A-homotopy theory of schemes

2 Homotopy category of a site with interval 2 2.1 Homotopy theory of simplicial sheaves . . . . . . . . . . . . . . 4 2.1.1 Simplicial sheaves . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 The simplicial model category structure . . . . . . . . 5 2.1.3 Local fibrations and resolution lemmas . . . . . . . . . 8 2.1.4 Homotopy limits and colimits. . . . . . . . . . . . . . . 12 2.1.5 Eilenberg-MacLane sheaves and Postnikoff towers . . . 14 2.1.6 Functoriality . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.7 Godement resolutions . . . . . . . . . . . . . . . . . . . 26 2.2 A localization theorem for simplicial sheaves . . . . . . . . . . 32 2.2.1 Basic definitions and main results . . . . . . . . . . . . 32 2.2.2 Elementary properties of classes WA and FA . . . . . . 33 2.2.3 A-model category structure theorem . . . . . . . . . . 37 2.2.4 Properness theorem . . . . . . . . . . . . . . . . . . . . 40 2.2.5 Localization of loop spaces . . . . . . . . . . . . . . . . 45 2.3 Homotopy category of a site with interval . . . . . . . . . . . . 49 2.3.1 Definitions, examples and the main theorem . . . . . . 49 2.3.2 The functor Sing∗ . . . . . . . . . . . . . . . . . . . . 51 2.3.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.4 An “explicit” I-resolution functor . . . . . . . . . . . . 57