for which
I can show that Ex Sop; Ab does not satisfy [AB4:5]. For a while, I
thought I could prove [AB4:5] for such Ex Sop; Ab . But there is a gap.
Included in Chapter 7, is the part of the argument that is correct.
The study of derived functors of co-Mittag-Leffler sequences in
abelian categories is of some independent interest, and the existence of
a Ex Sop; Ab satisfying [AB4] but not [AB4:5] is new and surprising.
The reader is referred to Proposition 1 in [29 ], or Lemma 1.15 on
page 213 of [20 ], to see just how striking it is. Since the results are
about abelian, as opposed to triangulated, categories, they have been
put in an appendix; see Appendix A.
The property [AB4:5] is extensively studied, for the abelian cate-
gories Ex Sop; Ab , in Chapter 7 and Appendix A. The study is incon-
clusive, but might be helpful to others. This occupies most of Chap-
ter 7. But the final section, Section 7.5, is quite unrelated.
Let S be a triangulated category closed under coproducts of < ff of
its objects. The category Ex Sop; Ab is the category of functors Sop -!
Ab, which respect products of fewer than ff objects. It is contained in
the category Cat Sop; Ab , of all additive functors Sop -! Ab. Let i be
the inclusion
i : Ex Sop; Ab-- - ! Cat Sop; Ab :
In Section 7.5, we prove that i has a left adjoint j. So far, this is
a special case of a theorem of Gabriel and Ulmer [16 ]. But more
interestingly, the functor j has left derived functors Lnj. And most
remarkably, if F is an object of Ex Sop; Ab , then iF is an object of
Cat Sop; Ab , and we prove that, for n 1,
Lnj{iF } = 0:
An easy consequence is that, given objects F and G in the abelian
category Ex Sop; Ab , the groups Extn(F; G) agree, whether we compute
them in Ex Sop; Ab or in the larger Cat Sop; Ab .
Chapter 8 has the proof of the two Brown representability theorems.
It shows how the earlier theory can be used to prove practical theorems.
1. INTRODUCTION 25
It is now time to give fairly precise statements of what we prove, and
explain the consequences.
In Theorems 1.16 and 1.18, we saw that if T is well-generated, then
the representability theorem holds for T, and also for Top, as long as
some abelian category has enough injectives. See Definition 1.19 for
what it means for a triangulated category T to satisfy the representabil-
ity theorem. These are true statements, but we prove more. Now it is
time to be precise about what we prove.
To say that T is well-generated asserts that, for some regular cardi-
nal ff, S = Tffis essentially small, and the natural homological functor
T --- ! Ex Sop; Ab
does not annihilate any object. A more precise statement of the theo-
rem we prove would be
Theorem 1.23. Suppose T is a triangulated category satisfying [TR5].
Suppose ff is a regular cardinal, and suppose S T is an ff-localising
subcategory. Suppose S is essentially small, and suppose the natural
homological functor
T --- ! Ex Sop; Ab
does not annihilate any object, and respects countable coproducts. Then
T satisfies the representability theorem.
Note that we do not assume, in the statement of the theorem, that
S = Tff. If S = Tff, the conditions placed on S amount to saying that
T is ff-compactly generated. For S = Tff, Theorem 1.23 reduces to
Theorem 1.16. The fact that we allow other S's is a generalisation.
Now let us analyse this.
If we assume that the map
T --- ! Ex Sop; Ab
preserves all coproducts, then the generalisation is in fact very minor.
Let us explain why. The point is that if
T --- ! Ex Sop; Ab
respects all coproducts, then S Tff. Recall that Tffis the largest of
all the S's for which coproducts are preserved. It contains all others,
in particular it contains S. It turns out that if ff @1, S Tff is
ff-localising, and the map
T --- ! Ex Sop; Ab
does not annihilate any object, then S = Tff. We know this because in
Theorem 8.3.3 we prove T = ~~~~, and Thomason's localisation theorem
26 CONTENTS
(Theorem 4.4.9) then tells us that Tff = ~~~~ff. But ~~~~ff= S, since
S Tffis ff-localising. If ff = @0 the statement is more delicate, and
what we said is true only up to splitting idempotents. Anyway, the
point is that, up to splitting idempotents when ff = @0, S = Tffis the
only choice. The only practical value of the seemingly more general
statement about arbitrary S's, is that it gives us a way to show that
some S is, in fact, equal to Tff.
But the fact that the functor
T --- ! Ex Sop; Ab
need only respect countable coproducts seems a genuine relaxation of
the hypothesis that T be well-generated. I use the word "seems" be-
cause I know of no example. I know no non-well-generated category to
which this applies. Still, we could define a triangulated category T sat-
isfying [TR5] to be @1-perfectly generated, if there exists an essentially
small @1-localising subcategory S T, so that the functor
T --- ! Ex Sop; Ab
does not annihilate any object, and preserves countable coproducts.
Theorem 1.23 applies, and we would deduce that T satisfies the repre-
sentability theorem. For a while I had hopes that maybe the dual of a
well-generated triangulated category would be @1-perfectly generated.
We will see in Section E.2 that the dual of D(Q) is not @1-perfectly
generated. I would like to thank Shelah for pointing out the cardinality
argument at the heart of Section E.2.
So we now know the precise statement of Theorem 8.3.3, which
asserts that @1-perfectly generated triangulated categories satisfy the
representability theorem. The statement for the dual, that is Theo-
rem 8.6.1, is precisely as we quotediit in Theoremj1.18. That is, if T
is ff-compactly generated and Ex {Tff}op; Ab has enough injectives,
then the representability theorem holds for Top. In other words, here
we do not get away with countable coproducts. We need to assume the
functor
T --- ! Ex Sop; Ab
preserves all coproducts.
Now we have made precise the two representability theorems we
prove. It is time to briefly review the applications. We have already
mentioned the application to adjoints. See Proposition 1.20; if S is an
ff-compactly generated triangulated category, and if Ex Sop; Ab has
enough injectives, then a triangulated functor S -! T has a left (respec-
tively right) adjoint if and only if it preserves products (respectively
1. INTRODUCTION 27
coproducts). Next we want to discuss what follows, if the category
Ex Sop; Ab has enough injectives.
First we should remind the reader. Let T be a triangulated cate-
gory satisfying [TR5]. When S is an ff-localising subcategory of T, the
homological functor
T --- ! Ex Sop; Ab
factors uniquely through Freyd's universal homological functor. It fac-
tors
T -- - ! A(T) -- ss-!Ex Sop; Ab
where ss is exact. The functor ss respects products and has a left adjoint
F
Ex Sop; Ab -- - ! A(T);
by Proposition 1.5. Also by Proposition 1.5, the unit of adjunction
" : 1--- ! ssF
is an isomorphism if and only if ss respects all coproducts. The inter-
esting case of this is S = Tff.
In particular, for S = Tff, the map ss does respect coproducts, the
unit of adjunction is an isomorphism, and by Gabriel's theory of local-
isation (in abelian categories), Ex Sop; Ab is a quotient of A(T) by the
class of objects annihilated by ss. See Section A.2 for a summary of
Gabriel's pertinent results.
So much is completely general. But we prove more. If S = Tffis
essentially small, if Ex Sop; Ab has enough injectives, and if T satisfies
the representability theorem, then the functor
A(T) --ss-!Ex Sop; Ab
also has a right adjoint
G
Ex Sop; Ab -- - ! A(T):
This is proved in Corollary 8.5.3. And it means the following. We
already knew that Ex Sop; Ab is a Gabriel quotient of A(T) by the
kernel of ss. We knew quite generally that there is a left adjoint to
ss. But under the hypotheses given above, which hold, for example, if
ff = @0, the functor ss also has a right adjoint G. The quotient is a
localizant-colocalizant one.
The existence of enough injectives in Ex Sop; Ab implies that ss :
A(T) - ! Ex Sop; Ab has a right adjoint G. Lemma 8.5.5 establishes
28 CONTENTS
that the existence of the right adjoint G implies that the category
Ex Sop; Ab has a cogenerator. We have implications
8 9 8 9
< Ex Sop; Ab = ae A(T) -! Ex Sop; Ab oe < Ex Sop; Ab =
has enough =) =) has a
: injectives ; has a right adjoint : cogenerator ;
And in the counterexamples of Sections C.4 and D.2, we see that in
general the category Ex Sop; Ab need not have a cogenerator. Thus the
right adjoint G need not exist, and Ex Sop; Ab may fail to have enough
injectives.
Injective objects and right adjoints are abstract and perhaps un-
inviting. It is therefore illuminating to rephrase everything in terms of
phantom maps.
A morphism f : x - ! y in T is called ff-phantom if its image
vanishes in Ex Sop; Ab . That is,
T (-; f)|S: T (-; x)|S--- ! T (-; y)|S
is the zero map. In Lemma 8.5.20, we prove that the right adjoint to
A(T) -! Ex Sop; Ab exists if and only if, for every object z 2 T, there
is a maximal ff-phantom map y - ! z. That is, every ff-phantom
x -! z must factor, non-uniquely, as
x --- ! y --- ! z:
In Lemma 8.5.17, we see that the category Ex Sop; Ab will have enough
injectives if and only if, for every object z 2 T, the maximal ff-phantom
map y -! z may be so chosen that, in the triangle
y --- ! z --- ! t --- ! y
the object t is orthogonal to the ff-phantom maps. Every ff-phantom
map x -! t vanishes.
If the category Ex Sop; Ab has enough injectives, there is a right
adjoint G to the functor ss : A(T) -! Ex Sop; Ab . Let I be an injective
cogenerator of Ex Sop; Ab . We may form the object GI. Since G has
an exact left adjoint ss, GI must be injective in A(T). That is, GI is
really an object in T A(T); the injective objects are direct summands
of objects in T, and as T satisfies [TR5], idempotents split in T. See
Proposition 1.6.8.
We call the object GI a Brown-Comenetz object, and denote it BC .
The Brown-Comenetz objects are somehow crucial to our proof that
the dual of T satisfies the representability theorem. Since Ex Sop; Ab
need not have enough injectives, it would be nice to have another
1. INTRODUCTION 29
proof, which does not so critically hinge on the existence of injectives
in Ex Sop; Ab .
The last chapter before the appendices is Chapter 9. It discusses
Bousfield localisation. It is relatively short, and exposes no new results.
Let me briefly tell the reader the contents of the Chapter.
Let T be a triangulated category, S T a triangulated subcategory.
We say that a Bousfield localisation functor exists for the pair S T
if the map T -! T=S has a right adjoint. Chapter 9 explores in some
detail what happens. It turns out that the adjoint
T=S --i-! T
is always fully faithful, which allows us to think of T=S as a subcategory
of T. The Verdier quotient
_T__
T=S
is naturally isomorphic to S, and the embedding
_T_
T=S = S --- ! T
is left adjoint to the quotient map
T -- - ! _T_T=S= S:
There are some parallels with Gabriel's constructions for quotients of
abelian categories. Anyway, we hope the reader finds the exposition of
Chapter 9 amusing, even if the results are basically all known. In the
second volume, which Voevodsky and the author still promise to write,
there will hopefully be more about Bousfield localisation.
There are also some appendices. The appendices contain two types
of results. The first is background which the reader will need, and
which is assembled here for convenience. There are several facts we
want to use about abelian categories, which go beyond the elemen-
tary homological algebra that is a prerequisite for the book. These
results may be found elsewhere, scattered around the literature. But
Appendix A offers the reader a condensed summary. See Section A.1
for locally presentable categories, Section A.2 for Gabriel's treatment
of localisation in abelian categories, and Sections A.3 and A.4 for the
derived functors of limits.
The remaining material in the appendices is new, often of indepen-
dent interest. Generally, if a result has no strong, direct bearing on
the development of the theory, it is left to an appendix. For example,
Appendix A contains more than just a summary of known facts about
abelian categories. The treatment of Mittag-Leffler sequences is new,
as is [AB4:5]. And the example of the category S for which Ex Sop; Ab
30 CONTENTS
satisfies [AB4] but not [AB4:5] is not only new, it is quite surprising. It
goes against the expectations in the literature. Since it is only tange-
tially related to the main subject of the book, the reader will find it
consigned to Section A.5.
Appendix A is about abelian categories, with some of the material
being old, some new. The remaining appendices offer results about
triangulated categories. These also divide into two types.
iThe main resultjof Appendix B characterises the functor T -!
Ex {Tff}op; Ab by a universal property. An abelian category A is said
to satisfy [AB5ff] if ff-filtered colimits are exact in A. We prove
Theorem 1.24. Let T be an ff-compactly generated triangulated
category. The coproduct-preserving homological functors H : T -! A,
where A is an abelian category satisfying [AB5ff], factor uniquely, up
to canonical equivalence, as
i j 9!
T --- ! Ex {Tff}op; Ab --- ! A;
i j
with the functor Ex {Tff}op; Ab - ! A coproduct-preserving and exact.
Natural transformations of coproduct-preserving homological functors
T -! A are in 1-to-1 correspondence withinatural transformationsjof
coproduct-preserving exact functors Ex {Tff}op; Ab - ! A.
Proof: Theorem B.2.5. 2
This result is the type that perhaps merits inclusion in the body of
the book. But we do not really use it elsewhere. It gives more evidence
that the construction of the categories Tffis natural. But beyond this,
it does not have a strong bearing on the development of the theory. At
least, not yet. For this reason, it was put in an appendix.
I would like to thank Christensen, who kept asking me about such
results. Christensen and Strickland, in [9 ], proved the assertion when T
is the category of spectra, and ff = @0. But their methods do not seem
to generalise, even to other T but with ff still @0. If not for Christensen's
prodding, I would probably never have obtained the result.
Now for the remaining three appendices. These are basically ex-
amples. In Appendix D, we work out in some detail what the general
theory says in the special case, where T is the category of spectra. We
began the Introduction with this, hence we will not repeat it. There
are two remaining Appedices, C and E. These mostly are about patho-
logical behavior. The reader is expected to know a little bit about the
derived category to read these examples. The body of the book does
1. INTRODUCTION 31
not discuss examples, and does not depend on knowing any. But in the
appendices, we assume some acquaintance with the derived category.
Appendix C has two results. First it proves that, in general, the ob-
jects of Freyd's universal category A(T) have classes, not sets, of subob-
jects. Very concretely, we show it for the object Z 2 D(Z) A D(Z) .
This result is certainly known to the experts, and in fact seems to have
been independently rediscovered several times. The earliest reference I
could find is Freyd's [14 ].
The second result is that the category Ex Sop; Ab need not have a
cogenerator. As we have seen above, this also means that there need
not be a right adjoint to ss : A(T) -! Ex Sop; Ab , and that Ex Sop; Ab
need not have enough injectives.
Appendix E offers examples of categories which are not well-generated.
In Corollary E.1.3, we see that if T is @0-compactly generated, then Top
cannot be well-generated. For this result, the reader does not need to
know any examples. In Section E.3, more specifically Summary E.3.3,
we see that if K(Z) is the homotopy category of chain complexes of
abelian groups, then neither K(Z) nor K(Z) op is well-generated. Sec-
tion E.2 treats a condition possibly weaker than well-generation. It
shows that the dual of D(Q), the derived category of vector spaces
over the field Q, is not even @1-perfectly generated.
One thing should be noted. In Corollary E.1.3, we prove that the
dual of an @0-compactly generated T cannot be well-generated. If T is
the homotopy category of spectra, we deduce that Top cannot be well-
generated. Hence T and Top cannot be equivalent. The assertion that
the homotopy category is not self-dual is an old theorem of Boardman,
[2 ]. What we have here is a generalisation. It is not the best generali-
sation one can prove, but in the interest of simplicity, it is the one we
give.
32 CONTENTS
CHAPTER 1
Definition and elementary properties of
triangulated categories
1.1. Pre-triangulated categories
Definition 1.1.1. Let C be an additive category and : C ! C be
an additive endofunctor of C. Assume throughout that the endofunctor
is invertible. A candidate triangle in C (with respect to ) is a
diagram of the form:
X --u-! Y --v-! Z --w-! X
such that the composites v Ou, w Ov and uOw are the zero morphisms.
A morphism of candidate triangles is a commutative diagram
X -- u-! Y --v-! Z --w- ! X
? ? ? ?
f?y g?y h?y f ?y
X0 -- u-! Y 0 --v-! Z0 --w- ! X0
where each row is a candidate triangle.
Definition 1.1.2. A pre-triangulated category T is an additive
category, together with an additive automorphism , and a class of
candidate triangles (with respect to ) called distinguished triangles.
The following conditions must hold:
TR0: Any candidate triangle which is isomorphic to a distinguished
triangle is a distinguished triangle. The candidate triangle
X -- 1-! X -- - ! 0 -- - ! X
is distinguished.
TR1: For any morphism f : X ! Y in T there exists a distin-
guished triangle of the form
f
X --- ! Y --- ! Z --- ! X
TR2: Consider the two candidate triangles
X --u-! Y --v-! Z --w-! X
33
34 1. ELEMENTARY PROPERTIES
and
Y ---v-! Z --w--! X ---u-! Y:
If one is a distinguished triangle, then so is the other.
TR3: For any commutative diagram of the form
X -- u-! Y --v-! Z --w- ! X
? ? ?
f?y g?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
where the rows are distinguished triangles, there is a morphism
h : Z ! Z0, not necessarily unique, which makes this diagram
commutative.
Remark 1.1.3. Parts of Definition 1.1.2 are known to be redun-
dant. For instance, it is not necessary to assume that distinguished
triangles are candidate triangles. In other words, we can assume that
the distinguished triangles are sequences
X --u-! Y --v-! Z --w-! X
without necessarily postulating that the composites v O u, w O v and
u O w vanish. It follows from the other axioms that the composites
v O u, w O v and u O w must be zero. Just consider the diagram
X --1-! X --- ! 0 --- ! X
? ? ?
1?y u?y 1?y
X --u-! Y --v-! Z --w-! X0
The bottom row is a distinguished triangle by hypothesis, the top by
[TR0]. But by [TR3] the diagram may be completed to a commutative
X --1-! X --- ! 0 --- ! X
? ? ? ?
