##
**Triangulated categories.**
*(English)*
Zbl 0974.18008

Annals of Mathematics Studies. 148. Princeton, NJ: Princeton University Press. vii, 449 p. (2001).

The present book gives a self-contained triangulated category treatment of Brown representability, Bousfield localization and connected topics. It is original on many points and contains many new and impressive theoretical achievements. Besides, it is extremely well written and can be read easily even by non-experts of the field. The theory is developed in the broad generality that one should need motivically (that is working with non-compactly generated categories). In a sequel, the author should discuss together with V. Voevodsky the applications to motives of his present results on triangulated categories.

The book is written axiomatically: it consists of the formal theory of triangulated categories developed almost without examples or applications to other theories. This lack of examples is a bit frustrating. On the other hand, the few applications that are given to the homotopy category of spectra (appendix D) are quite convincing and show how powerful the author’s approach is when facing classical problems such as localization with respect to homology. Here follows an account of the mathematical content of the book.

Chapter 1 contains the definitions and elementary properties of triangulated categories. Recall that their definition goes back to J.-L. Verdier [“Des catégories dérivées des catégories abéliennes”, Astérisque 239 (1996; Zbl 0882.18010)] and D. Puppe [“On the formal structure of stable homotopy theory”, in: Colloq. algebr. Topology, Aarhus 1962, 65-71 (1962; Zbl 0139.41106)] and was motivated by the examples of derived categories and stable homotopy. Verdier’s octahedral axiom is replaced here by an axiom involving mapping cones on maps of triangles. See also the author’s “Some new axioms for triangulated categories” [A. Neeman, J. Algebra 139, No. 1, 221-255 (1991; Zbl 0722.18003)] for a discussion of the axioms of triangulated categories.

Chapter 2 gives Verdier’s construction of the quotient of a triangulated category by a triangulated subcategory using Rickard’s characterization of thick subcategories as subcategories containing direct summands of their objects [J. Rickard, “Derived categories and stable equivalence”, J. Pure Appl. Algebra 61, No. 3, 303-317 (1989; Zbl 0685.16016)].

Chapter 3 discusses localizing subcategories and related set-theoretical questions involving infinite cardinals. For example, a subcategory \({\mathcal S}\) of a triangulated category \({\mathcal T}\) where coproducts exist is called \(\beta\)-localizing, where \(\beta\) is an infinite cardinal, if it is thick and closed with respect to the formation of coproducts of fewer than \(\beta\) of its objects. In the same way, the book, that as a whole insists very much on set-theoretical problems, introduces the concept of \(\beta\)-perfect classes to imitate some standard topological constructions involving transfinite induction on the number of cells of a complex.

Chapter 4 generalizes Thomason’s localization theorem. A \(\aleph_0\)-small object (usually known as a compact object) is such that any map from it into a coproduct factors through a finite coproduct. Thomason’s original theorem discusses the behavior of compactness with respect to Verdier quotients in the case of the derived category of the category of quasi-coherent sheaves on a semi-separated scheme [R. W. Thomason and T. Trobaugh, “Higher algebraic K-theory of schemes and of derived categories”, in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)]. Thomason’s result is generalized to all suitable triangulated categories and higher cardinals (that is, \(\aleph_0\)-small objects are replaced by suitable “\(\alpha\)-small objects”, where \(\alpha\) is any infinite cardinal, and so on).

Chapter 5 to 7 discuss the properties of remarkable categories of functors from triangulated categories to abelian groups: adjunctions between them, homological properties, universal properties for homological functors, subobjects, quotient objects, injectives... The results in chapter 5 originate in results by P. Freyd [“Stable homotopy”, in: Proc. Conf. categor. Algebra, La Jolla 1965, 121-172 (1966; Zbl 0195.52901)] and J.-L. Verdier [loc. cit.].

Chapter 8 discusses Brown representability, that is the conditions under which (co)homological functors are representable. The first theorem of this type goes back to E. H. Brown jun. [“Cohomology theories”, Ann. Math., II. Ser. 75, 467-484 (1962; Zbl 0101.40603)]. Brown representability for duals of triangulated categories is also discussed (on this point, see also A. Neeman [“Brown representability for the dual”, Invent. Math. 133, No. 1, 97-105 (1998; Zbl 0906.18002)]). Notice that J. Franke has also obtained recently, independently and by other means, general Brown representability theorems [“On the Brown representability theorem for triangulated categories”, to appear in Topology].

Chapter 9 studies Bousfield localization, that is the conditions under which the natural functor from a triangulated category to a Verdier quotient admits a right adjoint, and the properties of the adjunction. A. K. Bousfield’s localization theorems originate in [“The Boolean algebra of spectra”, Comment. Math. Helv. 54, 368-377 (1979; Zbl 0421.55002) and “The localization of spectra with respect to homology”, Topology 18, 257-281 (1979; Zbl 0417.55007)].

