Contents
Preface ix
Chapter 1. Model categories 1
1.1. The definition of a model category 2
1.2. The homotopy category 7
1.3. Quillen functors and derived functors 13
1.3.1. Quillen functors 14
1.3.2. Derived functors and naturality 16
1.3.3. Quillen equivalences 19
1.4. 2-categories and pseudo-2-functors 22
Chapter 2. Examples 27
2.1. Cofibrantly generated model categories 28
2.1.1. Ordinals, cardinals, and transfinite compositions 28
2.1.2. Relative /-cell complexes and the small object argument 30
2.1.3. Cofibrantly generated model categories 34
2.2. The stable category of modules 36
2.3. Chain complexes of modules over a ring 40
2.4. Topological spaces 49
2.5. Chain complexes of comodules over a Hopf algebra 60
2.5.1. The category of B-comodules 60
2.5.2. Weak equivalences 65
2.5.3. The model structure 67
Chapter 3. Simplicial sets 73
3.1. Simplicial sets 73
3.2. The model structure on simplicial sets 79
3.3. Anodyne extensions 81
3.4. Homotopy groups 83
3.5. Minimal fibrations 88
3.6. Fibrations and geometric realization 95
Chapter 4. Monoidal model categories 101
4.1. Closed monoidal categories and closed modules 101
4.2. Monoidal model categories and modules over them 107
4.3. The homotopy category of a monoidal model category 115
Chapter 5. Framings 119
5.1. Diagram categories 120
5.2. Diagrams over Reedy categories and framings 123
5.3. A lemma about bisimplicial sets 128
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