1?y u?y ?y 1?y
X --u-! Y --v-! Z --w-! X0
and we deduce that v Ou = 0. The vanishing of w Ov and uOw follows
from the above and axiom [TR2].
Similarly, it is not necessary to assume that the category T is ad-
ditive; something slightly less suffices. It suffices to assume that the
category T is pointed (there is a zero object), and that the Hom sets
are abelian groups. The fact that finite coproducts and products exist
and agree follows from the other axioms.
1.1. PRE-TRIANGULATED CATEGORIES 35
Notation 1.1.4. Let T be a pre-triangulated category. If we speak
of "triangles" in T, we mean distinguished triangles. When we mean
candidate triangles, the adjective will always be explicitly used.
Remark 1.1.5. It is clear from the definition that if T is a pre-
triangulated category, then so is its dual Top. For Top, the functor
gets replaced by -1.
Proposition 1.1.6. Let T be a pre-triangulated category. Then
the functor preserves products and coproducts. Let us state this pre-
cisely. Suppose {X ;` 2 } is a set of objects of T, and suppose the
categorical product 2 X exists in T. Then the natural map
( )
a a
{X } -! X
2 2
is an isomorphism. In other words, the natural maps
( )
a
X -! X
2
`
give 2 Q X the structure of a coproduct in the category T. Sim-
ilarly, if 2 X exists in T, then the natural map
( )
Y
X -! X
2
Q
give 2 X the structure of a product in the category T.
Proof: The point is that , being invertible, has both a right and a
left adjoint, namely -1. There are natural isomorphisms
-1
Hom (X; Y ) ' Hom X; Y
and
-1
Hom (X; Y ) ' Hom X; Y ;
induced by -1. A functor possessing a left adjoint respects products, a
functor possessing a right adjoint respects coproducts. Thus respects
both. 2
Because Top is a pre-triangulated category with -1 playing the
role of , it follows that -1 also respects products and coproducts.
36 1. ELEMENTARY PROPERTIES
Definition 1.1.7. Let T be a pre-triangulated category. Let H be
a functor from T to some abelian category A. The functor H is called
homological if, for every (distinguished) triangle
X --u-! Y --v-! Z --w-! X
the sequence
H(u) H(v)
H(X) --- ! H(Y ) --- ! H(Z)
is exact in the abelian category A.
Remark 1.1.8. Because of axiom [TR2], it follows that the se-
quence above can be continued indefinitely in both directions. In other
words, the infinite sequence
H(-1w) H(u) H(v) H(w)
H(-1Z) --- - - !H(X) -- - ! H(Y ) --- ! H(Z) -- - ! H(X)
is exact everywhere.
Remark 1.1.9. A homological functor on the pre-triangulated cat-
egory Top is called a cohomological functor on T. Thus, a cohomological
functor is a contravariant functor H : T ! A such that, for any triangle
X --u-! Y --v-! Z --w-! X
the sequence
H(v) H(u)
H(Z) --- ! H(Y ) -- - ! H(X)
is exact in the abelian category A.
Lemma 1.1.10. Let T be a pre-triangulated category, U be an ob-
ject of T. Then the representable functor Hom(U; -) is homological.
Proof. Suppose we are given a triangle
X --u-! Y --v-! Z -- w-! X:
We need to show the exactness of the sequence
Hom(U; X) -- - ! Hom(U; Y ) -- - ! Hom(U; Z)
We know in any case that the composite is zero. Let f 2 Hom(U; Y )
map to zero in Hom(U; Z). That is, let f : U ! Y be such that the
composite
f v
U --- ! Y --- ! Z
is zero. Then we have a commutative diagram
U --- ! 0 --- ! U ---1-! U
? ? ?
f?y ?y f ?y
Y ---v-! Z --w--! X ---u-! Y:
1.1. PRE-TRIANGULATED CATEGORIES 37
The bottom row is a triangle by [TR2], the top row by [TR0] and
[TR2]. There is therefore, by [TR3], a map h : U ! X such that
h : U ! X makes the diagram above commute. But this means
in particular that the square
U ---1-! U
? ?
h ?y f ?y
X --u--! Y
commutes, and hence f = u O h. Thus, we have produced an h 2
Hom(U; X) mapping to f 2 Hom(U; Y ). __
|__|
Remark 1.1.11. Recall that the dual of a pre-triangulated cate-
gory is pre-triangulated. It follows from Lemma 1.1.10 applied to the
dual of T that the functor Hom(-; U) is cohomological.
Definition 1.1.12. Let H : T ! A be a homological functor. The
functor H is called decent if
1.1.12.1. The abelian category A satisfies AB4*; that is, the
product of exact sequences is exact.
1.1.12.2. The functor H respects products. For any collec-
tion {X ; 2 } of objects X 2 T whose product exists in T,
the natural map
!
Y Y
H X -! H(X )
2 2
is an isomorphism.
Example 1.1.13. The functor Hom(U; -) : T ! Ab is a decent
homological functor. It is homological by Lemma 1.1.10, the abelian
category Ab of all abelian groups satisfies AB4*, and Hom(U; -) pre-
serves products.
Definition 1.1.14. Let T be a pre-triangulated category. A can-
didate triangle
X --u-! Y --v-! Z --w-! X
is called a pre-triangle if, for every decent homological functor H : T !
A, the long sequence
H(-1w) H(u) H(v) H(w)
H(-1Z) --- - - !H(X) -- - ! H(Y ) --- ! H(Z) --- ! H(X)
is exact.
38 1. ELEMENTARY PROPERTIES
Example 1.1.15. Every (distinguished) triangle is a pre-triangle.
A direct summand of a pre-triangle is a pre-triangle. The next little
lemma will show that an arbitrary product of pre-triangles is a pre-
triangle.
Caution 1.1.16. There are pre-triangles which are not distin-
guished. See for example the discussion of Case 2, pages 232-234 of
[22 ]. An example of a pre-triangle which is not a triangle is the map-
ping cone on the map of triangles in the middle of page 234, loc. cit.
Lemma 1.1.17. Let be an index set, and suppose that for every
2 we are given a pre-triangle
X --- ! Y -- - ! Z --- ! X :
Suppose further that the three products
Y Y Y
X ; Y ; Z
2 2 2
exist in T. The sequence
Y Y Y Y
X --- ! Y --- ! Z --- ! {X }
2 2 2 2
is identified, using Proposition 1.1.6, with
( )
Y Y Y Y
X -- - ! Y -- - ! Z --- ! X :
2 2 2 2
We assert that this candidate triangle is a pre-triangle. Thus, the
product of pre-triangles is a pre-triangle.
Proof: Let H : T ! A be a decent homological functor. Because for
each 2 the sequence
X -- - ! Y --- ! Z -- - ! X
is a pre-triangle, applying H we get a long exact sequence in A
H (-1Z ) --- ! H(X ) -- - ! H(Y ) --- ! H(Z ) --- ! H(X )
and because A is assumed to satisfy AB4*, the product of these se-
quences is exact. But now we are assuming H decent, and in particular
1.1. PRE-TRIANGULATED CATEGORIES 39
by 1.1.12.2 the maps
!
Y Y
H X --- ! H (X )
2 ! 2
Y Y
H Y --- ! H (Y )
2 ! 2
Y Y
H Z --- ! H (Z )
2 2
are all isomorphisms. This means that the functor H, applied to the
sequence
( )
Y Y Y Y
X --- ! Y --- ! Z --- ! X
2 2 2 2
gives a long exact sequence. This being true for all decent H, we deduce
that the sequence is a pre-triangle. 2
Lemma 1.1.18. Let H be a decent homological functor T ! A. Let
the diagram
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
be a morphism of pre-triangles. Suppose that for every n 2 Z, H(nf)
and H(ng) is an isomorphism. Then H(nh) are all isomorphisms.
Proof. Without loss, we are reduced to proving H(h) an isomor-
phism. But then the diagram
H(u) H(v) H(w) H(u)
H(X) -- - ! H(Y ) -- - ! H(Z) --- ! H(X) -- - ! H(Y )
? ? ? ? ?
H(f)?y H(g)?y H(h)?y H(f )?y H(g )?y
H(u0) 0 H(v0) 0 H(w0) 0 H(u0) 0
H(X0) -- - ! H(Y ) -- - ! H(Z ) --- ! H(X ) --- - ! H(Y )
is a commutative diagram in the abelian category A with exact rows._
By the 5-lemma, we deduce that H(h) is an isomorphism. |__|
40 1. ELEMENTARY PROPERTIES
Lemma 1.1.19. In the morphism of pre-triangles
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
if f and g are isomorphisms, then for any decent homological functor
H, H(h) is an isomorphism.
Proof. If f and g are isomorphisms, so are nf and ng for any n.
Hence Lemma 1.1.18 allows us to deduce that H(h) is an isomorphism. __
|__|
Proposition 1.1.20. If in the morphism of pre-triangles
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
both f and g are isomorphisms, then so is h.
Proof. By Lemma 1.1.19 we already know that for any decent
homological functor H : T ! A, H(h) is an isomorphism. By Exam-
ple 1.1.13, all representable functors Hom(U; -) are decent, for U 2 T.
We know therefore that the natural map
Hom(U; h) : Hom(U; Z) ! Hom(U; Z0)
is an isomorphism for every U. But then the map
Hom(-; h) : Hom(-; Z) ! Hom(-; Z0)
is an isomorphism. It follows from Yoneda's Lemma that h is an iso-__
morphism. |__|
Remark 1.1.21. Let u : X ! Y be given. By [TR1] it may be
completed to a triangle. Let
X --u-! Y --v-! Z --w-! X
and
0 w0
X --u-! Y --v-! Z0 --- ! X
be two distinguished triangles "completing" u. We have a diagram
X --u-! Y --v-! Z --w-! X
? ? ?
1?y 1?y 1?y
0 w0
X --u-! Y --v-! Z0 --- ! X
1.2. COROLLARIES OF PROPOSITION 1.1.20 41
which by [TR3] may be completed to a morphism of triangles
X --u-! Y --v-! Z --w-! X
? ? ? ?
1?y 1?y h?y 1?y
0 w0
X --u-! Y --v-! Z0 --- ! X
and since 1 : X ! X and 1 : Y ! Y are clearly isomorphisms,
Proposition 1.1.20 says that h is an isomorphism. It follows that Z
is well defined up to isomorphism. In fact, the entire triangle is well
defined up to isomorphism. But this isomorphism is not in general
canonical.
1.2. Corollaries of Proposition 1.1.20
In this section, we will group together some corollaries of Propo-
sition 1.1.20, which have in common that they concern products and
coproducts.
Proposition 1.2.1. Let T be a pre-triangulated category and
any index set. Suppose for every 2 we are given a (distinguished)
triangle
X -- - ! Y --- ! Z -- - ! X
in T. Suppose the three products
Y Y Y
X ; Y ; Z
2 2 2
exist in T. We know by Lemma 1.1.17 that the product is a pre-triangle
( )
Y Y Y Y
X --- ! Y --- ! Z --- ! X
2 2 2 2
We assert that it is a distinguished triangle.
Proof. By [TR1], the map
Y Y
X --- ! Y
2 2
can be completed to a triangle
( )
Y Y Y
X --- ! Y --- ! Q -- - ! X :
2 2 2
42 1. ELEMENTARY PROPERTIES
For each 2 , we get a diagram where the rows are triangles
( )
Y Y Y
X -- - ! Y -- - ! Q --- ! X
2 ? 2 ? 2 ?
? ? ?
y y y
X -- - ! Y -- - ! Z --- ! X
By [TR3] we may complete this to a morphism of triangles
( )
Y Y Y
X -- - ! Y -- - ! Q --- ! X
2 ? 2 ? ? 2 ?
? ? ? ?
y y y y
X -- - ! Y -- - ! Z --- ! X
Taking the product of all these maps, we get a morphism
( )
Y Y Y
X --- ! Y --- ! Q --- ! X
2 ? 2 ? ? 2 ?
1?y 1?y ?y 1?y
( )
Y Y Y Y
X --- ! Y --- ! Z --- ! X
2 2 2 2
Both rows are pre-triangles. The top row because it is a triangle, the
bottom row by Lemma 1.1.17. It follows from Proposition 1.1.20 that
this map is an isomorphism of the top row (a distinguished triangle) __
with the bottom, which is therefore a triangle. |__|
Remark 1.2.2. Dually, the coproduct of distinguished triangles is
distinguished.
Proposition 1.2.3. Let T be a pre-triangulated category. Let
X --- ! Y --- ! Z --- ! X
X0 -- - ! Y 0 --- ! Z0 -- - ! X0
be candidate triangles. Suppose the direct sum
X X0 --- ! Y Y 0 --- ! Z Z0 --- ! X X0
is a distinguished triangle. Then so are the summands.
1.2. COROLLARIES OF PROPOSITION 1.1.20 43
Proof. The situation being symmetric, it suffices to prove that
X --- ! Y --- ! Z --- ! X
is a triangle. Since it is the direct summand of a pre-triangle, it is in
any case a pre-triangle, by Example 1.1.15. Let
X --- ! Y --- ! Q --- ! X
be a distinguished triangle. The diagram
0 1X --- ! Y --- ! Q --- ! X
? 0 1 ? 0 1?
@ 1 A?y @ 1 A ?y @ 1 A?y
0 0 0
X X0 --- ! Y Y 0 --- ! Z Z0 --- ! X X0
may be completed to a morphism of triangles
0 1X --- ! Y --- ! Q --- ! X
? 0 1 ? ? 0 1?
@ 1 A?y @ 1 A ?y ?y @ 1 A?y
0 0 0
X X0 --- ! Y Y 0 --- ! Z Z0 --- ! X X0
If we compose this with the projection
X X0 --- ! Y Y 0 --- ! Z Z0 --- ! X X0
i j?? i j?? i j?? i j??
1 0 y 1 0 y 1 0 y 1 0 y
X --- ! Y --- ! Z --- ! X
we get a morphism of pre-triangles
X --- ! Y --- ! Q --- ! X
? ? ? ?
1?y 1?y h?y 1?y
X --- ! Y --- ! Z --- ! X
where the bottom is a pre-triangle because it is the direct summand
of a triangle, and the top is a triangle. Once again, Proposition 1.1.20
implies that h is an isomorphism. Thus the bottom row is isomorphic
to the top row, which is a distinguished triangle. By [TR0], the bottom__
row is also a triangle. |__|
Next we give two rather trivial corollaries, which nevertheless are use-
ful.
Corollary 1.2.4. The map f : X ! Y is an isomorphism if
and only if, for some Z (necessarily isomorphic to zero), the candidate
triangle
f 0 0
X --- ! Y --- ! Z --- ! X
is distinguished.
44 1. ELEMENTARY PROPERTIES
Proof: If f is an isomorphism, then the diagram below
X --1-! X --- ! 0 --- ! X
? ? ? ?
1?y f?y ?y 1?y
f
X --- ! Y --- ! 0 --- ! X
defines an isomorphism of candidate triangles. The top is distinguished,
hence so is the bottom. Thus, we can take Z = 0.
Conversely, assume
f 0 0
X --- ! Y --- ! Z --- ! X
is a distinguished triangle. It is the sum of the two candidate triangles
f
X -- - ! Y -- - ! 0 -- - ! X
and
0 --- ! 0 --- ! Z --- ! 0;
which by Proposition 1.2.3 must both be triangles. But then the dia-
gram
X --1-! X --- ! 0 --- ! X
? ? ? ?
1?y f?y ?y 1?y
f
X --- ! Y --- ! 0 --- ! X
gives a morphism of triangles. We know that 1 : X ! X and 1 : 0 ! 0
are isomorphisms. Proposition 1.1.20 implies that the morphism f :
X ! Y also is. Now in the morphism of triangles
X --1-! X --- ! 0 --- ! X
? ? ? ?
1?y f?y ?y 1?y
f 0 0
X --- ! Y --- ! Z --- ! X
we know that 1 : X ! X and f : X ! Y are isomorphisms, hence so
is 0 : 0 ! Z. Thus, Z is isomorphic to zero. 2
Corollary 1.2.5. Any triangle of the form
X --- ! Y --- ! Z --0-! X
is isomorphic to
X -- - ! X Z --- ! Z --0-! X;
that is, if the map Z ! X vanishes, then the triangle splits.
1.2. COROLLARIES OF PROPOSITION 1.1.20 45
Proof: By [TR0] and [TR2] both
X --1-! X --- ! 0 -- - ! X;
and
0 --- ! Z --1--! Z -- - ! 0
are triangles. From Proposition 1.2.1 we learn that so is the direct sum
X -- - ! X Z --- ! Z --0-! X:
But now the diagram
X --- ! X Z -- - ! Z ---0! X
? ? ?
1?y 1?y 1?y
X --- ! Y -- - ! Z ---0! X
can be completed to a morphism of triangles, which must be an iso-
morphism by Proposition 1.1.20. 2
It is often necessary to know not only that Y is isomorphic to X Z,
but also to give an explicit isomorphism. We observe
Lemma 1.2.6. Let us be given a triangle
X --u-! Y --v-! Z -- w-! X:
If v0 : Z -! Y is a map such that
0 v
Z --v-! Y --- ! Z
composes to the identity on Z, then the map of triangles
X --- ! X Z -- - ! Z --- ! X
? i j? ? ?
1?y u v0 ?y 1?y 1?y
X --u-! Y -- v-! Z --w-! X
is an isomorphism.
Proof: It is a map of triangles, where two of the vertical maps are
isomorphisms; by Proposition 1.1.20, so is the third. 2
Remark 1.2.7. Dually, given a triangle
X --u-! Y --v-! Z --w-! X
and a map u0: Y - ! X so that
0
X --u-! Y --u-! X
46 1. ELEMENTARY PROPERTIES
is the identity, then the map of triangles
X --u-! Y -- v-! Z --w-! X
? 0 u0 1? ? ?