The book ends with five appendices, respectively on: abelian categories, homological functors, counterexamples, localization in the homotopy category of spectra, examples of non-perfectly generated categories.

The book is written axiomatically: it consists of the formal theory of triangulated categories developed almost without examples or applications to other theories. This lack of examples is a bit frustrating. On the other hand, the few applications that are given to the homotopy category of spectra (appendix D) are quite convincing and show how powerful the author’s approach is when facing classical problems such as localization with respect to homology. Here follows an account of the mathematical content of the book.

Chapter 1 contains the definitions and elementary properties of triangulated categories. Recall that their definition goes back to J.-L. Verdier [“Des catégories dérivées des catégories abéliennes”, Astérisque 239 (1996; Zbl 0882.18010)] and D. Puppe [“On the formal structure of stable homotopy theory”, in: Colloq. algebr. Topology, Aarhus 1962, 65-71 (1962; Zbl 0139.41106)] and was motivated by the examples of derived categories and stable homotopy. Verdier’s octahedral axiom is replaced here by an axiom involving mapping cones on maps of triangles. See also the author’s “Some new axioms for triangulated categories” [A. Neeman, J. Algebra 139, No. 1, 221-255 (1991; Zbl 0722.18003)] for a discussion of the axioms of triangulated categories.

Chapter 2 gives Verdier’s construction of the quotient of a triangulated category by a triangulated subcategory using Rickard’s characterization of thick subcategories as subcategories containing direct summands of their objects [J. Rickard, “Derived categories and stable equivalence”, J. Pure Appl. Algebra 61, No. 3, 303-317 (1989; Zbl 0685.16016)].

Chapter 3 discusses localizing subcategories and related set-theoretical questions involving infinite cardinals. For example, a subcategory \({\mathcal S}\) of a triangulated category \({\mathcal T}\) where coproducts exist is called \(\beta\)-localizing, where \(\beta\) is an infinite cardinal, if it is thick and closed with respect to the formation of coproducts of fewer than \(\beta\) of its objects. In the same way, the book, that as a whole insists very much on set-theoretical problems, introduces the concept of \(\beta\)-perfect classes to imitate some standard topological constructions involving transfinite induction on the number of cells of a complex.

Chapter 4 generalizes Thomason’s localization theorem. A \(\aleph_0\)-small object (usually known as a compact object) is such that any map from it into a coproduct factors through a finite coproduct. Thomason’s original theorem discusses the behavior of compactness with respect to Verdier quotients in the case of the derived category of the category of quasi-coherent sheaves on a semi-separated scheme [R. W. Thomason and T. Trobaugh, “Higher algebraic K-theory of schemes and of derived categories”, in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)]. Thomason’s result is generalized to all suitable triangulated categories and higher cardinals (that is, \(\aleph_0\)-small objects are replaced by suitable “\(\alpha\)-small objects”, where \(\alpha\) is any infinite cardinal, and so on).

Chapter 5 to 7 discuss the properties of remarkable categories of functors from triangulated categories to abelian groups: adjunctions between them, homological properties, universal properties for homological functors, subobjects, quotient objects, injectives... The results in chapter 5 originate in results by P. Freyd [“Stable homotopy”, in: Proc. Conf. categor. Algebra, La Jolla 1965, 121-172 (1966; Zbl 0195.52901)] and J.-L. Verdier [loc. cit.].

Chapter 8 discusses Brown representability, that is the conditions under which (co)homological functors are representable. The first theorem of this type goes back to E. H. Brown jun. [“Cohomology theories”, Ann. Math., II. Ser. 75, 467-484 (1962; Zbl 0101.40603)]. Brown representability for duals of triangulated categories is also discussed (on this point, see also A. Neeman [“Brown representability for the dual”, Invent. Math. 133, No. 1, 97-105 (1998; Zbl 0906.18002)]). Notice that J. Franke has also obtained recently, independently and by other means, general Brown representability theorems [“On the Brown representability theorem for triangulated categories”, to appear in Topology].

Chapter 9 studies Bousfield localization, that is the conditions under which the natural functor from a triangulated category to a Verdier quotient admits a right adjoint, and the properties of the adjunction. A. K. Bousfield’s localization theorems originate in [“The Boolean algebra of spectra”, Comment. Math. Helv. 54, 368-377 (1979; Zbl 0421.55002) and “The localization of spectra with respect to homology”, Topology 18, 257-281 (1979; Zbl 0417.55007)].

The book ends with five appendices, respectively on: abelian categories, homological functors, counterexamples, localization in the homotopy category of spectra, examples of non-perfectly generated categories.

Reviewer: Frédéric Patras (Nice)

### MSC:

18E30 | Derived categories, triangulated categories (MSC2010) |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |