Contents
Introduction 1
Chapter I. Model approximations and bounded diagrams 5
1. Notation 5
2. Model categories 6
3. Left derived functors 12
4. Left derived functors of colimits and left Kan extensions 14
5. Model approximations 16
6. Spaces and small categories 19
7. The pull-back process and local properties 22
8. Colimits of diagrams indexed by spaces 22
9. Left Kan extensions 24
10. Bounded diagrams 26
Chapter II. Homotopy theory of diagrams 29
11. Statements of the main results 29
12. Cofibrations 30
13. Funb(K,M) as a model category 32
14. Ocolimit of bounded diagrams 35
15. Bousfield-Kan approximation of Fun(I, C) 36
16. Homotopy colimits and homotopy left Kan extensions 37
17. Relative boundedness 38
18. Reduction process 40
19. Relative cofibrations 44
20. Cofibrations and colimits 46
21.
Funbf(L,
M) as a model category 48
22. Cones 49
23. Diagrams indexed by cones I 51
Chapter III. Properties of homotopy colimits 54
24. Fubini theorem 54
25. Bounded diagrams indexed by Grothendieck constructions 57
26. Thomason's theorem 59
27. Etale spaces 63
28. Diagrams indexed by cones II 67
29. Homotopy colimits as etale spaces 69
30. Cofinality 70
31. Homotopy limits 71
Appendix A. Left Kan extensions preserve boundedness 75
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