1?y @ A?y 1?y 1?y
v
X --- ! X Z -- - ! Z --- ! X
is an isomorphism. In other words, given a triangle
X --u-! Y --v-! Z -- w-! X;
we get a canonical isomorphism
Y ' X Z
whenever we give either a factoring of the identity on Z as
0 v
Z --v-! Y --- ! Z
or a factoring of the identity on X as
0
X -- u-! Y --u- ! X:
Lemma 1.2.8. Suppose we have a triangle
X --u-! Y --v-! Z -- w-! X;
a factoring of the identity on Z as
0 v
Z -- v-! Y -- - ! Z;
and a factoring of the identity on X as
0
X -- u-! Y -- u-! X:
Then we have two isomorphisms
Y ' X Z;
one from each factoring. These two isomorphisms will agree if and only
if the composite
0 u0
Z --v-! Y -- - ! X
vanishes.
Proof: By Lemma 1.2.6, the map v0 : Z -! Y gives an isomorphism
i j
u v0
X Z --- - - - !Y;
and by the dual of Lemma 1.2.6, the map u0 : Y -! X gives an
isomorphism
0 0 1
@ u A
v
Y --- - - !X Z:
1.3. MAPPING CONES 47
We need to decide whether the isomorphisms agree, meaning whether
they are inverse to each other. To do this, it suffices to check whether
0 0 1
i j @ u A
u v0 v
X Z -- - - - - !Y --- - - !X Z
composes to the identity on X Z. But the composite is clearly
0 0 0 0 1
@ u u u v A
vu vv0
X Z --- - - - - - - -X! Z:
On the other hand, we know that u0u = 1 and vv0 = 1, and vu = 0
since it is the composite of two maps in a triangle. This makes the
matrix
0 0 0 1
@ 1 u v A
0 1
X Z -- - - - - - -X! Z;
and it will be the identity precisely if u0v0 vanishes. 2
1.3. Mapping cones, and the definition of triangulated
categories
Definition 1.3.1. Let T be a pre-triangulated category. Suppose
that we are given a morphism of candidate triangles
X -- u-! Y --v-! Z --w- ! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
There is a way to form a new candidate triangle out of this data. It is
the diagram
-v 0 -w 0 -u 0
g u0 h v0 f w0
Y X0 ____________-Z Y 0 ____________- X Z0 ____________-Y X0 *
* :
This new candidate triangle is called the mapping cone on a map of
candidate triangles.
48 1. ELEMENTARY PROPERTIES
Definition 1.3.2. Two maps of candidate triangles
X -- u-! Y --v-! Z --w- ! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
and
X -- u-! Y --v-! Z --w- ! X
? ? ? ?
f0?y g0?y h0?y f0?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
are called homotopic if they differ by a homotopy; that is, if there exist
, and below
X ________-u Y ________-v Z ________-w X
0 0 0
X0 ________-u Y 0 ________-v Z0 ________-w X0 ;
with
f - f0 = u + -1{w0} g - g0 = v + u0 h - h0= w + v0:
Lemma 1.3.3. Up to isomorphism, the mapping cone depends not
on the morphism of candidate triangles
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
but only on the homotopy equivalence class of this morphism. If the
map above is homotopic to the map
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f0?y g0?y h0?y f0 ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
as in Definition 1.3.2, then the mapping cones are isomorphic candidate
triangles.
1.3. MAPPING CONES 49
Proof: The diagram below is commutative
-v 0 -w 0 -u 0
g u0 h v0 f w0
Y X0 ____________-Z Y 0 ____________- X Z0 ____________-Y X0
| | | |
| | | |
| | | |
| | | |
1 |0 1 |0 1 |0 1 | 0
| | | |
|1 |1 |1 | 1
| | | |
| | | |
| | | |
|? |? |? |?
Y X0 ____________-Z Y 0 ____________- X Z0 ____________-Y X0
-v 0 -w 0 -u 0
g0 u0 h0 v0 f0 w0
and the vertical maps are isomorphisms; hence we have an isomorphism
of the top row with the bottom row. 2
The following is an elementary fact of homological algebra, whose
proof we leave to the reader.
Lemma 1.3.4. Suppose F : C ! D and F 0: C ! D are two
morphisms of candidate triangles. Suppose F and F 0are homotopic.
Then for any map G : C0 ! C and any map H : D ! D0, the
composites H O F O G and H O F 0O G are homotopic. 2
Definition 1.3.5. A candidate triangle C is called contractible if
the identity map 1 : C ! C is homotopic to the zero map 0 : C ! C.
Lemma 1.3.6. If C is a contractible candidate triangle, then any
map from F : C ! D or F 0: D ! C of candidate triangles is homo-
topic to the zero map.
Proof. The two cases being dual, we can restrict attention to F .
But F = F O 1C , and since C is contractible, 1C is homotopic to 0. But_
then F is homotopic to F O 0 = 0. |__|
Lemma 1.3.7. If C is a contractible candidate triangle, then C is
a pre-triangle.
Proof. We need to show that if H is any decent homological func-
tor, and C is the candidate triangle
X --- ! Y --- ! Z --- ! X
50 1. ELEMENTARY PROPERTIES
then the long sequence
H(-1Z) --- ! H(X) --- ! H(Y ) --- ! H(Z) -- - ! H(X)
is exact. In fact, this is true not only for decent homological functors
H, but for any additive functor. The point is that the identity on C is
homotopic to the zero map. There are , and as below
X ________-u Y ________-v Z ________-w X
X ________-u Y ________-v Z ________-w X ;
with
1X = u + -1{w0} 1Y = v + u 1Z = w + v:
Applying any additive functor H to this, we deduce that the identity
on the sequence
H(-1Z) --- ! H(X) --- ! H(Y ) --- ! H(Z) -- - ! H(X)
__
is chain homotopic to the zero map; hence the sequence is exact. |__|
Proposition 1.3.8. Let C be a contractible candidate triangle.
Then C is a distinguished triangle.
Proof. If C is the sequence
X --u-! Y --v-! Z --w-! X
then we can complete u : X ! Y to a triangle
0 w0
X --u-! Y --v-! Q --- ! X
Because C is contractible, there are the three maps
X ________-u Y ________-v Z ________-w X
X ________-u Y ________-v Z ________-w X ;
giving the homotopy of 1C to the zero map. Consider the map w.
Clearly, u O {w} = 0 , since u O w = 0. But because Hom(X; -)
is a homological functor, when we apply it to the triangle
0 w0
X --u-! Y --v-! Q --- ! X
1.3. MAPPING CONES 51
we get an exact sequence. Since u kills w, there must be a map
0: X ! Q with w00= w.
Now form the map of pre-triangles
X --u-! Y --v-! Z --w-! X
? ? ? ?
1?y 1?y 0w+v0 ?y 1?y
0 w0
X --u-! Y --v-! Q --- ! X
with 0 as above. The reader will easily show that it is a map of pre-
triangles; the diagram commutes. But as 1 : X ! X and 1 : Y ! Y
are isomorphisms, so is 0w + v0 by Proposition 1.1.20. We deduce
that the top candidate triangle is isomorphic to the bottom, but the __
bottom is a distinguished triangle. |__|
From now on, when we speak of contractible candidate triangles, we
will call them contractible triangles. Since we know by Proposition 1.3.8
that they are all distinguished, there is no risk of confusion. Remember
that, in this book, the word "triangle", with no adjective preceding it,
means distinguished triangle.
Lemma 1.3.9. Let the diagram
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
be a map of pre-triangles in the pre-triangulated category T. Then the
mapping cone is a pre-triangle.
Proof. Let H be a decent homological functor. We need to show
that H takes the mapping cone to an exact sequence. But because each
row is a pre-triangle, we have two exact sequences
H(-1Z) --- ! H(X) --- ! H(Y ) --- ! H(Z) -- - ! H(X)
and
H(-1Z0) -- - ! H(X0) --- ! H(Y 0) --- ! H(Z0) --- ! H(X0)
and a map between them. The mapping cone on this map of exact
sequences is exact. But it agrees with what we get if we apply H to
the candidate triangle
-v 0 -w 0 -u 0
g u0 h v0 f w0
Y X0 ____________-Z Y 0 ____________- X Z0 ____________-Y X0 *
* :
52 1. ELEMENTARY PROPERTIES
__
Hence the candidate triangle is a pre-triangle. |__|
Now suppose we are given a morphism of triangles
X -- u-! Y --v-! Z --w- ! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
Then the mapping cone is a pre-triangle by Lemma 1.3.9. It turns out
that it need not always be a triangle. Let us first analyse the trivial
cases.
Lemma 1.3.10. The mapping cone on the zero map between trian-
gles is a triangle.
Proof. Consider the zero map
X -- u-! Y --v-! Z --w- ! X
? ? ? ?
0?y 0?y 0?y 0?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
The mapping cone is the sequence
-v 0 -w 0 -u 0
0 u0 0 v0 0 w0
Y X0 ____________-Z Y 0 ____________- X Z0 ____________-Y X0 *
* :
This is nothing other than the direct sum of the sequences
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
and
Y ---v-! Z --w--! X ---u-! Y:
The first row is a triangle by hypothesis. The second row is a triangle
by [TR2], and because
X --u-! Y --v-! Z --w-! X
__
is. The direct sum is therefore a triangle, by Proposition 1.2.1. |__|
Because the mapping cone does not change, up to isomorphism, if we
replace a map by a homotopic one, we immediately deduce
1.3. MAPPING CONES 53
Corollary 1.3.11. Given a map of triangles
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
if the map is homotopic to zero, the mapping cone is a triangle. 2
Corollary 1.3.12. If either
X --u-! Y --v-! Z --w-! X
or
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
is a contractible triangle, and the other is a triangle, then the mapping
cone on any map
X -- u-! Y --v-! Z -- w-! X
? ? ? ?
f?y g?y h?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
is a triangle.
Proof. By Lemma 1.3.6, the map is homotopic to the zero map. __
Then by Corollary 1.3.11, the mapping cone is a triangle. |__|
This is as far as one gets without further assumptions. Now we come
to the main definition of this section:
Definition 1.3.13. Let T be a pre-triangulated category. Then T
is triangulated if it satisfies the further hypothesis
TR4': Given any diagram
X -- u-! Y --v-! Z -- w-! X
? ? ?
f?y g?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
where the rows are triangles, there is, by [TR3], a way to choose
an h : Z ! Z0 to make the diagram commutative. This h may
be chosen so that the mapping cone
-v 0 -w 0 -u 0
g u0 h v0 f w0
Y X0 ____________-Z Y 0 ____________- X Z0 ____________-Y X0
54 1. ELEMENTARY PROPERTIES
is a triangle.
Definition 1.3.14. A morphism of triangles will be called good if
its mapping cone is a triangle.
Remark 1.3.15. [TR4'] can be restated as saying that any dia-
gram
X -- u-! Y --v-! Z --w- ! X
? ? ?
f?y g?y f ?y
0 v0 w0
X0 -- u-! Y 0 --- ! Z0 -- - ! X0
where the rows are distinguished triangles, may be completed to a good
morphism of triangles.
The authors do not know an example of a pre-triangulated category
which is not triangulated.
1.4. Elementary properties of triangulated categories
We begin with the definition of homotopy cartesian squares.
Definition 1.4.1. Let T be a triangulated category. Then a com-
mutative square
f
Y --- ! Z
? ?
g?y ?yg0
Y 0 --- ! Z0
f0
is called homotopy cartesian if there is a distinguished triangle
0 1
@ g A i j
f -f0 g0 @
Y --- - ! Y 0 Z --- - - - - -Z!0 --- ! Y
for some @ : Z0 ! Y .
Notation 1.4.2. If
f
Y --- ! Z
? ?
g?y ?yg0
Y 0 --- ! Z0
f0
1.4. PROPERTIES OF TRIANGULATED CATEGORIES 55
is a homotopy cartesian square, we call Y the homotopy pullback of
Z
?
? 0
y g
Y 0 --- ! Z0
f0
and Z0 the homotopy pushout of
f
Y --- ! Z
?
g?y
Y 0
It follows from [TR1] that any diagram
f
Y --- ! Z
?
g?y
Y 0
has a homotopy pushout; The morphism
Y --- ! Y 0 Z
can be completed to a triangle
0 1
@ g A
f
Y -- - - ! Y 0 Z --- ! Z0 -- - ! Y
and this triangle defines a homotopy cartesian square
f
Y -- - ! Z
? ?
g?y ?yg0
Y 0-- - ! Z0:
f0
By Remark 1.1.21, the homotopy pushout is unique up to non-canonical
isomorphism. Also, any commutative square
f
Y --- ! Z
? ?
g?y ?y
Y 0 --- ! P
56 1. ELEMENTARY PROPERTIES
corresponds to a map Y 0 Z ! P so that the composite
0 1
@ g A
f
Y -- - - ! Y 0 Z --- ! P
vanishes. But Hom(-; P ) is a cohomological functor, and in particular
takes the triangle
0 1
@ g A
f
Y -- - - ! Y 0 Z --- ! Z0 -- - ! Y
to a long exact sequence. We are given a map in Hom(Y 0 Z; P )
whose image in Hom(Y; P ) vanishes. It must therefore come from
Hom(Z0; P ). There is a map Z0 ! P (non-unique) which maps the
square
f
Y --- ! Z
? ?
g?y ?yg0
Y 0 --- ! Z0
f0
to the square
f
Y --- ! Z
? ?
g?y ?y
Y 0 --- ! P:
Dually, homotopy pullbacks always exist and are unique up to non-
canonical isomorphism. And given a commutative square
P --- ! Z
? ?
? ? 0
y y g
Y 0 --- ! Z0
f0
there is always a map P ! Y mapping this square to the homotopy
pullback square
f
Y -- - ! Z
? ?
g?y ?yg0
Y 0-- - ! Z0:
f0
1.4. PROPERTIES OF TRIANGULATED CATEGORIES 57
Lemma 1.4.3. Let the following be a commutative diagram with
triangles for rows
f
X -- - ! Y --- ! Z --- ! X
? ? ?
1?y g?y 1?y
gf 0 0
X -- - ! Y --- ! Z --- ! X:
It may be completed to a morphism of triangles
f
X -- - ! Y --- ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0 w0
X -- - ! Y --- ! Z --- ! X:
so that
Y --- ! Z
? ?
? ?
y y
Y 0 --- ! Z0
is homotopy cartesian. In fact, the differential @ : Z0 ! Y may be
chosen to be the composite
0 f
Z0 --w-! X --- ! Y:
Proof: By [TR4'] we may complete
f
X -- - ! Y --- ! Z --- ! X
? ? ?
1?y g?y 1?y
gf 0 0
X -- - ! Y --- ! Z --- ! X:
to a good morphism of triangles
f
X -- - ! Y --- ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0 w0
X -- - ! Y --- ! Z --- ! X:
Then the mapping cone is a triangle
X Y --- ! Y 0 Z --- ! X Z0 --- ! X Y:
An elementary computation allow us to show that this triangle is iso-
morphic to the direct sum of the two candidate triangles
X -- - ! 0 --- ! X -- - ! X
58 1. ELEMENTARY PROPERTIES
and
Y --- ! Y 0 Z --- ! Z0 -- - ! Y:
By Proposition 1.2.3, each direct summand of a triangle is a triangle.
In particular,
Y -- - ! Y 0 Z -- - ! Z0 -- - ! Y
must be a distinguished triangle. It is also easy to compute that the
differential is as in the statement of the Lemma. 2
Lemma 1.4.4. Let
Y --- ! Z
? ?
g?y h?y
Y 0 --- ! Z0
be a homotopy cartesian square. If
g 0 00
Y -- - ! Y --- ! Y --- ! Y
is a triangle, then there is a triangle
Z -- h-! Z0 --- ! Y 00-- - ! Z
which completes the homotopy cartesian square to a map of triangles
g 0 00
Y --- ! Y -- - ! Y -- - ! Y
? ? ? ?
? ? ? ?
y y 1y y
Z --h-! Z0 -- - ! Y 00-- - ! Z:
That is, the differential Z0 ! Y is the composite
Z0 -- - ! Y 00-- - ! Y:
Proof: We know that the square
Y --- ! Z
? ?
g?y h?y
Y 0 --- ! Z0
is homotopy cartesian, in other words we have a triangle
Y --- ! Y 0 Z --- ! Z0 -- - ! Y:
But then the diagram
Y --- ! Y 0 Z --- ! Z0 --- ! Y
? ? ?
1?y ?y 1?y
Y --- ! Y 0 --- ! Y 00--- ! Y
1.4. PROPERTIES OF TRIANGULATED CATEGORIES 59
may be completed to a good morphism of triangles
Y --- ! Y 0 Z --- ! Z0 --- ! Y
? ? ? ?
1?y ?y ?y 1?y
Y --- ! Y 0 --- ! Y 00--- ! Y
and in particular the mapping cone is a triangle. The mapping cone
can easily be written as a direct sum of the three candidate triangles
Y 0 --1-! Y 0-- - ! 0 --- ! Y 0
Y --- ! 0 -- - ! Y --1-! Y
Z --- ! Z0 -- - ! Y 00--- ! Z
which must therefore all be triangles; in particular,
Z -- - ! Z0 --- ! Y 00-- - ! Z
is a distinguished triangle. Computing the various maps, we deduce
the map of triangles
g 0 00
Y --- ! Y -- - ! Y -- - ! Y
? ? ? ?
? ? ? ?
y y 1y y
Z --h-! Z0 -- - ! Y 00-- - ! Z:
of the Lemma. 2
Remark 1.4.5. Combining Lemmas 1.4.3 and 1.4.4, we have that
good maps of triangles
f
X -- - ! Y --- ! Z -- - ! X
? ? ? ?
