The Homotopy Theory of (∞,1)-Categories

Abstract......Page 5
Sammendrag......Page 7
Acknowledgements......Page 13
Introduction and plan......Page 19
Ideas of higher categories......Page 22
Ideas of higher morphisms......Page 23
Inductive interpretation of higher categories......Page 26
Higher invertible morphisms and (n,k)-categories......Page 27
Higher categories as ``weakly'' enriched categories......Page 30
Simplicial and topological categories......Page 35
Definition of topological categories......Page 36
The homotopy category of a topological category......Page 37
Simplicial categories......Page 40
Quasi-categories......Page 41
Definition of quasi-categories......Page 42
The homotopy category of a quasi-category......Page 46
Mapping spaces and higher morphisms......Page 49
Functors and homotopy coherent diagrams......Page 55
Contractable spaces of choices......Page 58
Comparison of quasi-categories with simplicial categories......Page 60
Functors and subcategories......Page 65
Functors of (,1)-categories......Page 66
Subcategories of quasi-categories......Page 67
Constructions......Page 68
Join constructions......Page 69
Overcategories and undercategories......Page 72
Initial objects and terminal objects......Page 75
Limits and colimits......Page 78
The notion of (,1)-categorical stability......Page 85
Pointed (,1)-categories......Page 86
Fibers and cofibers......Page 87
Definition of stability......Page 91
Constructions of suspension functor and loop functor......Page 92
Suspension functor......Page 93
Loop functor......Page 95
Suspension and loop functor on stable (,1)-categories......Page 96
The homotopy category of stable (,1)-categories......Page 98
Additive structure......Page 99
Triangulated structure......Page 108
Enclosing comments......Page 117
Definition of monoidal categories......Page 121
Enriched categories......Page 123
Additive categories......Page 126
Definition of triangulated categories......Page 128
The delta-category......Page 133
Definition of simplicial sets......Page 136
Horn and spheres......Page 138
Construction of simplicial spheres......Page 139
Definition of simplicial horns......Page 140
The nerve functor......Page 142
Relations between categories and nerves......Page 143
Bibliography......Page 144