E-Book Overview
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.
E-Book Content
Contents 0. Acknowledgements 1. Introduction
3 3
Chapter 1. Definition and elementary properties of triangulated categories 1.1. Pre–triangulated categories 1.2. Corollaries of Proposition 1.1.20 1.3. Mapping cones, and the definition of triangulated categories 1.4. Elementary properties of triangulated categories 1.5. Triangulated subcategories 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
29 29 37 45 52 60 63 68 70
Chapter 2. Triangulated functors and localizations of triangulated categories 73 2.1. Verdier localization and thick subcategories 73 2.2. Sets and classes 99 2.3. History of the results in Chapter 2 100 Chapter 3. Perfection of classes 3.1. Cardinals 3.2. Generated subcategories 3.3. Perfect classes 3.4. History of the results in Chapter 3
103 103 103 110 122
Chapter 4. Small objects, and