1?y g?y h?y 1?y
gf 0 0
X -- - ! Y --- ! Z -- - ! X
are in closely related to homotopy cartesian squares
Y -- - ! Z
? ?
g?y h?y
Y 0-- - ! Z0:
One can pass back and forth from one to the other, of course not
uniquely. By Lemma 1.4.3 a good map gives a homotopy cartesian
60 1. ELEMENTARY PROPERTIES
square, and by Lemma 1.4.4, a homotopy cartesian square
Y --- ! Z
? ?
g?y h?y
Y 0 --- ! Z0
and a triangle
f
X --- ! Y --- ! Z --- ! X
give a good morphism of triangles
f
X -- - ! Y --- ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0
X -- - ! Y --- ! Z --- ! X:
Putting together Lemmas 1.4.3 and 1.4.4 slightly differently, we
deduce
Proposition 1.4.6. Let T be a triangulated category. Let f : X !
Y and g : Y ! Y 0be two, composable morphisms. Let us be given
triangles
f
X --- ! Y --- ! Z --- ! X
gf 0 0
X --- ! Y --- ! Z --- ! X
g 0 00
Y --- ! Y --- ! Y --- ! Y:
Then we can complete this to a commutative diagram
f
X --- ! Y -- - ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0
X --- ! Y -- - ! Z --- ! X
? ? ? ?
? ? ? ?
y y y y
0 --- ! Y 00-- 1-! Y 00 --- ! 0
? ? ? ?
? ? ? ?
y y y y
f 2
X --- ! Y -- - ! Z --- ! X
where the first and second row and second column are our given three
triangles, and every row and column in the diagram is a distinguished
1.4. PROPERTIES OF TRIANGULATED CATEGORIES 61
triangle. Furthermore, the square
Y --- ! Z
? ?
? ?
y y
Y 0 --- ! Z0
is homotopy cartesian, with differential being given by the equal com-
posites
Z0 --- ! X --- ! Y;
Z0 --- ! Y 00 --- ! Y:
Proof: By Lemma 1.4.3, the diagram
f
X -- - ! Y --- ! Z -- - ! X
? ? ?
1?y g?y 1?y
gf 0 0
X -- - ! Y --- ! Z -- - ! X
may be completed to a morphism of triangles
f
X -- - ! Y --- ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0
X -- - ! Y --- ! Z --- ! X;
so that
Y --- ! Z
? ?
? ?
y y
Y 0 --- ! Z0
is homotopy cartesian. By Lemma 1.4.4, the homotopy cartesian square
Y --- ! Z
? ?
? ?
y y
Y 0 --- ! Z0
and the triangle
Y -- - ! Y 0 --- ! Y 00--- ! Y
may be completed to a map of triangles
g 0 00
Y --- ! Y -- - ! Y -- - ! Y
? ? ? ?
? ? ? ?
y y 1y y
Z --h-! Z0 -- - ! Y 00-- - ! Z;
62 1. ELEMENTARY PROPERTIES
and the diagram
f
X --- ! Y -- - ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0
X --- ! Y -- - ! Z --- ! X
? ? ? ?
? ? ? ?
y y y y
0 --- ! Y 00-- 1-! Y 00 --- ! 0
? ? ? ?
? ? ? ?
y y y y
f 2
X --- ! Y -- - ! Z --- ! X
just assembles all this information together. 2
Remark 1.4.7. Proposition 1.4.6 is generally known as [TR4], or
the Octahedral Axiom. The diagram whose existence the Proposition
asserts is known as an octahedron. The reason for this name is that we
do in fact have 8 triangles which can be assembled to an octahedron.
The rows and the columns of the diagram give 4 distinguished triangles.
But there are also 4 commutative triangles
f
X --- ! Y and Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0
X --- ! Y Z --- ! X
Y 0 --- ! Y 00 and Y 00--- ! Y
? ? ? ?
? ? ? ?
y 1y 1y y
Z0 --h-! Y 00 Y 00--- ! Z:
These 8 triangles make an octahedron, and the reader is referred else-
where for the study of its symmetries.
It may be shown that a pre-triangulated category satisfies [TR4] if
and only if it satisfies [TR4']. We have shown the "if". The proof that
any pre-triangulated category satisfying [TR4] also satisfies [TR4'] may
be found in [22 ], Theorem 1.8.
1.5. Triangulated subcategories
Definition 1.5.1. Let T be a triangulated category. A full addi-
tive subcategory S in T is called a triangulated subcategory if every
1.5. TRIANGULATED SUBCATEGORIES 63
object isomorphic to an object of S is in S, if S = S, and if for any
distinguished triangle
X --- ! Y --- ! Z --- ! X
such that the objects X and Y are in S, the object Z is also in S.
Remark 1.5.2. From [TR2] we easily deduce that if S is a trian-
gulated subcategory of T and
X --- ! Y --- ! Z --- ! X
is a triangle in T, then if any two of the objects X, Y or Z are in S, so
is the third.
Definition 1.5.3. Let T be a triangulated category, S a triangu-
lated subcategory. We define a collection of morphisms MorS T by
the following rule. A morphism f : X ! Y belongs to MorS if and
only if, in some triangle
f
X --- ! Y --- ! Z -- - ! X;
the object Z lies in S.
Remark 1.5.4. Note that it is irrelevant which particular triangle
f
X --- ! Y --- ! Z --- ! X
we take. By Remark 1.1.21 the object Z is unique up to isomorphism,
and by Definition 1.5.1, S contains all objects in T isomorphic to its
objects.
Lemma 1.5.5. Every isomorphism f : X ! Y is in MorS.
Proof: Let f : X ! Y be an isomorphism. By Corollary 1.2.4, the
diagram
f
X -- - ! Y -- - ! 0 -- - ! X
is a triangle in T. But since S is, among other things, an additive
subcategory of T, 0 must be in S. Thus f is in MorS. 2
Lemma 1.5.6. Let f : X ! Y and g : Y ! Y 0be two morphisms
in T. If any two of f : X ! Y , g : Y ! Y 0and gf : X ! Y 0lie in
MorS, then so does the third.
64 1. ELEMENTARY PROPERTIES
Proof: By Proposition 1.4.6, there is a diagram of triangles
f
X --- ! Y -- - ! Z --- ! X
? ? ? ?
1?y g?y ?y 1?y
gf 0 0
X --- ! Y -- - ! Z --- ! X
? ? ? ?
? ? ? ?
y y y y
0 --- ! Y 00-- 1-! Y 00 --- ! 0
? ? ? ?
? ? ? ?
y y y y
f 2
X --- ! Y -- - ! Z --- ! X
Now f lies in MorS if and only if Z lies in S, gf lies in MorS if and
only if Z0 lies in S, and g lies in MorS if and only if Y 00lies in S. From
the triangle
Z -- - ! Z0 --- ! Y 00-- - ! Z
we learn that if any two of Z, Z0 and Y 00lie in S, then so does the third.
2
Lemma 1.5.7. If S is a triangulated subcategory of T, then there
is a subcategory of T whose objects are all the objects of T, and whose
morphisms are the ones in MorS.
Proof: Let X be an object of T. From Lemma 1.5.5 we learn that the
identity morphisms 1 : X ! X, being isomorphisms, lie in MorS.
But by Lemma 1.5.6 we know that the composite of two morphisms
in MorS is again in MorS. Thus MorS is a subcategory of T. 2
Lemma 1.5.8. Let the square
f
Y --- ! Z
? ?
g?y ?yg0
Y 0 --- ! Z0
f0
be a homotopy cartesian square. Then f is in MorS if and only if f0
is, and g is in MorS if and only if g0 is. Another way to phrase this is
that homotopy pushout and homotopy pullback of morphisms in MorS
give morphisms in MorS.
Proof: The two statement being transposes of each other, we will
prove only that g is in MorS if and only if g0 is. But by Lemma 1.4.4
1.6. HOMOTOPY LIMITS 65
the homotopy cartesian square above may be completed to a morphism
of triangles
g 0 00
Y --- ! Y -- - ! Y -- - ! Y
? ? ? ?
f?y ?yf0 1?y f ?y
Z --- ! Z0 -- - ! Y 00-- - ! Z:
g0
Now Y 00will lie in S precisely if both maps g and g0 are in MorS. This
proves that g is in MorS if and only if g0 is. 2
1.6. Direct sums and products, and homotopy limits and
colimits
Definition 1.6.1. Let ff be an infinite cardinal. A triangulated
category T is said to satisfy [TR5(ff)] if, in addition to the other axioms,
the following holds.
TR5(ff): For any set of cardinality < ff,aand any collection
{X ; 2 }, of objects of T, the coproduct X exists in T. If
2
T satisfies [TR5(ff)] for all infinite cardinals ff, we say T satisfies
[TR5].
Definition 1.6.2. If the dual triangulated category Top satisfies
[TR5(ff)], we say T satisfies [TR5*(ff)]. If the dual satisfies [TR5], we
say that T satisfies [TR5*].
Remark 1.6.3. It follows from Proposition 1.2.1 and Remark 1.2.2
that coproducts and products of triangles are triangles. More precisely,
if T satisfies [TR5*], then given any set of triangles, the product exists,
and is a triangle by Proposition 1.2.1. Dually, if T satisfies [TR5], the
coproduct of any collection of triangles exists and is a triangle.
Definition 1.6.4. Let T be a triangulated category satisfying [TR5(@1)];
that is, countable coproducts exist in T. Let
j1 j2 j3
X0 --- ! X1 --- ! X2 --- ! . . .
be a sequence of objects and morphisms in T. The homotopy colimit of
the sequence, denoted Hocolim_-Xi, is by definition given, up to non-
canonical isomorphism, by the triangle
1a 1 ( 1 )
1 - shift a a
Xi --- - - - ! Xi --- ! Hocolim_-Xi --- ! Xi
i=0 i=0 i=0
66 1. ELEMENTARY PROPERTIES
a1 1
shift a
where the shift map Xi -- - ! Xi is the direct sum of ji+1 :
i=0 i=0
Xi ! Xi+1. In other words, the map {1 - shift} is the infinite matrix
0 1
1X0 0 0 0 . . .
B -j1 1X1 0 0 . . .C
B C
B 0 -j2 1X2 0 . . .C
B C
@ 0 0 -j3 1X3 . . .A
.. . . .
. .. .. ..
Lemma 1.6.5. If we have two sequences
X0 --- ! X1 --- ! X2 --- ! . . .
and
Y0 --- ! Y1 -- - ! Y2 --- ! . . .
then, non-canonically,
Hocolim_-{Xi Yi} = Hocolim_-Xi Hocolim_-Yi :
Proof: Because the direct sum of two triangles is a triangle by Propo-
sition 1.1.20, there is a triangle
( 1 ) ( ) ( ) ( )
a 1a 1 - shift a1 1a
Xi Yi -- - - - - ! Xi Yi
i=0 i=0 i=0 ? i=0
?
y
Hocolim_-Xi Hocolim_-Yi
and this triangle identifies
Hocolim__-Xi Hocolim_-Yi = Hocolim_-{Xi Yi} :
2
Lemma 1.6.6. Let X be an object of T, and let
X -- 1-! X -- 1-! X -- 1-! . . .
be the sequence where all the maps are identities on X. Then
Hocolim_-X = X;
even canonically.
Proof: The point is that the map
a1 1
1 - shift a
X --- - - - ! X
i=0 i=0
1.6. HOMOTOPY LIMITS 67
is split. Perhaps a simpler way to say this is that the map
( 1 ) i j
a i0 {1-shift} a1
X X --- - - - - - - - - - -X!
i=0 i=0
` 1
is an isomorphism, where i0 : X - ! i=0X is the inclusion into the
zeroth summand. In other words, the candidate triangle
1a 1 ( 1 )
1 - shift a pr 0 a
X --- - - - ! X -- - ! X -- - ! X
i=0 i=0 i=0
` 1
where pr : i=0X -! X is the map which is 1 on every summand, is
isomorphic to the sum of the two triangles
a1 a1 (a1 )
X -- 1-! X --- ! 0 --- ! X
i=0 i=0 i=0
and
0 -- - ! X -- 1-! X -- - ! 0:
Hence X is identified as Hocolim_-X. 2
Lemma 1.6.7. If in the sequence
X0 --0-! X1 --0-! X2 --0-! . . .
all the maps are zero, then Hocolim_-Xi = 0.
Proof: The point is that then the shift map in
1a 1
1 - shift a
Xi --- - - - ! Xi
i=0 i=0
vanishes. But by [TR0] there is a triangle
a1 a1 (a1 )
Xi -- 1-! Xi --- ! 0 -- - ! Xi
i=0 i=0 i=0
and this identifies 0 as Hocolim_-Xi. 2
Proposition 1.6.8. Suppose T is a triangulated category satisfy-
ing [TR5(@1)]. Let X be an object of T, and suppose e : X ! X is
idempotent; that is, e2 = e. Then e splits in T. There are morphisms
f and g below
f g
X --- ! Y --- ! X
with gf = e and fg = 1Y .
68 1. ELEMENTARY PROPERTIES
Proof: Cosider the two sequences
X -- e-! X -- e-! X -- e-! . . .
and
X --1-e-!X --1-e-!X --1-e-!. . .
Let Y be the homotopy colimit of the first, and Z the homotopy colimit
of the second. We will denote this by writing Y = Hocolim_-(e) and
Z = Hocolim_-(1 - e).
By Lemma 1.6.5, Y Z is the homotopy colimit of the direct sum
of the two sequences, that is of
0 1 0 1 0 1
@ e 0 A @ e 0 A @ e 0 A
0 1-e 0 1-e 0 1-e
X X --- - - - - -X! X --- - - - - -X! X --- - - - - -.!. .
But the following is a map of sequences
0 1 0 1 0 1
BBe 0 CC BBe 0 CC BBe 0 CC
@ 0 1-e A @ 0 1-e A @ 0 1-e A
0 X 1X -- - - - - - !X X --- - - - - !X X -- - - - - - !. . .
? 0 1? 0 1?
BB e 1-e CC? BB e 1-eCC? BB e 1-e CC?
@ 1-e e Ay @ 1-e e Ay @ 1-e e Ay
X X ---0- - -1- !X X ---0- - -1- !X X ---0- - -1- !. . .
@ 1 0 A @ 1 0 A @ 1 0 A
0 0 0 0 0 0
and in fact, the vertical maps are isomorphisms. The map
e 1-e
1-e e : X X -! X X
is its own inverse; its square is easily computed to be the identity.
It follows that the homotopy limits of the two sequences are the
same. Thus Y Z is the homotopy limit of the bottom row, and the
bottom row decomposes as the direct sum of the two sequences
X -- 1-! X -- 1-! X -- 1-! . . .
and
X -- 0-! X -- 0-! X -- 0-! . . .
By Lemma 1.6.6, the homotopy colimit of the first sequence is X, while
by Lemma 1.6.7, the homotopy colimit of the second sequence is 0. The
homotopy colimit of the sum, which is Y Z, is therefore isomorphic
to X 0 = X.
1.6. HOMOTOPY LIMITS 69
More concretely, consider the maps of sequences
X ---e! X ---e! X -- e-! . . .
? ? ?
e?y e?y e?y
X ---1! X ---1! X -- 1-! . . .
and
X -1-e--!X --1-e-!X --1-e-!. . .
? ? ?
1-e?y 1-e?y 1-e?y
X ---1! X ---1! X -- 1-! . . .
What we have shown is that the induced maps on homotopy colimits,
that is g : Y ! X and g0 : Z ! X can be chosen so that the sum
Y Z ! X is an isomorphism.
In the sequence
X -- e-! X -- e-! X -- e-! . . .
defining Y as the homotopy colimit, we get a map f : X ! Y , just the
map from a finite term to the colimit. In the sequence
X -- 1-! X -- 1-! X -- 1-! . . .
the map from the finite terms to the homotopy colimit is the identity.
We deduce a commutative square
f
X -- - ! Y
? ?
e?y g?y
X -- 1-! X:
Similarly, from the other sequence we deduce a commutative square
f0
X -- - ! Z
? ?
1-e?y g0?y
X -- 1-! X:
In other words, we conclude in total that e = gf and 1 - e = g0f0. The
composite
0 1
@ f A i j
f0 g g0
X --- - - !Y Z --- - - - !X
70 1. ELEMENTARY PROPERTIES
is e + (1 - e) = 1. Since we know that the map Y Z ! X is an
isomorphism, it follows that the map X ! Y Z above is its (two-
sided) inverse. The composite in the other order is also the identity.
In particular, fg = 1Y and f0g0 = 1Z . 2
Remark 1.6.9. Dually, if T satisfies [TR5*(@1)], then idempotents
also split.
Remark 1.6.10. In this entire section, we have used nothing more
than countable coproducts and products.
1.7. Some weak "functoriality" for homotopy limits and
colimits
In the last section we saw the definition and elementary properties
of the homotopy colimit of a sequence. We also saw how homotopy
colimits can be useful; for example, they prove that idempotents split.
See Proposition 1.6.8. Since homotopy colimits are defined by a trian-
gle, they are not in any reasonable sense functorial. But they do have
some good properties. Let us mention one here.
Lemma 1.7.1. Let T be a triangulated category satisfying [TR5(@1)].
Suppose we are given a sequence of objects and morphisms in T
X0 --- ! X1 --- ! X2 --- ! . . .
Suppose we take any increasing sequence of integers
0 i0 < i1 < i2 < i3 < . . .
Then we can form the subsequence
Xi0 --- ! Xi1 --- ! Xi2 -- - ! . . .
The two sequences have isomorphic homotopy colimits.
Proof: For each integer n, define jn to be the smallest im such that
n im . Then there is a map of sequences
X0 -- - ! X1 --- ! X2 -- - ! . . .
? ? ?
? ? ?
y y y
Xj0 -- - ! Xj1 --- ! Xj2 -- - ! . . .
This seems much more complicated than it actually is. If we start with
the sequence
0 < 2 < 5 < . . .
1.7. "FUNCTORIALITY" OF LIMITS 71
our map of sequences is simply
X0 -- - ! X1 -- - ! X2 --- ! X3 --- ! X4 --- ! X5 --- ! . . .
? ? ? ? ? ?
? ? ? ? ? ?
y y y y y y
X0 -- - ! X2 -- - ! X2 --- ! X5 --- ! X5 --- ! X5 --- !
