Homotopical Algebraic Geometry I: Topos theory

Introduction......Page 2
The homotopy theory of S-categories......Page 9
Simplicial localization......Page 11
Model categories of diagrams......Page 12
Diagrams......Page 13
Restricted diagrams......Page 14
The Yoneda embedding......Page 15
Comma S-categories......Page 17
S-topologies and S-sites......Page 18
Simplicial objects and hypercovers......Page 20
Local equivalences......Page 23
The local model structure......Page 26
Functoriality......Page 34
Injective model structure and stacks of morphisms......Page 35
Truncated stacks and truncation functors......Page 37
Model topoi......Page 39
Stacks over pseudo-model categories......Page 42
Model categories of pre-stacks on a pseudo-model category......Page 43
The Yoneda embedding of a pseudo-model category......Page 45
Model pre-topologies and pseudo-model sites......Page 47
Simplicial objects and hypercovers......Page 51
Local equivalences......Page 52
The local model structure......Page 53
Comparison between the S-theory and the pseudo-model theory......Page 55
Functoriality......Page 56
A Giraud\'s theorem for model topoi......Page 58
Étale K-theory of commutative S-algebras......Page 61
S-modules, S-algebras and their algebraic K-theory......Page 62
The étale topology on commutative S-algebras......Page 64
Étale K-theory of commutative S-algebras......Page 66
U-cofibrantly generated model categories......Page 67
U-cellular and U-combinatorial model categories......Page 68