The Homotopy Theory of (∞,1)-Categories

Description

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