The map of sequences yields a commutative square
a1 1
1 - shift a
Xn --- - - - ! Xn
n=0? n=0?
? ?
y y
1a 1
1 - shift a
Xjn --- - - - ! Xjn
n=0 n=0
and the point is that this commutative square is homotopy cartesian
(see Definition 1.4.1). In fact, the zero map is the differential. In the
sequence
a1 ( a1 ) ( 1a ) a1
Xn -- - ! Xn Xjn -- - ! Xjn
n=0 n=0 n=0 n=0
both maps are split, expressing the middle as the direct sum of the two
outside terms. We leave it to the reader to check this fact.
By Lemma 1.4.4 the homotopy commutative square may be com-
pleted to a morphism of triangles
1a 1 ( 1 )
1 - shift a a
Xn -- - - - - ! Xn -- - ! Y -- - ! Xn
n=0? n=0? ? n=0 ?
? ? ? ?
y y y y
a1 1 ( 1 )
1 - shift a a
Xjn -- - - - - ! Xjn -- - ! Y -- - ! Xjn
n=0 n=0 n=0
and Y is identified as both Hocolim_-Xn and Hocolim_-Xjn.
Now we need to identify Hocolim_-Xjn with Hocolim_-Xin; recall
that the sequence
Xj0 --- ! Xj1 --- ! Xj2 --- ! . . .
is obtained from the sequence
Xi0 --- ! Xi1 --- ! Xi2 -- - ! . . .
72 1. ELEMENTARY PROPERTIES
by repeating many of the terms. The reader can check that the triangle
a1 1 ( 1 )
1 - shift a a
Xjn -- - - - - ! Xjn -- - ! Y -- - ! Xjn
n=0 n=0 n=0
is isomorphic to the direct sum of two candidate triangles
1a 1 ( 1 )
1 - shift a a
Xin --- - - - ! Xin -- - ! Y -- - ! Xin
n=0 n=0 8 n=0 9
1a a1 < 1a =
Xin --- ! Xin -- - ! 0 -- - ! Xin
in-1fiwill stand for the smallest S, S a triangulated subcategory of T
satisfying
3.2.1.1. The objects in S lie in S.
3.2.1.2. Any coproduct of fewer than fi objects of S lies in
S.
3.2.1.3. The subcategory S T is thick.
Remark 3.2.2. The subcategory ~~~~ fiis well-defined. It is the
intersection of all the subcategories S of T satisfying 3.2.1.1, 3.2.1.2
and 3.2.1.3.
105
106 3. PERFECTION OF CLASSES
The key fact we want to prove in this section is that, as long as S is
a set, for all infinite fi the subcategory ~~~~fiis essentially small. That
is, there is only a set of isomorphism classes of objects of ~~~~fi. The
first observation is
Lemma 3.2.3. If fi fl and S T is a set of objects, then ~~~~fi
~~~~fl.
Proof: ~~~~flsatisfies 3.2.1.1 and 3.2.1.3, which do not involve fl, as
well as 3.2.1.2 for fl; that is the coproduct of fewer than fl objects in
~~~~flis in ~~~~fl. But then the coproduct of fewer than fi objects of ~~~~fl
certainly must be in ~~~~ fl. But ~~~~ fiis minimal with this property;
hence ~~~~fi ~~~~fl. 2
Lemma 3.2.4. Let S be a set of objects in a triangulated category
T. Suppose T has small Hom-sets. Then_the_smallest triangulated
subcategory of T containing S, denoted T (S), is essentially small. Fur-
thermore, in the course_of the proof we will construct a small category
T (S) equivalent to T (S), containing S.
Proof: Define T1(S) to be the full subcategory whose objects are
S [{0}. Because S is a set the category T1(S) has only a set of objects,
and since T has small Hom-sets, T1(S) is small. Now inductively define
Tn(S). Suppose T1(S); T2(S); : :;:Tn(S) have already been defined, and
are small. To define Tn+1(S) choose, for every morphism f : x ! y in
Tn(S), one object of T in the isomorphism class of z in the triangle
f
x --- ! y --- ! z --- ! x
Call this object Cf. Let Tn+1(S) be the smallest full subcategory con-
taining Tn(S) and allSthe Cf's.
Now put T (S) = 1n=1Tn(S). Given any morphism f : x ! y in
T (S), it lies in some Tn(S). But then Tn+1(S) contains an object Cf
which fits in a triangle
f
x --- ! y --- ! Cf --- ! x
Thus the full subcategory of T whose objects are all_the isomorphs_
of the objects of T (S)_is triangulated. Call it T (S). Since T (S) is
equivalent_to T (S), T (S) is essentially small. It is easy to see that
T(S) is in fact the smallest triangulated subcategory containing S. 2
Proposition 3.2.5. Let T be triangulated category with small Hom-
sets, satisfying [TR5]. Suppose S is some set of objects in T. Let fi be
an infinite cardinal. Then the category ~~~~fiis essentially small.
3.2. GENERATED SUBCATEGORIES 107
Proof: By Lemma 3.2.3, if fi fl, then ~~~~ fi ~~~~ fl. It therefore
clearly suffices to show that if fi is an infinite cardinal and fl is the
successor of fi, then ~~~~flis essentially small. But fl, being a successor
cardinal, is regular (see Section 3.1). Furthermore, fl > fi @0 (@0
is the first infinite cardinal). Replacing fi by its successor fl, we may
therefore assume fi regular, and fi > @0.
Now observe that any subcategory S satisfying 3.2.1.2 automatically
satisfies 3.2.1.3; after all fi > @0 together with 3.2.1.2 implies that the
category S is closed with respect to countable coproducts. But then
by Proposition 1.6.8 all idempotents in S split, and the subcategory S
must be thick.
In other words, for fi > @0 3.2.1.3 is redundant; ~~~~fiis the small-
est subcategory satisfying 3.2.1.1 and 3.2.1.2. Now we will show ~~~~fi
essentially small. But first a little notation.
Let S be a set of objects in T, containing the zero object. Let M
be a set of cardinality fi. Consider the set
Y
S;
2M
of all sequences of objects in S of length fi. Consider the subset
Y
G S;
2M
consisting of all sequences such that the cardinality of the set of non-
zero terms is < fi. For each sequence in G, choose a representative in
the isomorphism class of the coproduct of the sequence. Let CP (S)
be the full subcategory of T, whose objects are the union of S and our
choices of isomorphism representatives of coproducts of length < fi.
Then CP (S) is small, contains S, and contains an object isomorphic
to the coproduct of any < fi objects of S.
Now we proceed by transfinite induction. Inductively define Si,
for all ordinals i. Set S0 = T (S) (a small category equivalent to the
smallest triangulated subcategory containing S, as in Lemma 3.2.4).
For a successor ordinal i + 1, define Si+1 to be T (CP (Si)); that is, first
choose something isomorphic to any coproduct of fewer than fi objects
in Si and throw it in, then close with respect to triangles. For i a limit
ordinal, define Si to be the union
[
Si = Sj:
j~~*fi. But since Sfi
is equivalent to the small category Sfi, it is essentially small. It follows
that the smaller subcategory *~~fiis essentially small. 2
Definition 3.2.6. Let fi be an infinite cardinal. Let T be a tri-
angulated category satisfying [TR5]. A subcategory S T is called
fi-localising if it is thick and closed with respect to the formation of
coproducts of fewer than fi of its objects. That means that the coprod-
uct of fewer than fi objects of S exists, as a coproduct in T, and is an
object of S. A triangulated subcategory S T is called localising if it
3.2. GENERATED SUBCATEGORIES 109
is fi-localising for all fi. Equivalently, S is localising if it is closed un-
der the formation of all small T-coproducts of its objects. That is, if
{X ; 2 } is a family of objects in S, then the T-coproduct
a
X
2
is an object of S.
Remark 3.2.7. If fi > @0, then any triangulated subcategory S
T closed under the formation of coproducts of fewer than fi of its objects
is automatically thick, hence localising. It is redundant to assume
S thick. The point is that if fi > @0, then S conatins all countable
coproducts of its objects. By Proposition 1.6.8 every idempotent in S
splits, and hence S contains all direct summands of its objects.
Example 3.2.8. Let T be a triangulated category satisfying [TR5].
Let S be a class of objects of T. Then ~~~~fiis fi-localising, whereas
[ fi
~~~~ = ~~~~
fi
is localising, as in Definition 3.2.6. The statement for ~~~~fiis really part
of its definition, as the smallest fi-localising subcategory with certain
properties. The statement for ~~~~ requires a little proof. Let us give
the proof.
Let {X ; 2 } be any collection of objects in ~~~~. Let the cardi-
nality of be fi, and suppose that for each 2 , the object X lies
in ~~~~fi, for some cardinal fi . Let fl be a cardinal greater than the
sum of fi and the fi 's. The X 's are all in ~~~~fl, and since fl > fi, the
coproduct of the X 's, that is of fi objects of ~~~~fl, must lie in ~~~~fl.
Hence in ~~~~, which is therefore localising.
In fact, ~~~~ is the smallest localising subcategory containing S.
Any localising subcategory containing S will satisfy 3.2.1.1, 3.2.1.2 and
3.2.1.3 for all infinite fi. Therefore it must contain ~~~~fifor all fi, and
hence ~~~~, the union.
Definition 3.2.9. Let T be a triangulated category satisfying [TR5].
Let S be a class of objects of T. Then the union of all the ~~~~fi's, that
is
[ fi
~~~~
fi
110 3. PERFECTION OF CLASSES
will be denoted ~~~~ . Note that even when S is a small set and T has
small Hom-sets, in which case Proposition 3.2.5 tells us that the cat-
egories ~~~~fiare all essentially small, the category ~~~~ will usually be
gigantic. It is called the localising subcategory generated by S.
Lemma 3.2.10. Let fi be an infinite cardinal. Let T be a triangu-
lated category closed under the formation of coproducts of fewer than
fi of its objects. Let S be a fi-localising subcategory of T. Then T=S is
closed with respect to the formation of coproducts of fewer than fi of its
objects, and the universal functor F : T -! T=S preserves coproducts.
Proof: (cf. proof of Lemma 2.1.29). Since the objects of T and T=S
are the same, it suffices to show that the coproduct in T of fewer than
fi objects is also the coproduct in T=S. Let be a set of cardinality less
than fi, {X ; 2 } a collection of objects of T. Form their coproduct
in T, which we know exists,
a
X :
2
We need to show this is also a coproduct in T=S.
Let Y be an arbitrary object of T. We need to show that any
collection of morphisms in T=S
{X -! Y; 2 }
`
factors uniquely through 2 X . That is, we need to show the exis-
tence and uniqueness of a factorisation.
The morphisms X -! Y in T=S can be represented by diagrams
P -- - ! Y
?
f ?y
X
where f : P -! X are in MorS. That is, in the triangle
f
P --- ! X --- ! Z --- ! P
the object Z must be in S. But now the coproduct is a triangle
a ` 2 f a a a
P -- - - - ! X --- ! Z -- - ! P
2 2 2 2
`
and 2 Z , being the coproduct of fewer than fi objects in S, must
be in S. The map
a ` 2 f a
P -- - - - ! X
2 2
3.2. GENERATED SUBCATEGORIES 111
therefore lies in MorC. The diagram
a
P --- ! Y
2 ?
` ?
2 f y
a
X
2
`
represents a morphism in T=S of the form 2 X - ! Y . Its com-
posites with the inclusions
a
i : X -! X
2
are computed by the commutative diagrams
a
P --- ! P -- - ! Y
? 2 ?
f ?y `2 f?y
a
X --- ! X
2
to be the given maps
P -- - ! Y
?
f ?y
X
a
We have therefore factored X -! Y through X .
2
Now for the uniqueness of the factorisation. Suppose we are given
a map in T=S of the form
a
OE : X -! Y
2
so that the composites OE O i with every
a
i : X -! X
2
112 3. PERFECTION OF CLASSES
vanish. We need to show the map OE vanishes. Represent OE as a diagram
Y
?
g?y
a
X --- ! Q
2
with g : Y - ! Q in MorS. Then for each , the composite with i is
represented by
Y
?
g?y
X -- - ! Q
and must vanish in T=S. By Lemma 2.1.26, there must then exist Z 2 S
so that X -! Q factors as
X -! Z -! Q:
But then
a
X -! Q
2
factors as
a a
X -! Z -! Q
2 2
`
and 2 Z , being a coproduct of fewer than fi objects of S, must lie
in S. Lemma 2.1.26 tells us that the map
Y
?
g?y
a
X --- ! Q
2
must then vanish in T=S. 2
The following is an immediate corollary.
Corollary 3.2.11. Let T be a triangulated category satisfying [TR5].
Let S be a localising subcategory. Then T=S satisfies [TR5], and the uni-
versal functor T -! T=S preserves coproducts.
3.3. PERFECT CLASSES 113
3.3. Perfect classes
Definition 3.3.1. Let T be a triangulated category satisfying [TR5].
Let fi be an infinite cardinal. A class of objects S T is called fi-perfect
if the following hold
3.3.1.1. S contains 0.
3.3.1.2. Suppose {X ; 2 } is a collection of objects in T.
Suppose the cardinality of is less than fi. Let k be an object of
S. Then any map
a
k --- ! X
2
factors as
a a
k --- ! k --- ! X
2 2
with k 2 S. More precisely there is, for each 2 , an object
k 2 S and a map f : k -! X , so that the map factors as
a ` 2 f a
k --- ! k --- - - ! X :
2 2
3.3.1.3. Suppose again that is a set of cardinality < fi.
Suppose k and the k 's, 2 , are objects of S and the X 's are
any objects of T, and the composite
a `2 f a
k --- ! k --- - - ! X
2 2
vanishes. Then it is possible to factor each f : k -! X as
g h
k -- - ! l -- - ! X
so that l 2 S, and the composite
a ` 2 g a
k --- ! k --- - - ! l
2 2
already vanishes.
We begin with a trivial lemma.
Lemma 3.3.2. Let T be a triangulated category satisfying [TR5].
Suppose fi is an infinite cardinal, and T is a fi-perfect class in T. Sup-
pose that S T is an equivalent class; that is, any object of T is
isomorphic to some object of S. Then S is also fi-perfect.
114 3. PERFECTION OF CLASSES
Proof: Trivial. 2
Lemma 3.3.3. Let T be a triangulated category satisfying [TR5].
Let fi be an infinite cardinal. Let S be a fi-perfect class of T. Let T be
the class of all objects in T which are direct summands of objects of S.
Then T is also fi-perfect.
Proof: Suppose we are given an object k 2 T , a set of cardinality
< fi, a family of X 's in T and a map
a
k --- ! X :
2
Because k 2 T , there must exist some k0 2 T so that k k0 2 S; T is
defined to be the class of all direct summands of objects of S. Consider
the composite
i j
1 0 a
k k0 --- - - - !k --- ! X :
2
Since k k0 2 S and S is fi-perfect, this factors
a `2 f a
k k0 -- - ! k --- - - ! X
2 2
with k 2 S T . But then
0 1
@ 1 A
0 a ` 2 f a
k -- - - ! k k0 --- ! k --- - - ! X
2 2
is a factorisation of the original map
a
k --- ! X :
2
Suppose now that we are given a vanishing composite
a `2 f a
k --- ! k --- - - ! X
2 2
with k; k 2 T . Choose k0 so that k k0 2 S, and for each choose k0
with k k0 2 S. Then we have a vanishing composite
0 1
i j ` 2 @ 1 A ` i j
1 0 a 0 a 2 f 0 a
k k0 --- - - - !k --- ! k --- - - - - - ! k k0 --- - - - - - - - !X
2 2 2
3.3. PERFECT CLASSES 115
Because S is fi-perfect, for each the map
f 0 : k k0 -! X
must factor as
k k0 -! l -! X
with l 2 S, where the composite
0 1
i j ` 2 @ 1 A
1 0 a 0 a a
k k0 --- - - - !k --- ! k --- - - - - - ! k k0 --- ! l
2 2 2
already vanishes. But then
0 1 0 1
@ 1 A i j `2 @ 1 A
0 1 0 a 0 a
k --- - ! k k0 --- - - - !k -- - ! k --- - - - - - ! k k0
2 2 ?
?
y
a
l
2
also vanishes, completing the proof of the fi-perfection of T . 2
Definition 3.3.4. Let T be a triangulated category satisfying [TR5].
Let S T be a triangulated subcategory. Let fi be an infinite cardinal.
An object k 2 T is called fi-good if the following holds.
3.3.4.1. Let {X ; 2 } be a family of objects of T. Sup-
pose the cardinality of is less than fi. Then any map
a
k --- ! X
2
has a factorisation
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 S.
More intuitively, 3.3.1.2 sort of holds for the single object k; it holds
where the k may be taken in S.
Lemma 3.3.5. Let T be a triangulated category satisfying [TR5],
and let S be a triangulated subcategory. If k is a fi-good object of T,
then the following automatically holds.
116 3. PERFECTION OF CLASSES
3.3.5.1. Suppose is an index set of cardinality < fi. Given
a vanishing composite
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 S, then it is possible to factor each f : k -! X as
g h
k -- - ! l -- - ! X
so that l 2 S, and the composite
a ` 2 g a
k --- ! k --- - - ! l
2 2
also vanishes.
Proof: Assume that is a set whose cardinality is less than fi. Suppose
we are given a vanishing composite
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 S.
For each ,consider the triangle
f
Y --- ! k -- - ! X --- ! Y :
Summing these triangles, we obtain a triangle
a a `2 f a a
Y -- - ! k --- - - ! X --- ! Y :
2 2 2 2
Since the composite
a `2 f a
k --- ! k --- - - ! X
2 2
vanishes, the map
a
k -- - ! k
2
must factor as
a a
k -- - ! Y -- - ! k
2 2
but then the hypothesis that k is good guarantees that the map
a
k -- - ! Y
2
3.3. PERFECT CLASSES 117
factors as
a a
k --- ! j --- ! Y
2 2
with j 2 S. The composite
f
j --- ! Y -- - ! k --- ! X
clearly vanishes, and if we define l by forming the triangle
g
j -- - ! k --- ! l --- ! j ;
then clearly k -! X factors as
g h
k --- ! l --- ! X :
Since j and k are in S and S is triangulated, l must also be in S. And
the composite
a a
k --- ! k --- ! l
2 2
must vanish, since it is
a a a
k --- ! j --- ! k -- - ! l
2 2 2
and
g
j --- ! k --- ! l
are two morphisms in a triangle. 2
Remark 3.3.6. From Lemma 3.3.5 we learn that if S is a trian-
gulated subcategory of T all of whose objects are fi-good, then S is
a fi-perfect class; more precisely, the collection of all objects of S is
a fi-perfect class. For any k 2 S, if it satisfies 3.2.1.2 then 3.2.1.3 is
automatic. The next couple of Lemmas allow us to cut the work even
shorter. One need not check that every object of S is fi-good. It is
enough to check all the objects of a large enough generating class.
Before the next Lemmas, we should perhaps remind the reader of __
the notation of Section 3.2. Let S be a class of objects of T. Then T (S)
is the smallest triangulated subcategory containing S. For an infinite
cardinal ff, ~~~~ffis the smallest ff-localising subcategory containing S.
More explicitly, ~~~~ffis the minimal thick subcategory S T such that
3.2.1.1: S contains S.
3.2.1.2: S is closed under the formation of coproducts of fewer
than ff of its objects.
118 3. PERFECTION OF CLASSES
Recall that by Remark 3.2.7, if ff > @0 it is redundant to assume S is
thick; 3.2.1.2 already tells us that idempotents split in S.
Lemma 3.3.7. Let ff and fi be infinite cardinals. Let T be a tri-
angulated category satisfying [TR5]. Let S be a class of objects of T.
Suppose that every object k 2 S is fi-good, as an object of the triangu-
lated subcategory ~~~~ff T. Then the objects of ~~~~ffform a fi-perfect
class.
A similar Lemma, whose proof is nearly identical, is
Lemma 3.3.8. Let fi be an infinite cardinal. Let T be a triangulated
category satisfying [TR5]. Let S be a class of objects of T. Suppose that
every_object k 2 S is fi-good_as_an object of the triangulated subcategory
T(S) T. Then the objects of T (S) form a fi-perfect class.
Proofs of Lemmas 3.3.7 and 3.3.8. Before we start the proof,_let_
us make one observation. Assume Lemma 3.3.8; that is assume T (S)
is fi-perfect.__By Lemma 3.3.3, the collection of all direct summands
of objects of T (S)_is also fi-perfect; but this is precisely ~~~~ @0, the
thick closure of T (S). Without loss, we may therefore assume ff > @0
in Lemma 3.3.7; the case ff = @0 is an immediate consequence of
Lemma 3.3.8.
The proofs of the two Lemmas are so nearly the same, that we will
give them together. Consider the following two full subcategories R T
and S T, given by
S = {k 2 ~~~~ff|k is fi-good};
__
R = {k 2 T (S)|k is fi-good}:
__ ff
It suffices to show_that R contains T (S) and S contains ~~~~ . To show
that R contains T (S), it suffices to prove that R is triangulated and
contains S. To show that S contains ~~~~ff, it suffices to prove that S is a
thick subcategory of T satisfying 3.2.1.1 and 3.2.1.2. Since the argument
for R is similar and slightly simpler, we leave it to the reader. We
assume therefore that ff > @0, and we will prove that S is triangulated,
and satisfies 3.2.1.1 and 3.2.1.2. Since ff > @0, S would then be closed
under the formation of countable coproducts, hence idempotents would
split in S. It is redundant to prove the thickness.
We need to prove three things, of which 3.2.1.1 is the hypothesis
of the Lemma. By hypothesis we know that all the objects in S are
fi-good as objects of ~~~~ff. Hence S S, S being the subcategory of
all fi-good objects.
Proof that S satisfies 3.2.1.2. We need to show that, if {k ; 2 M}
is a collection of fewer than ff objects of S, then the coproduct is in S.
3.3. PERFECT CLASSES 119
Let {X ; 2 } be a collection of objects of T, with of cardinality
< fi. Suppose that we are given a map
a a
k -- - ! X :
2M 2
That is, for each 2 M, we have a map
a
k -- - ! X :
2
Because k 2 S, this map factors as
a ` 2 f{;} a
k --- ! k{;} --- - - - - ! X
2 2
with k{;} 2 ~~~~ff. It follows that our map
a a
k --- ! X
2M 2
factors as
a a a `2;2M f{;} a
k --- ! k{;} --- - - - - - - - !X
2M 2 2M 2
and as ~~~~ffis closed under the formation of coproducts of fewer than
ff of its objects,
a ff
k{;} 2 ~~~~ :
2M
__
|__|
Proof that S is triangulated. Note that the category ~~~~ffis trian-
gulated, in particular closed under suspension. It follows that factoring
a
k --- ! X
2
as a composite
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 ~~~~ffis the same as factoring
a
k --- ! X
2
as a composite
a `2 f a
k --- ! k -- - - - ! X
2 2
120 3. PERFECTION OF CLASSES
with k 2 ~~~~ff. Thus S satisfies S = S.
Now suppose f : k ! l is a morphism in S. Complete it to a triangle
f
k --- ! l --- ! m --- ! k:
We know that k and l are in S. We must prove that so is m.
Let {X ; 2 } be a set of objects of T, with of cardinality < fi.
We deduce an exact sequence
! ! !
a a a
Hom m; X --- ! Hom l; X --- ! Hom k; X
2 2 2
Suppose now that we are given a map
a
m -- h-! X
2
By the exact sequence, this gives a map
a
l --- ! X
2
so that the composite
a
k --- ! l-- - ! X
2
vanishes. But l is in S, meaning it is fi-good; hence there exists a
factorisation
a ` 2 f a
l --- ! l --- - - ! X
2 2
with l 2 ~~~~ff. Furthermore, the composite
a `2 f a
k --- ! l --- ! l --- - - ! X
2 2
vanishes, and hence since k 2 S, we deduce by Lemma 3.3.5 that
f
l --- ! X
factors as
g h
l --- ! m --- ! X
so that the composite
a `2 g a
k --- ! l --- ! l --- - - ! m
2 2
3.3. PERFECT CLASSES 121
vanishes. But then the map
a
l --- ! m
2
factors through m; there is a map
g a
m -- - ! m
2
giving it. Now the composite
` a
g a 2 h
m --- ! m --- - - ! X
2 2
need not agree with the given map
a
m --h-! X :
2
But by construction, the composites with l ! m agree. The difference
therefore factors as a map
a
m --- ! k --- ! X :
2
But k 2 S, and hence the map
a
k --- ! X
2
factors as
a `2 h0 a
k -- - ! m0 --- - - ! X
2 2
with m0 2 ~~~~ff, and the map h factors as
i j
a ` 2 h h0 a
m --- ! {m m0} -- - - - - - - - - - !X
2 2
__
with {m m0} 2 ~~~~ff. |__|
Theorem 3.3.9. Let ff and fi be infinite cardinals. Let T be a
triangulated category satisfying [TR5]. Let {Si; i 2 I} be a family of
fi-perfect classes of T. Recall that if [Si is the union
[
Si;
i2I
__
then T ([Si) is the smallest triangulated subcategory of T containing
[Si, and <[Si> ffis the smallest thick, ff-localising subcategory S T
containing [Si.
122 3. PERFECTION OF CLASSES
Our theorem asserts_that_for any fi-perfect Si's as above, the col-
lection of objects of T ([Si) is a fi-perfect class. Furthermore, for any
infinite ff, the objects of <[Si>ffalso form a fi-perfect class.
Proof: Since the proofs are virtually identical, we will prove the state-
ment about <[Si> ff. By Lemma 3.3.8, it suffices to show that any
k 2 [i2ISi is fi-good. More explicitly, let {X ; 2 } be a collection
of objects of T, where has cardinality < fi. Given any map
a
k --- ! X
2
we must show there is a factorisation
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 <[Si>ff.
So take k 2 [i2ISi. For some i 2 I, k must lie in Si. But then a
map
a
k --- ! X
2
as above must factor as
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 Si, hence k 2 <[Si>ff. 2
Corollary 3.3.10. Let fi be an infinite cardinal. Let T be a trian-
gulated category satisfying [TR5]. Let S be a triangulated subcategory.
The collection of fi-perfect classes Si S of T has a unique maximal
member S. Futhermore, any fi-perfect class R of T, whose objects lie
in S, is contained in the maximal class S.
Proof: Take the collection of {Si; i 2 I} to be the class of all fi-perfect
classes in T, all of whose_objects lie in S. Then by Theorem 3.3.9, the
objects of the category T ([Si) form an fi-perfect class._On the other
hand,_S is triangulated and contains [Si, hence also T ([Si). Thus
T([Si) is contained_in S, is fi-perfect, and contains all the fi-perfect
Si's. Putting S = T ([Si), we clearly have that S is maximal. 2
Definition 3.3.11. Let fi be an infinite cardinal. Let T be a trian-
gulated category satisfying [TR5]. Let S be a triangulated subcategory.
Then the full subcategory whose object class is the maximal fi-perfect
class S of Corollary 3.3.10 will be called Sfi.
3.3. PERFECT CLASSES 123
Corollary 3.3.12. Let S be a triangulated subcategory of a tri-
angulated category T. Suppose T satisfies [TR5]. Let fi be an infinite
cardinal. Then Sfiis triangulated.
Proof: S is a triangulated subcategory containing Sfi, and hence S
contains the minimal triangulated subcategory containing Sfi. That is,
__
Sfi T (Sfi) S:
__
But Sfiis a fi-perfect class. By Theorem 3.3.9, so_is T (Sfi). Because Sfi
is the maximal fi-perfect class, it must contain T (Sfi), hence is equal
to it, hence is triangulated. 2
Corollary 3.3.13. Let S be a thick subcategory of a triangulated
category T satisfying [TR5]. Let fi be an infinite cardinal. Then the
category Sfiis thick.
Proof: Let T be the class of all objects isomorphic to direct summands
of objects of Sfi. Since S contains Sfiand is thick, it must contain T .
We have
Sfi T S:
But T is fi-perfect by Lemma 3.3.3, and by the maximality of Sfiwe
must have T Sfi. Hence the two are equal, and Sfiis thick. 2
Corollary 3.3.14. Let ff and fi be infinite cardinals. Suppose S
is an ff-localising subcategory of a triangulated category T satisfying
[TR5]. That is, S is a thick subcategory closed under the formation of
coproducts of fewer than ff of its objects. Then the subcategory Sfiis
also ff-localising.
Proof: S is ff-localising and contains Sfi, and hence contains ff, the
smallest ff-localising subcategory containing Sfi. We get an inclusion
Sfi ff S:
On the other hand, Sfiis a fi-perfect class. By Theorem 3.3.9, so is
ff. By the maximality of Sfiwe must have
ff Sfi:
Hence the two are equal and Sfiis ff-localising. 2
Remark 3.3.15. If S is fl-perfect, and fl > fi, then S is also fi-
perfect. We deduce that for any S T, Sfl, being fi-perfect, must be
contained in the maximal fi-perfect class Sfi. If R S T, then Rfiis
a fi-perfect class in S, hence contained in the maximal Sfi.
124 3. PERFECTION OF CLASSES
Example 3.3.16. Let T be a triangulated category. Let k be any
object of T. Then the class S = {0; k} of only two objects is @0 perfect.
Given any set of cardinality < @0 (that is, a finite set), and a map
a
k --- ! X
2
we can factor it as
a ` 2 f a
k --- ! k -- - - - ! X
2 2
where is the diagonal map from k to kn. If the composite vanishes,
then in fact each f must vanish, and the map factors as
a a a
k --- ! k --- ! 0 --- ! X :
2 2 2
It follows that if S is any triangulated subcategory of T, then S@0 = S.
The case fi = @0 is the trivial case for perfection.
3.4. History of the results in Chapter 3
The results of Section 3.2 are very standard. The definition of
localising subcategories (Definition 3.2.6) is probably due to Bousfield,
[4 ], [6 ] and [5 ]. Let T be a triangulated category satisfying [TR5]. As in
Definition 3.2.1, let ~~~~fibe the smallest fi-localising subcategory of T
containing the set S of objects. The fact that ~~~~fiis essentially small
(Proposition 3.2.5) is obvious. The fact that quotients by localising
subcategories respect coproducts (Corollary 3.2.11) may be found in
Bokstedt-Neeman [3 ].
Section 3.3 is completely new. This book introduces the notion of
perfect classes to imitate some standard constructions involving trans-
finite induction on the number of cells of a complex. Whatever the
motivation, the definition is new, and in the rest of the book, we will
attempt to explain what one can do with it.
CHAPTER 4
Small objects, and Thomason's localisation
theorem
4.1. Small objects
Definition 4.1.1. Let T be a triangulated category satisfying [TR5]
(that is, coproducts exist). Let ff be an infinite cardinal. An object k 2 T
is called ff-small if, for any collection {X ; 2 } of objects of T, any
map
a
k --- ! X
2
factors through some coproduct of cardinality strictly less than ff. In
other words, there exists a subset 0 , where the cardinality of 0 is
strictly less than ff, and the map above factors as
a a
k --- ! X -- - ! X :
20 2
Example 4.1.2. The special case where ff = @0 is of great interest.
An object k 2 T is called @0-small if for any infinite coproduct in T,
say the coproduct of a family {X ; 2 } of objects of T, any map
a
k --- ! X
2
factors through a finite coproduct. That is, there is a finite subset
{X1; X2; : :;:Xn} {X ; 2 }
and a factorisation
an a
k --- ! Xi --- ! X :
i=1 2
Expressing this still another way, the natural map
!
a a
Hom(k; X ) --- ! Hom k; X
2 2
is an isomorphism.
125
126 4. THOMASON'S LOCALISATION
Definition 4.1.3. Let ff be an infinite cardinal. Let T be a trian-
gulated category satisfying [TR5]. The full subcategory whose objects
are all the ff-small objects of T will be denoted T(ff).
Lemma 4.1.4. Let ff be an infinite cardinal. Let T be a triangulated
category satisfying [TR5]. Then the subcategory T(ff) T is triangu-
lated.
Proof: To begin with, observe that k 2 T(ff)if and only if k 2 T(ff).
This comes about from the identity
! ( ) !
a a
Hom k; X = Hom k; -1 X
2 2 !
a
= Hom k; -1X
2
where the second equality is the fact that the suspension functor re-
spects coproducts, i.e. Proposition 1.1.6.
Let k ! l be a morphism in T(ff). It may be completed to a triangle
in T. There is a triangle in T
k ! l ! m ! k:
We know that k and l are ff-small. To show that T(ff)is triangulated,
we need to show that so is m.
Let {X ; 2 } be a set of objects of T. Because
!
a
Hom -; X
2
is a cohomological functor on T, we deduce an exact sequence
! ! !
a a a
Hom m; X --- ! Hom l; X --- ! Hom k; X
2 2 2
Suppose now that we are given a map
a
m -- h-! X
2
By the exact sequence, this gives a map
a
l --- ! X
2
so that the composite
a
k --- ! l-- - ! X
2
4.1. SMALL OBJECTS 127
vanishes. But l is ff-small; hence there exists a subset 0 of
cardinality < ff, so that the map
a
l --- ! X
2
factors as
a a
l --- ! X --- ! X :
20 2
Of course, we know that the composite
a a
k --- ! l --- ! X --- ! X
20 2
vanishes. On the other hand, the map
a a
X --- ! X
20 2
is the inclusion of a direct summand, hence a monomorphism in T. We
deduce that the composite
a
k -- - ! l -- - ! X
20
is already zero, and therefore that
a
l-- - ! X
20
factors through m. We therefore have produced a map
g a
m --- ! X
20
so that the composite
g a a
m --- ! X --- ! X
20 2
differs from the map
a
m -- h-! X
2
by a map vanishing on l. On the other hand, the exact sequence
! ! !
a a a
Hom k; X --- ! Hom m; X -- - ! Hom l; X
2 2 2
tells us that the difference factors as
a
m --- ! k --- ! X :
2
128 4. THOMASON'S LOCALISATION
But k 2 T(ff)implies k 2 T(ff), and we deduce that there is a subset
00 , of cardinality < ff, so that the map
a
k --- ! X
2
factors as
a a
k --- ! X -- - ! X :
200 2
Putting this all together, one easily deduces that the given map
a
m -- h-! X
2
factors as
a a
m -- - ! X --- ! X
20[00 2
and because ff is infinite and 0 and 00are of cardinality < ff, the
cardinality of 0[ 00is also < ff. Therefore T(ff)is triangulated. 2
Lemma 4.1.5. Suppose ff is a regular cardinal. That is, ff is not
the sum of fewer than ff cardinals, each of which is less than ff. Then
T(ff)is ff-localising. That is, the coproduct of fewer than ff objects of
T(ff)is an object of T(ff).
Proof: Let {k ; 2 M} be a collection of objects in T(ff), where the
cardinality of M is less than ff. Let {X ; 2 } be an arbitrary
collection of objects of T. Suppose we are given a map
a a
k -- - ! X :
2M 2
This means that, for every 2 M, we are given a map
a
k -- - ! X :
2
Because k is ff-small, for each there exists an , with cardi-
nality < ff, so that the map
a
k --- ! X
2
factors as
a a
k -- - ! X --- ! X :
2 2
4.1. SMALL OBJECTS 129
Thus the map
a a
k --- ! X
2M 2
factors as
a a a
k -- - ! X -- - ! X ;
2M 2[2M 2
and the cardinality of [2M is bounded by the sum of the cardinal-
ities of over all 2 M, which is a sum of fewer than ff cardinals,
each less than ff. Because ff is regular, this sum is less than ff. 2
Lemma 4.1.6. Let ff be an infinite cardinal. The category T(ff)is
thick.
Proof: By Lemma 4.1.4 we know that T(ff)is a triangulated subcate-
gory. To prove it thick, we need to show that any direct summand of
an object in T(ff)is in T(ff).
Let k, l be objects of T and assume that the direct sum k l is
ff-small. We wish to show that k is ff-small. Take any map
a
k --h-! X
2
and consider the map
i j
h 0 a
k m --- - - - ! X
2
Because k m is small, there is a subset 0 of cardinality < ff, so
that the above factors as
a a
k m --- ! X --- ! X
20 2
and hence the given map
a
k --h-! X
2
factors as
0 1
@ 1 A
0 a a
k --- - ! k m -- - ! X -- - ! X
20 2
and in particular, it factors through a coproduct of fewer than ff terms.
2
130 4. THOMASON'S LOCALISATION
Remark 4.1.7. If ff is a regular cardinal greater than @0, Lemma 4.1.6
is redundant. By Lemma 4.1.5, T(ff)is ff-localising. But since ff > @0,
Remark 3.2.7 tells us that idempotents split in T(ff), and T(ff)must be
thick.
4.2. Compact objects
Let T be a triangulated category satisfying [TR5]. In Section 4.1 we
learned how to construct, for each infinite cardinal ff, a triangulated
subcategory T(ff)of ff-small objects in T. In Section 3.3, we learned
that given any triangulated subcategory S T and an infinite cardinal
fi, there is a way to construct a triangulated subcategory Sfi S. In
this section, the idea will be to combine the constructions and study
{T(ff)}fi.
Lemma 4.2.1. Let T be a triangulated category satisfying [TR5].
Let ff be an infinite cardinal. Let S be an ff-perfect class of ff-small
objects. Then S is also fi-perfect for all infinite fi.
Proof: Suppose k is an object in S, and {X ; 2 } a family of fewer
than fi objects of T. Because k is ff-small, any map
a
k --- ! X
2
factors as
a a
k -- - ! X --- ! X
20 2
with the cardinality of 0 being ff. Since S is ff-perfect, the map
a
k --- ! X
20
factors as
a ` 20 f a
k -- - ! k --- - - ! X
20 20
with k 2 S. For 62 0, define k = 0. Then we deduce a factorisation
a ` 2 f a
k --- ! k --- - - ! X ;
2 2
where of course most of the k 's vanish, but in any case they are all in
S.
Suppose now that we have a vanishing composite
a `2 f a
k --- ! k --- - - ! X
2 2
4.2. COMPACT OBJECTS 131
with k and k all in S and of cardinality < fi. Because k is ff-small,
the map
a
k -- - ! k
2
must factor as
a a
k -- - ! k --- ! k
20 2
where the cardinality of 0 is < ff. The composite
a ` 20 f a
k -- - ! k --- - - ! X
20 20
vanishes, and since S is ff-perfect, we deduce that for each 2 0, the
map f : k -! X factors as
g h
k -- - ! l -- - ! X
so that l 2 S and the composite
a ` 20 g a
k --- ! k -- - - - ! l
20 20
already vanishes. For 62 0, define g : k - ! l to be the identity.
Then we still have the vanishing of
a `2 g a
k -- - ! k --- - - ! l :
2 2
2
Definition 4.2.2. Let T be a triangulated category satisfying [TR5].
Let ff be an infinite cardinal. Define a triangulated subcategory Tffby
(ff)
Tff= T ff:
Lemma 4.2.3. If ff < fi are infinite cardinals, then Tff Tfi.
Proof: Tff is an ff-perfect class, whose objects are ff-small. By
Lemma 4.2.1, Tff must be fi-perfect. Since its objects are ff-small,
they are also fi-small. Thus Tffis a fi-perfect class in T(fi), hence it
must be contained in the maximal one, T(fi)fi. 2
Lemma 4.2.4. For every infinite ff, Tffis thick.
Proof: By Lemma 4.1.6, T(ff)is thick. By Corollary 3.3.13, for every
infinite cardinal fi, T(ff)fiis also thick. In particular, letting fi = ff,
Tffis thick. 2
132 4. THOMASON'S LOCALISATION
Lemma 4.2.5. Let ff be a regular cardinal. Then Tffis ff-localising.
Proof: ff is regular, and hence Lemma 4.1.5 says that T(ff)is ff-
localising. But then Corollary 3.3.14 asserts that, for any infinite fi,
T(ff)fiis also ff-localising. Letting fi = ff, we deduce that Tff is
ff-localising. 2
Remark 4.2.6. In the special case ff = @0, all classes are ff-
perfect; see Example 3.3.16. Thus
(@ ) (@ )
T 0 @0 = T 0 :
In other words,
T@0 = T(@0):
Definition 4.2.7. The objects of Tffwill be called the ff-compact
objects of T. In the case ff = @0, the objects of T@0 will be called the
compact objects. They are fi-compact for any infinite fi. We will per-
mit ourselves to write Tc for T@0; the superscript c stands for compact.
4.3. Maps factor through ~~~~fi
Reminder 4.3.1. Let T be a triangulated category satisfying [TR5].
Let S be a class of objects of T. We remind the reader of Defini-
tion 3.2.9; ~~~~ stands for the localising subcategory generated by S,
that is
[ fi
~~~~ = ~~~~ :
fi
~~~~fiis the smallest thick subcategory containing S and closed with
respect to forming the coproducts of fewer than fi of its objects; See
Definition 3.2.1.
Lemma 4.3.2. Let T be a triangulated category satisfying [TR5].
Let fi be a regular cardinal. Suppose S is a class of objects of Tfi. That
is, S Tfi. Then the subcategory ~~~~fiis also contained in Tfi.
Proof: If fi is regular, then by Lemma 4.2.5 Tfiis fi-localising. By
hypothesis, S is contained in Tfi. But ~~~~fiis the minimal fi-localising
subcategory containing S, hence ~~~~fi Tfi. 2
The main theorem of this section is the following.
Theorem 4.3.3. Let T be a triangulated category satisfying [TR5].
Let fi be a regular cardinal. Let S be some class of objects in Tfi. Let
x be a fi-compact object of T. Let z be an object of ~~~~ . Suppose
4.3. MAPS FACTOR THROUGH ~~~~fi 133
f : x -! z is a morphism in T. Then there exists an object y 2 ~~~~fi
so that f factors as
x -! y -! z:
Proof: We define a full subcategory S of T as follows. If Ob(S) is the
class of objects of S, then
ae fi fioe
Ob(S) = z 2 Ob(T) | 8x 2 Ob(T ); 8f : x -! z; 9y 2 ~~~~ :
and a factorisation of f asx -! y -! z
It suffices to prove that ~~~~ S. To do this, we will show that S
contains S, is triangulated, and contains all coproducts of its objects.
Since ~~~~ is minimal with these properties (see Example 3.2.8), it will
follow that ~~~~ S.
The fact that S contains S is obvious. Take any objects z 2 S and
x 2 Tfi, and any morphism x -! z. Since we know that z 2 S, clearly
z 2 ~~~~ fi, the smallest thick category containing S and closed with
respect to coproducts of fewer than fi of its objects. Put y = z, and
factor f : x -! z as
f 1
x --- ! z --- ! z:
Equally clearly, z 2 S if and only if z 2 S. After all, x -! z can
be factored as
x -- - ! y --- ! z
if and only if -1x -! z can be factored as
-1x --- ! -1y -- - ! z
and x 2 Tfiiff -1x 2 Tfi, y 2 ~~~~fiiff -1y 2 ~~~~fi.
Suppose now that OE : z -! z0 is a morphism in S. Complete it to
a triangle
z --- ! z0 -- - ! z00-- - ! z:
We know that z and z0 are in S. To show that S is triangulated, we
must establish that z00is also in S. Choose any x 2 Tfi, and any map
f : x -! z00. We need to factor it as
x -! y -! z00;
with y 2 ~~~~fi.
First of all, the composite
x -! z00-! z
gives a map from x 2 Tfito z 2 S, which must factor as
x -! y -! z;
134 4. THOMASON'S LOCALISATION
with y 2 ~~~~fi. Now the composite
x -! z00-! z -! z0
clearly vanishes, and is equal to the composite
x -! y -! z -! z0:
Complete x -! y to a triangle
x -! y -! C -! x;
then the map y -! z0 must factor through C. We deduce a commu-
tative square
y --- ! C
? ?
? ?
y y
z --- ! z0:
Now, in the triangle defining C, the other two objects are x and y. By
hypothesis, x 2 Tfi. By construction, y 2 ~~~~fi, and by Lemma 4.3.2,
~~~~fi Tfi. Because x and y are both in Tfi, so is C. Since z0 2 S, we
deduce that the map C -! z0 factors as
C -! y0 -! z0
with y0 2 ~~~~fi. Our commutative square above gets replaced by an-
other,
y --- ! y0
? ?
? ?
y y
z --- ! z0;
where the top row involves only objects in ~~~~fi. Note also that since
the composite
x -! y -! C
vanishes, so does the longer composite
x -! y -! C -! y0:
Now complete the commutative square
y -- - ! y0
? ?
? ?
y y
z -- - ! z0
4.3. MAPS FACTOR THROUGH ~~~~fi 135
to a map of triangles
-1y00 --- ! y -- - ! y0 --- ! y00
? ? ? ?
? ? ? ?
y y y y
z00 --- ! z -- - ! z0 --- ! z00
and note that, because y and y0 are in ~~~~fi, so is y00, and because the
composite
x -! y -! y0
vanishes, the map x -! y factors as
x -! -1y00-! y:
We deduce a commutative diagram
x --- ! -1y00 --- ! y
? ?
? ?
y y
z00 --- ! z:
The composite
x -! -1y00-! z00
is not the map f : x -! z00we began with. But when we compose
x -! -1y00-! z00-! z
we do get the given map
f 00
x -- - ! z --- ! z:
In other words, the difference between the composite
x -! -1y00-! z00
and f : x -! z00factors as x -! z0 -! z00. We know that z0 2 S, and
hence x -! z0 must factor as
x -! __y-! z0
with __y2 ~~~~fi. But then f : x -! y factors as
x --- ! {__y -1y00} --- ! z00;
and __y -1y00is in ~~~~fi. Thus z002 S, as required.
It remains to prove that S is closed with respect to the formation
of coproducts of its objects. Let {z ; 2 } be a set of objects of S.
We wish to show that
a
z
2
136 4. THOMASON'S LOCALISATION
is an object of S. Rephrasing this again, we wish to show that if x 2 Tfi
and
a
f : x -! z
2
is any map, then there is a factorisation
a
x --- ! y --- ! z
2
with y 2 ~~~~fi.
Take therefore any map
a
f : x -! z :
2
Now recall that x 2 Tfi T(fi). In particular, x is fi-small. Therefore
there is a subset 0 , where the cardinality of 0 is less than fi, so
that f factors as
g a a
x --- ! z -- - ! z :
20 2
Now x is in fact not only fi-small, but also fi-compact. In other words,
x belongs to the fi-perfect class Tfi. Therefore the map g factors as
a `20 h a
x --- ! x --- - - ! z
20 20
for some collection of h : x -! z , with x in Tfi.
But, for each 2 0, we have x 2 Tfi, z 2 S and a map h :
x -! z . It follows that, for each , we may choose a y 2 ~~~~fiand
a factorisation of h : x -! z as
x -! y -! z ;
with y 2 ~~~~fi. But then f factorises as
a a
x --- ! y -- - ! z ;
20 2
` fi
and 20 y , being a coproduct of fewer than fi objects of ~~~~ , must
lie in ~~~~fi. 2
4.4. MAPS IN THE QUOTIENT 137
4.4. Maps in the quotient
As we have said in Section 2.2, one of the problems with Verdier's
construction of the quotient is that one ends up with a category in
which the Hom-sets need not be small. Suppose T is a triangulated
category with small Hom-sets, and S is a triangulated subcategory.
Then the quotient T=S of Theorem 2.1.8 is a category, which in general
need not have small Hom-sets. Nevertheless, we can already give one
criterion that guarantees the smallness of the Hom-sets. The criterion
is Corollary 4.4.3.
Then we will further explore some of the consequences of the ma-
chinery that has been developed so far. We lead up to Thomason's
localisation theorem (Theorem 4.4.9), which will give a summary of
the results in this Section.
Proposition 4.4.1. Let T be a triangulated category. Let fi be a
regular cardinal. Let S be a subclass of the objects of Tfi. Let y 2 T be
an arbitrary object, x 2 T a fi-compact object (i.e. x 2 Tfi). Then
n o
{T=~~~~ } (x; y) = T=~~~~ fi (x; y):
In other words, the maps x -! y are the same in the Verdier quotient
categories T=~~~~ and T=~~~~ fi.
Proof: There is a natural map
n o
OE : T=~~~~ fi (x; y) -! {T=~~~~ }(x; y);
and we want to prove it an isomorphism. We need to show it injective
and surjective. Let us begin by proving it surjective.
Proof that OE is surjective. Let x -! y be a morphism in T=~~~~ .
That is, an equivalence class of diagrams
p --ff-! y
?
f?y
x
with f in Mor~~~~. In the triangle
p -! x -! z -! p
we must have z 2 ~~~~. On the other hand, from the hypothesis of the
Lemma, x 2 Tfi, S Tfiand z 2 ~~~~. By Theorem 4.3.3 we know then
that there is a z0 2 ~~~~fiso that x -! z factors as
x -! z0 -! z:
138 4. THOMASON'S LOCALISATION
We deduce a map of triangles
fg 0 0
p0 -- - ! x --- ! z --- ! p
? ? ? ?
g?y 1?y ?y ?y
f
p -- - ! x --- ! z --- ! p
The morphism fg : p0 -! x lies in Mor~~~~fisince z0 2 ~~~~ fi. The
diagram
ffg
p0 --- ! y
?
fg?y
x
is therefore a morphism in T=~~~~ fi, whose image under OE is clearly
equivalent to the given morphism
p --ff-! y
?
f?y
x
Hence the surjectivity of OE.
Proof that OE is injective. Let the diagram
p --ff-! y
?
f?y
x
represent a morphism in T=~~~~ fiwhose image under OE is zero. Because
the diagram is a morphism in T=~~~~ fi, f must be in Mor~~~~fi. In the
triangle
p -! x -! z -! p;
we must have z 2 ~~~~fi. On the other hand, fi is regular, and Lemma 4.3.2
tells us that ~~~~fi Tfi. We were given that x 2 Tfi, and deduce that
p, the third edge of the triangle, must also be in Tfi.
We also assume that the image under OE of the morphism represented
by the diagram
p --ff-! y
?
f?y
x
4.4. MAPS IN THE QUOTIENT 139
vanishes. In other words, the diagram is equivalent, in T=~~~~ , to the
diagram
p --0-! y
?
f?y
x
By Lemma 2.1.26, the map ff : p -! y must factor as
p -! z -! y
with z 2 ~~~~ . But we have just shown that p 2 Tfi. As z 2 ~~~~ , we
conclude from Theorem 4.3.3 that p ! z factors as
p -! z0 -! z
with z0 2 ~~~~fi. Since we have factored p ! y through z0 2 ~~~~fi, it
follows, again from Lemma 2.1.26, that the class of
p --ff-! y
?
f?y
x
vanishes already in T=~~~~ fi. 2
An immediate corollary is
Corollary 4.4.2. With the notation as in Proposition 4.4.1, the
natural functor
Tfi=~~~~ fi-! T=~~~~
is fully faithful.
Proof: Proposition 4.4.1 asserts that for all x 2 Tfi, y 2 T,
n o
{T=~~~~ } (x; y) = T=~~~~ fi (x; y):
Corollary 4.4.2 is the weaker assertion that this holds if y 2 Tfias well.
2
All of this becomes useful when we have T = [fiTfi. We then know
Corollary 4.4.3. Suppose T is a triangulated category with small
Hom-sets, satisfying [TR5]. Suppose T = [fiTfi; that is, every object
of T is fi-compact for some fi. Suppose S is a set of objects in Tff, for
some infinite cardinal ff. Suppose T has small Hom-sets. Then the
category T=~~~~ also has small Hom-sets.
140 4. THOMASON'S LOCALISATION
Proof: Let x and y be objects of T, we need to show that
{T=~~~~ }(x; y)
is a set. But by hypothesis, T = [fiTfi. Therefore x must lie in some
Tfi. Recall that if fi < fl, then by Lemma 4.2.3, Tfi Tfl. We may
therefore choose fi so that
4.4.3.1. x is fi-compact.
4.4.3.2. fi is regular.
4.4.3.3. fi > ff, where S Tff, ff @0 given.
Then x and S both lie in Tfi, and by Proposition 4.4.1,
n o
{T=~~~~ } (x; y) = T=~~~~ fi (x; y):
On the other hand, S was a set, and Proposition 3.2.5 gives us that the
category ~~~~fiis essentially small. But by Proposition 2.2.1, the Verdier
quotient of T by ansmall (oroessentially small) category has small Hom-
sets. Therefore T=~~~~ fi (x; y) is a small set; { T=~~~~ }(x; y), being
equal to it, is also a set. 2
The situation of Corollary 4.4.3 is worth analysing more closely.
Lemma 4.4.4. Suppose T is a triangulated category satisfying [TR5].
Suppose T = [fiTfi; that is, every object of T is fi-compact for some fi.
Suppose S is a class of objects in Tff, for some infinite cardinal ff. Then
for any regular fi ff, the image of Tfiin T=S satisfies
Tfi {T=~~~~ }fi:
In other words, every object of Tfiis fi-small even in T=~~~~ , and in
fact the class Tfiis a fi-perfect class of T=~~~~ , hence contained in the
maximal one inside {T=~~~~ }(fi).
