Description
The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories"--the "well generated triangulated categories"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.
Chapter
1.6. Direct sums and products, and homotopy limits and colimits
1.7. Some weak "functoriality" for homotopy limits and colimits
1.8. History of the results in Chapter 1
Chapter 2. Triangulated functors and localizations of triangulated categories
2.1. Verdier localization and thick subcategories
2.3. History of the results in Chapter 2
Chapter 3. Perfection of classes
3.2. Generated subcategories
3.4. History of the results in Chapter 3
Chapter 4. Small objects, and Thomason's localisation theorem
4.3. Maps factor through (S)^β
4.4. Maps in the quotient
4.5. A refinement in the countable case
4.6. History of the results in Chapter 4
Chapter 5. The category A(S)
5.1. The abelian category A(S)
5.2. Subobjects and quotient objects in A(S)
5.3. The functoriality of A(S)
5.4. History of the results in Chapter 5
Chapter 6. The category εx (S^op, Ab)
6.1. εx(S^op, Ab) is an abelian category satisfying [AB3] and [AB3*]
6.3. εx(S^op, Ab) satisfies [AB4] and [AB4*], but not [AB5] or [AB5*]
6.4. Projectives and injectives in the category εx (S^op, Ab)
6.5. The relation between A(J) and εx ( {J^α}^op, Ab)
6.6. History of the results of Chapter 6
Chapter 7. Homological properties of εx(S^op, Ab)
7.1. εx(S^op, Ab) as a locally presentable category
7.2. Homological objects in εx(S^op, Ab)
7.3. A technical lemma and some consequences
7.4. The derived functors of colimits in εx(S^op, Ab)
7.5. The adjoint to the inclusion of εx(S^op, Ab)
7.6. History of the results in Chapter 7
Chapter 8. Brown representability
8.2. Brown representability
8.3. The first representability theorem
8.4. Corollaries of Brown representability
8.5. Applications in the presence of injectives
8.6. The second representability theorem: Brown representability for the dual
8. 7. History of the results in Chapter 8
Chapter 9. Bousfield localisation
9.2. The six gluing functors
9.3. History of the results in Chapter 9
Appendix A. Abelian categories
A.1. Locally presentable categories
A.2. Formal properties of quotients
A.3. Derived functors of limits
A.4. Derived functors of limits via injectives
A.5. A Mittag—Leffier sequence with non—vanishing lim^n
A.6. History of the results of Appendix A
Appendix B. Homological functors into [AB5^α] categories
B.2. Abelian categories satisfying [AB5^α]
B.3. History of the results in Appendix B
Appendix C. Counterexamples concerning the abelian category A(J)
C.1. The submodules p^i M
C.3. The category A(S) is not well—powered
C.4. A category εx (S^op, Ab) without a cogenerator
C.5. History of the results of Appendix C
Appendix D. Where J is the homotopy category of spectra
D.1. Localisation with respect to homology
D.2. The lack of injectives
D.3. History of the results in Appendix D
Appendix E. Examples of non—perfectly—generated categories
E.1. If J is N0—compactly generated, J^op is not even well—generated
E.2. An example of a non N1—perfectly generated J
E.3. For J = K(Z), neither J nor J^op is well—generated
E.4. History of the results in Appendix E
Triangulated Categories. (AM-148)
：Triangulated Categories. (AM-148)
( Annals of Mathematics Studies )
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