# Triangulated Categories. (AM-148)

AMNON NEEMAN
Pages: 449
https://www.jstor.org/stable/j.ctt7zvcqc

1. Front Matter
(pp. i-iv)
(pp. v-2)
3. 0. Acknowledgements
(pp. 3-3)
4. 1. Introduction
(pp. 3-28)

Before describing the contents of this book, let me explain its origins. The book began as a joint project between the author and Voevodsky. The idea was to assemble coherently the facts about triangulated categories, that might be relevant in the applications to motives. Since the presumed reader would be interested in applications, Voevodsky 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,...

5. CHAPTER 1 Definition and elementary properties of triangulated categories
(pp. 29-72)

Definition 1.1.1.Let$\script {C}$be an additive category and$\Sigma:\script{C}\rightarrow \script{C}$be an additive endofuntor of$\script {C}$.Assume throughout that the endofunctorΣis invertible. Acandidate triangle in$\script {C}$(with respect toΣ) is a diagram of the form:$X\overset{u}{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\,Y\overset{\upsilon }{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\,Z\overset{w}{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\,\Sigma X$such that the composites v o u, w o v andΣu o w are the zero morphisms.

A morphism of candidate triangles is a commutative diagram$\begin{matrix} X & \overset{u}{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\, & Y & \overset{\upsilon }{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\, & Z & \overset{w}{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\, & \Sigma X \\ f\big\downarrow & {} & g\big\downarrow & {} & h\big\downarrow & {} & {\Sigma f}\big\downarrow \\ {{X}'} & \overset{{{u}'}}{\mathop{\xrightarrow{\hspace*{0.75cm}}}}\, & {{Y}'} & \overset{{{\upsilon }'}}{\mathop{\xrightarrow{\hspace*{0.75cm}} }}\, & {{Z}'} & \overset{{{w}'}}{\mathop{\xrightarrow{\hspace*{0.75cm}} }}\, & \Sigma {X}' \\ \end{matrix}$where each row is a candidate triangle.

Definition 1.1.2. A pre–triangluated category$\script{T}$is an additive category, together with an additive automorphism Σ, and a class of candidate triangles (with respect to Σ) called...

6. CHAPTER 2 Triangulated functors and localizations of triangulated categories
(pp. 73-102)

Definition 2.1.1.Let$\script {D}_{1}, \script {D}_{2}$be triangulated categories. A triangulated functor$F:\script {D}_{1} \rightarrow \script {D}_{2}$is an additive functor$F:\script {D}_{1} \rightarrow \script {D}_{2}$together with natural isomorphisms$\phi_{X}:F(\Sigma(X)) \longrightarrow \Sigma(F(X))$such that for any distinguished triangle$X \overset{u}\longrightarrow Y \overset{v}\longrightarrow Z \overset{w}\longrightarrow \Sigma X$in$\script {D}_{1}$the candidate triangle$F(X) \overset {F(u)}\longrightarrow F(Y) \overset {F(v)}\longrightarrow F(Z) \overset {\phi_{X}oF(w)}\longrightarrow \Sigma(F(X))$is a distinguished triangle in$\script {D}_{2}$.

We remind the reader of the definition of triangulated subcategories (see Section 1.5)

Definition 1.5.1Let$\script {D}$be a triangulated category. A full additive subcategory$\script {C}$in$\script {D}$is called atriangulated subcategoryif every object isomorphic to an object of$\script {C}$is in$\script {C}$, and the inclusion$\script {C} \rightarrow \script {D}$is a triangulated functor, as in Definition 2.1.1. We assume further that$\phi_{X}:1(\Sigma(X)) \longrightarrow \Sigma(1(X))$is the identity onΣX.

Remark 2.1.2. To say that...

7. CHAPTER 3 Perfection of classes
(pp. 103-122)

Let us briefly review some standard definitions for large cardinals. A cardinal α is calledsingularif α can be written as a sum of fewer than α cardinals, all smaller than α.

Let$\aleph_{n}$be thenthinfinite cardinal. Thus,$\aleph_{0}$is the zeroth, that is the countable cardinal,$\aleph_{1}$the next, and so on. The cardinal$\aleph_{w}$is defined to be the smallest cardinal bigger than$\aleph_{n}$for alln. Clearly,$\aleph_{w}=\sum^{\infty}_{n=1} \aleph_{n}$is a countable union of cardinals, each strictly smaller than$\aleph_{w}$. Hence$\aleph_{w}$is an example of a singular cardinal.

A cardinal which is not...

8. CHAPTER 4 Small objects, and Thomason’s localisation theorem
(pp. 123-152)

Definition 4.1.1.Let$\script {T}$be a triangulated category satisfying [TR5] (that is, coproducts exist). Let α be an infinite cardinal. An object$k \in \script {T}$is calledα-smallif, for any collection$\{X_{\lambda}; \lambda \epsilon \Lambda\}$of objects of$\script {T}$, any map$k \longrightarrow \coprod_{ \lambda \epsilon \Lambda} X_{ \lambda}$factors through some coproduct of cardinality strictly less thanα.In other words, there exists a subsetΛʹ ⊂ Λ,where the cardinality ofΛʹis strictly less thanα,and the map above factors as$k \longrightarrow \coprod_{ \lambda \epsilon \Lambda'} X_{ \lambda} \longrightarrow \coprod_{ \lambda \epsilon \Lambda} X_{ \lambda}$.

Example 4.1.2. The special case where$\alpha=\aleph_{0}$is of great interest. An object$k \in \script {T}$is called$\aleph_{0}$-smallif for any infinite coproduct in$\script {T}$, say the coproduct...