Proof: We need to show that Tfiis consists of fi-small objects of
T=~~~~ , and that Tfiis fi-perfect in T=~~~~ . Let k be an object of Tfi. Let
{X ; 2 } be a set of objects of T=~~~~ , which of course are the same
as objects of T. Let us be given a map in T=~~~~
a
k --- ! X :
2
To show that k is fi-small in T=~~~~ , we need to factor the map as
a a
k -- - ! X --- ! X
20 2
4.4. MAPS IN THE QUOTIENT 141
for some subset 0 of cardinality < fi.
To show further that Tfiis a fi-perfect, note first that the objects
of Tfiform a subcategory equivalent to a triangulated subcategory. By
Corollary 4.4.2, the subcategory
Tfi=~~~~ fi T=~~~~
is a full subcategory, and since it is closed with respect to the formation
of triangles it is equivalent to a triangulated subcategory of T=~~~~ . By
Remark 3.3.6, to check that the objects of a triangulated subcategory
form a fi-perfect class, one needs only show that each object is fi-good.
In other words, we need to further prove that the map
a
k --- ! X
20
can be factored as
a ` 20 f a
k -- - ! k --- - - ! X
20 20
with k 2 Tfi.
Let us begin therefore with a map in T=~~~~
a
k --- ! X :
2
`
By Corollary 3.2.11, the coproduct 2 X exists in T=~~~~ ; in fact, it
agrees with the coproduct in T. We are given a map in T=~~~~
a
k --- ! X :
2
`
It is a map in T=~~~~ from an object in k 2 Tfito an object 2 X 2 T,
and by Proposition 4.4.1,
! !
a n fio a
{T=~~~~ } k; X = T=~~~~ k; X :
2 2
That is, the map
a
k --- ! X
2
142 4. THOMASON'S LOCALISATION
comes from a morphism in T=~~~~ fi. It therefore is represented by a
diagram
a
p -- - ! X
? 2
f?y
k
where f 2 Mor~~~~fi. This means that there is a triangle
p -! k -! z -! p
with z 2 ~~~~ fi Tfi. Since k is also assumed in Tfi, it follows that
p 2 Tfi. But then the fi-smallness of p 2 T guarantees that the map in
T
a
p -! X
2
factors as
a a
p -! X -! X
20 2
where 0 has cardinality less than fi. The fi-compactness of p in
T says that the map in T
a
p -! X
20
factors as
a a
p -! k -! X
20 20
with k 2 Tfi. In other words, in T=~~~~ we factored the map
a
k -! X
2
as
a a a
k -! k -! X -! X :
20 20 2
2
Even the seemingly stupid case, where ~~~~ = T, is worth considering
further.
4.4. MAPS IN THE QUOTIENT 143
Lemma 4.4.5. Let T be a triangulated category satisfying [TR5].
Let S be a class of objects in Tff, for some infinite ff. Suppose ~~~~ = T.
Let fi be a regular cardinal ff. Then the inclusion ~~~~fi Tfiis an
equality. In other words, every object of Tfiis in ~~~~fi.
Proof: Let x be an object of Tfi. We need to prove that x is in ~~~~fi.
The identity map 1 : x -! x is a morphism from x 2 Tfito x 2 ~~~~
(we are assuming ~~~~ = T). By Theorem 4.3.3, it factors through some
object y 2 ~~~~fi. Thus x is a direct summand of y 2 ~~~~fi. But ~~~~fiis
thick, hence x 2 ~~~~fi. 2
Proposition 4.4.6. Let T be a triangulated category satisfying [TR5].
Let T Tffand S Tffbe two classes of ff-compact objects, ff an in-
finite cardinal. Suppose that = T. Suppose that fi ff is a regular
cardinal. Then the inclusion
Tfi=~~~~ fi {T=~~~~ }fi
is almost an equivalence; every object of {T=~~~~ }fiis isomorphic to a
direct summand of something in the image. That is, the fi-compact ob-
jects of T=~~~~ are, up to splitting idempotents, the images of fi-compact
objects of T.
Proof: Note that the map T -! T=~~~~ takes Tfito fi-compact objects
of T=~~~~ , by Lemma 4.4.4. Hence there is a well defined map
Tfi=~~~~ fi-! {T=~~~~ }fi:
The fact that this map is fully faithful is a consequence of Corol-
lary 4.4.2. We may therefore view
Tfi=~~~~ fi {T=~~~~ }fi;
________
as a fully faithful embedding of categories. Let Tfi=~~~~ fibe the thick
closure of Tfi=~~~~ fi. Since {T=~~~~ }fiis a thick subcategory of T=~~~~
containing Tfi=~~~~ fi, we deduce
________
Tfi=~~~~ fi {T=~~~~ }fi:
We need to prove the opposite inclusion,
________
Tfi=~~~~ fi {T=~~~~ }fi:
Now Tfiis a triangulated subcategory of T, and contains all coprod-
ucts of fewer that fi of its objects. By Lemma 3.2.10, coproducts_in
T and T=~~~~ agree. Therefore Tfi=~~~~ fi, and hence also Tfi=~~~~ fi, are
closed under the formation of coproducts in T=~~~~ of fewer than fi of
144 4. THOMASON'S LOCALISATION
________
their objects. Then Tfi=~~~~ fiis thick, and contains the coproducts of
fewer than fi of its objects. Since T Tff Tfi= Tfi=~~~~ fi,
________
Tfi=~~~~ fi fi
where fi T=~~~~ is the smallest fi-localising, thick subcategory
containing T .
On the other hand, we assume that = T; that is, T is the smallest
localising subcategory of T containing T . We deduce that T=~~~~ is the
smallest localising subcategory of T=~~~~ containing T . If we view T as
a subclass of T=~~~~ , we still get = T=~~~~ : By Lemma 4.4.5, applied
to the class T in the category T=~~~~ , we get
fi= {T=~~~~ }fi:
Hence
________
Tfi=~~~~ fi {T=~~~~ }fi:
Thus the two subcategories are equal, as stated. 2
Remark 4.4.7. In Proposition 4.4.6, the statement can be im-
proved if fi is not only regular but also fi > @0. Since in that case,
Tfi=~~~~ fiis closed under the formation of coproducts of countably many
objects, idempotents in Tfi=~~~~ fimust split. Therefore, if fi > @0 then
every object of {T=~~~~ }fiis isomorphic in T=~~~~ to an object in Tfi=~~~~ fi.
There is no need to split idempotents; the categories {T=~~~~ }fiand
Tfi=~~~~ fiagree, up to extending Tfi=~~~~ fito include every object in T=~~~~
isomorphic to an object of Tfi=~~~~ fi.
Lemma 4.4.8. Let T be a triangulated category satisfying [TR5].
Let S Tff be a class of ff-compact objects, ff an infinite cardinal.
Let S = ~~~~ be the localising subcategory generated by S. Suppose that
fi ff is a regular cardinal. Then there is an inclusion
S \ Tfi Sfi:
Proof: We will show that S \ Tfiis a fi-perfect class of objects in S(fi);
hence it must be contained in the maximal such, Sfi.
Let k 2 S\Tfibe any object. Let {X ; 2 } be a family of objects
in S = ~~~~. Suppose we are given a map
a
k --- ! X :
2
4.4. MAPS IN THE QUOTIENT 145
Because k 2 Tfi, it is fi-small in T, and hence there must be a subset
0 of cardinality < fi, so that the map factors as
a a
k -- - ! X -- - ! X :
20 2
This proves that k is fi-small in S.
Now consider a map
a
k --- ! X :
2
Because k 2 Tfi, this map must factor as
a `2 f a
k --- ! k --- - - ! X
2 2
with k 2 Tfi. On the other hand, f : k - ! X is a map from an
object of Tfito an object of ~~~~. We know from Theorem 4.3.3 that it
must factor as
k -! k0 -! X
where k0 2 ~~~~fi S \ Tfi. Thus we have factored
a
k --- ! X
2
as
a a
k --- ! k0 --- ! X :
2 2
Since S \ Tfiis triangulated, it follows from Remark 3.3.6 that this is
all we need to check. 2
Summarising the work of this Section, we get
Theorem 4.4.9. Let S be a triangulated category satisfying [TR5],
R S a localising subcategory. Write T for the Verdier quotient S=R.
Suppose there is an infinite cardinal ff, a class of objects S Sff
and another class of objects R R \ Sff, so that
R = and S = ~~~~:
Then for any regular fi ff,
fi= Rfi= R \ Sfi;
~~~~ fi= Sfi:
146 4. THOMASON'S LOCALISATION
The natural map
Sfi=Rfi-! T
factors as
Sfi=Rfi-! Tfi T;
and the functor
Sfi=Rfi-! Tfi
is fully faithful. If fi > @0, the functor
Sfi=Rfi-! Tfi
is an equivalence of categories. If fi = @0, then every object of Tfiis a
direct summand of an object in Sfi=Rfi.
Proof: From Lemma 4.4.8 we deduce an inclusion R \ Sfi Rfi. Triv-
ially, we have an inclusion fi R\Sfi. Combining, we have inclusions
fi R \ Sfi Rfi:
By Lemma 4.4.5 applied to R R \ Sfi Rfi, we have
fi= Rfi:
Hence equality must hold throughout, and we have
fi= R \ Sfi= Rfi:
By Lemma 4.4.5 applied to S Sffwe have
~~~~ fi= Sfi:
That the natural map
Sfi=Rfi-! T
factors as
Sfi=Rfi-! Tfi T
is the statement that the image of a fi-compact object of S is fi-compact
in T, that is Lemma 4.4.4. That the functor
Sfi=Rfi-! Tfi
is fully faithful is Corollary 4.4.2. That the functor
Sfi=Rfi-! Tfi
is an equivalence if fi > @0 is Remark 4.4.7. The statement that, for
fi = @0, every object of Tfiis a direct summand of an object of the full
subcategory Sfi=Rfifollows from Proposition 4.4.6. 2
4.5. A REFINEMENT IN THE COUNTABLE CASE 147
4.5. A refinement in the countable case
The classical, most useful case of Thomason's localisation theorem
is the case fi = @0. As stated, the theorem says that S@0=R@0 is embed-
ded fully faithfully in T@0, and that the embedding is an equivalence
up to splitting idempotents. But one gets a refinement, which we will
discuss in this section. We begin with some definitions.
Definition 4.5.1. Let T be an essentially small category. Define
Z(T) to be the free abelian group on isomorphism classes of the objects
of T. The object X of T, viewed as an element in Z(T), will be denoted
[X].
Definition 4.5.2. Let T be an essentially small additive category.
One defines a group A(T) Z(T) to be the subgroup generated by all
[X Y ] - [X] - [Y ]
where X, Y are objects of T.
Definition 4.5.3. Let T be an essentially small additive category.
Then the set of isomorphism classes of objects of T forms an abelian
semigroup, with addition being given by direct sum. Define KA (T) to
be the group completion of T with respect to this addition. That is,
elements of KA (T) are equivalence classes of formal differences [X] -
[Y ], where [X] and [Y ] are isomorphism classes of objects of T. We
declare [X] - [Y ] equivalent to [X0] - [Y 0] if, for some object P 2 T,
there is an isomorphism
X Y 0 P ' X0 Y P:
One proves easily that this is an equivalence relation, and that KA (T)
is naturally an abelian group.
Remark 4.5.4. This is usually called the Grothendieck group of
the symmetric monoidal category T. The subscript A here is to distin-
guish KA (T) from K0(T), whose definition will come in Definition 4.5.8.
Lemma 4.5.5. Let T be an essentially small additive category. Then
there is a natural isomorphism
Z(T)_
-! KA (T):
A(T)
Proof: Define a map
Z(T) -! KA (T)
148 4. THOMASON'S LOCALISATION
to be the following. It sends an element of Z(T), that is a linear com-
bination
Xn Xm
[Xi] - [Yj];
i=1 j=1
to the element "
Mn # "Mm #
Xi - Yj :
i=1 j=1
This is clearly surjective, and the kernel is A(T); hence the identification
Z(T)_
' KA (T):
A(T)
2
Definition 4.5.6. Let T be an essentially small triagulated cate-
gory. One defines a subgroup T (T) Z(T) to be the subgroup generated
by sums
[Y ] - [X] - [Z]
where
X -! Y - ! Z -! X
is a triangle in T.
Lemma 4.5.7. Let T be an essentially small triangulated category.
Then A(T) T (T).
Proof: Let X and Y be objects of T. Then the following is a triangle
X -! X Y - ! Y - ! X
establishing that
[X Y ] - [X] - [Y ]
is in T (T). 2
Definition 4.5.8. Let T be an essentially small triagulated cate-
gory. One defines a group K0(T), the Grothendieck group of T, to be
Z(T)
K0(T) = _____:
T (T)
Remark 4.5.9. In particular, from the triangle
X -! 0 -! X -! X
in T, we learn that [X] + [X] vanishes in K0(T).
4.5. A REFINEMENT IN THE COUNTABLE CASE 149
Lemma 4.5.10. Suppose T is an essentially small triangulated cat-
egory, S T a triangulated subcategory. Suppose T is the thick closure
of S. That is, every object of T is a direct summand of an object of S.
Then in Z(T), which of course contains Z(S), one gets
T (T) = A(T) + T (S):
Proof: Clearly, A(T) and T (S) are both subgroups of T (T), and hence
T (T) A(T) + T (S):
The problem is to show the reverse inclusion. Choose any of the gen-
erators of T (T), that is
[Y ] - [X] - [Z]
where
X -! Y - ! Z -! X
is a triangle in T. By hypothesis, T is the thick closure of S; there is an
object A 2 T so that X A 2 S, and there is an object B 2 T so that
Z B 2 S. We deduce a triangle in T
X A -! Y A B -! Z B -! {X A} ;
and as X A 2 S and Z B 2 S and S is triangulated, the entire
triangle lies in S. Hence
[Y A B] - [X A] - [Z B]
is an element of T (S). As
[Y A B] - [Y ] - [A] - [B]; [X A] - [X] - [A];
[Z B] - [Z] - [B]
all lie in A(T), the identity
[Y ] - [X] - [Z] = {[Y A B] - [X A] - [Z B]}
- {[Y A B] - [Y ] - [A] - [B]}
+ {[X A] - [X] - [A]}
+ {[Z B] - [Z] - [B]}
shows that an arbitrary generator [Y ] - [X] - [Z] of T (T) lies in A(T) +
T (S). Hence
T (T) A(T) + T (S);
and we are done. 2
150 4. THOMASON'S LOCALISATION
Proposition 4.5.11. Let T be an essentially small triangulated
category. Let S T be a triangulated subcategory. Suppose the thick
closure of S is all of T. Then the natural map K0(S) - ! K0(T) is
a monomorphism. Furthermore, if X 2 T is an object so that [X] 2
K0(T) lies in the image of K0(S) -! K0(T), then X 2 S.
Proof: We need to show that the map f0 : K0(S) -! K0(T) is injec-
tive, and analyse when an element [X] 2 K0(T) lies in the image of f0.
But f0 is identified as the map
Z(S)_ Z(T) Z(T)
- ! _____ = ____________ :
T (S) T (T) A(T) + T (S)
We are using the fact that T (T) = A(T) + T (S), that is Lemma 4.5.10.
It therefore suffices to show that the map
Z(S) Z(T)
fA : _____ -! _____
A(S) A(T)
is injective. The map f0 is obtained from fA by further dividing by
T (S)=A(S). In other words, we are reduced to showing the injectivity
of
fA : KA (S) -! KA (T)
and analysing its cokernel, which is isomorphic to the cokernel of f0 :
K0(S) -! K0(T).
Let [X] - [Y ] be an element of the kernel of fA ; that is, X and Y
are objects of S, and in T there exists an object P and an isomorphism
X P ' Y P:
But since T is the thick closure of S, there must exist an object P 02 T
so that P P 02 S. Then the fact that there is an isomorphism
X P P 0' Y P P 0
says that [X]-[Y ] vanishes already in KA (S); the kernel of fA vanishes.
Finally, assume that X is an object of T so that [X] lies in the image
of the map f0 : K0(S) -! K0(T), or equivalently in the image of the
map fA : KA (S) - ! KA (T). Then there exist B and C in S and an
identity in KA (T)
[X] = [B] - [C]:
This means that there is an object P 2 T and an isomorphism
X C P ' B P:
Find an object P 02 T so that P P 0lies in S. We have an isomorphism
X C P P 0' B P P 0:
4.5. A REFINEMENT IN THE COUNTABLE CASE 151
Replacing C 2 S by C P P 02 S and B 2 S by B P P 02 S, we
may say that there are objects A and B in S and an isomorphism
X C ' B:
Now consider the triangle
C -! B -! X -! C:
It is a triangle in T, but since C and B are in S and S is triangulated,
the triangle lies in S. Hence X 2 S. 2
A very useful special case of this is
Corollary 4.5.12. Let S be a triangulated subcategory of a tri-
angulated category T. Suppose T is the thick closure of S. Then for any
object X 2 T, the object X X lies in S.
Proof: If T is small, this is an immediate corollary of Lemma 4.5.13,
once one observes that X X vanishes in K0(T). One can reduce
the general case to the essentially small case; but since a direct proof
is very simple, let us give one.
Since X 2 T and T is the thick closure of S, there exists Y in T
with {X Y } 2 S. The suspension of {X Y } is also in S; that is
{X Y } 2 S. There are three distinguished triangles in T
Y --- ! 0 --- ! Y --- ! Y
X --- ! X --- ! 0 --- ! X
0 --- ! X --- ! X --- ! 0
The direct sum is, by Proposition 1.2.1, a triangle in T
X Y --- ! X X -- - ! X Y -- - ! X Y:
But two of the terms, namely X Y and X Y , lie in S. Hence so
does the third term X X. 2
One more Lemma before we get to the main point.
Lemma 4.5.13. As in Theorem 4.4.9, let S be a triangulated cate-
gory satisfying [TR5]. But unlike Theorem 4.4.9, we now insist that S
have small Hom-sets. Let R S be a localising subcategory. Write T
for the Verdier quotient S=R.
Let fi be a regular cardinal. Suppose there is a set (not just a class,
as in Theorem 4.4.9) of objects S Sfiand another set of objects
R R \ Sfi, so that
R = and S = ~~