9. CHAPTER 5 The category A($\script (S)$)
(pp. 153-182)

Let$\script {S}$be an additive category. We do not assume that$\script {S}$is essentially small. We define

Definition 5.1.1The category$\script {C}at(\script {S}^{op}, \script {A}b)$has for its objects all the additive functors$F : S^{op} \longrightarrow \script {A}b$.

The morphisms in$\script {C}at(\script {S}^{op}, \script {A}b)$are the natural transormations.

It is well-known that$\script {C}at(\script {S}^{op}, \script {A}b)$is an abelian category. We remind the reader what sequences are exact in$\script {C}at(\script {S}^{op}, \script {A}b)$. Suppose we are given a sequence$0 \longrightarrow F'(-) \longrightarrow F(-) \longrightarrow F''(-) \longrightarrow 0$of objects and morphisms in$\script {C}at(\script {S}^{op}, \script {A}b)$, that is functors and natural transformations$\script {C}at(\script {S}^{op} \longrightarrow \script {A}b$. This sequence is exact in$\script {C}at(\script {S}^{op}, \script {A}b)$if and only if, for every$s \epsilon \script {S}$, the sequence of abelian groups$0 \longrightarrow F'(s) \longrightarrow F(s) \longrightarrow F''(s) \longrightarrow 0$is...

10. CHAPTER 6 The category $\script {E}x(S^o^p, \script {A}b)$
(pp. 183-220)

Let α be a regular cardinal. Throughout this Chapter, we fix a choice of such a cardinal α. Let$\script {S}$be a category, satisfying the following hypotheses

Hypotheses 6.1.1.The category$\script {S}$is said to satisfy hypothesis 6.1.1if

6.1.1.1.$\script {S}$is an essentially small additive category.

6.1.1.2.The coproduct of fewer thanαobjects of$\script {S}$exists in$\script {S}$.

6.1.1.3.Homotopy pullback squares exist in$\script {S}$.That is, given a diagram in$\script {S}$.$\begin{matrix} {} & {} & x \\ {} & {} & \big\downarrow \\ {{x}'} & \xrightarrow{\hspace*{0.75cm}} & y \\ \end{matrix}$it may be completed to a commutative square$\begin{matrix} {p} & {\xrightarrow{\hspace*{0.75cm}}} & x \\ {\big\downarrow} & {} & \big\downarrow \\ {{x}'} & \xrightarrow{\hspace*{0.75cm}} & y \\ \end{matrix}$so that any commutative square$\begin{matrix} {s} & {\xrightarrow{\hspace*{0.75cm}}} & x \\ {\big\downarrow} & {} & \big\downarrow \\ {{x}'} & \xrightarrow{\hspace*{0.75cm}} & y \\ \end{matrix}$is induced by a (non-unique) maps sp. The object p is called...

11. CHAPTER 7 Homological properties of $\script {E}x(S^o^p, \script {A}b)$
(pp. 221-272)

We have learned, in the previous Chapter, some of the basic properties of the categories$\script {E}x(S^o^p, \script {A}b)$. In Appendix C, more specifically in Section C.4, we can see that in general the categories$\script {E}x(S^o^p, \script {A}b)$need not have enough injectives; in fact, they can fail to have cogenerators. See also Lemma 6.4.6 for the fact that, if$\script {E}x(S^o^p, \script {A}b)$fails to have a cogenerator, it certainly cannot have enough injectives.

But nevertheless, something positive is true. This Chapter will be devoted to proving the positive results we have. These positive results are fragmented and inconclusive. They are included for the benefit of...

12. CHAPTER 8 Brown representability
(pp. 273-308)

In this Chapter, all categories are assumed to have small Hom sets. Sometimes we will explicitly remind the reader of this; even when we do not, it is assumed. Let us make some definitions about possible sets of generators for$\script {T}$.

Definition 8.1.1(cf. Definition 6.2.8). Let$\script {T}$be a triangulated category satisfying [TR5]. A set T of objects of$\script {T}$is called a generating set if

8.1.1.1.$\{Hom(T,x)=0\} \Longrightarrow \{x=0\};$that is, if x$\script {T}$satisfiestT, Hom (t, x= 0)then x is isomorphic in$\script {T}$to 0.

8.1.1.2.Up to isomorphisms, T is closed under suspension and...

13. CHAPTER 9 Bousfield localisation
(pp. 309-320)

Let$\script {T}$be a triangulated category,$\script {S} \subset \script {T}$a triangulated subcategory. In Theorem 2.1.8 we learned how to construct the Verdier quotient$\script {T} / \script {S}$. There is a natural localisation map$F:\script {T}\longrightarrow \script {T} / \script {S}$. In Example 8.4.5 we learned that under suitable hypotheses, the functorFhas a right adjoint. We remind the reader of the hypoteses.

Suppose$\script {T}$is a triangulated category with small Hom-sets, satisfying [TR5]. Suppose further that the representability theorem holds for$\script {T}$. Let$\script {S}$be a localising subcategory. Assume that the Verdier quotient$\script {T} / \script {S}$is a category with small Hom-sets. In Example 8.4.5 we saw that...

14. APPENDIX A. Abelian categories
(pp. 321-368)
15. APPENDIX B. Homological functors into [AB5α] categories
(pp. 369-386)
16. APPENDIX C. Counterexamples concerning the abelian category A($\script {T}$)
(pp. 387-406)
17. APPENDIX D. Where $\script {T}$ is the homotopy category of spectra
(pp. 407-426)
18. APPENDIX E. Examples of non—perfectly—generated categories
(pp. 427-442)
19. Bibliography
(pp. 443-444)
20. Index
(pp. 445-451)
21. Back Matter
(pp. 452-452)