1 Introduction ................................................. 1
Part I. Preliminaries
2 Homotopy Theory of Simplicial Sets .......................... 15
2.1 Simplicial Sets ........................................ 15
2.2 Model Structure for Simplicial Sets .................... 21
2.3 Projective Model Structure for Diagrams ................ 25
3 Some Topos Theory ........................................... 29
3.1 Grothendieck Topologies ................................ 31
3.2 Exactness Properties ................................... 38
3.3 Geometric Morphisms .................................... 42
3.4 Points ................................................. 46
3.5 Boolean Localization ................................... 49
Part II. Simplicial Presheaves and Simplicial Sheaves
4 Local Weak Equivalences ..................................... 59
4.1 Local Weak Equivalences ................................ 60
4.2 Local Fibrations ....................................... 69
4.3 First Applications of Boolean Localization ............. 77
5 Local Model Structures ...................................... 91
5.1 The Injective Model Structure .......................... 93
5.2 Injective Fibrations .................................. 100
5.3 Geometric and Site Morphisms .......................... 107
5.4 Descent Theorems ...................................... 116
5.5 Intermediate Model Structures ......................... 126
5.6 Postnikov Sections and n-Types ........................ 131
6 Cocycles ................................................... 139
6.1 Cocycle Categories .................................... 142
6.2 The Verdier Hypercovering Theorem ..................... 150
7 Localization Theories ...................................... 159
7.1 General Theory ........................................ 161
7.2 Localization Theorems for Simplicial Presheaves ....... 174
7.3 Properness ............................................ 185
Part III. Sheaf Cohomology Theory
8 Homology Sheaves and Cohomology Groups ..................... 191
8.1 Chain Complexes ....................................... 194
8.2 The Derived Category .................................. 202
8.3 Abelian Sheaf Cohomology .............................. 207
8.4 Products and Pairings ................................. 223
8.5 Localized Chain Complexes ............................. 227
8.6 Linear Simplicial Presheaves .......................... 235
9 Non-abelian Cohomology ..................................... 247
9.1 Torsors ............................................... 251
9.2 Stacks and Homotopy Theory ............................ 267
9.3 Groupoids Enriched in Simplicial Sets ................. 280
9.4 Presheaves of Groupoids Enriched in Simplicial Sets ... 304
9.5 Extensions and Gerbes ................................. 318
Part IV. Stable Homotopy Theory
10 Spectra and T-spectra ...................................... 337
10.1 Presheaves of Spectra ................................. 344
10.2 T-spectra and Localization ............................ 360
10.3 Stable Model Structures for T-spectra ................. 368
10.4 Shifts and Suspensions ................................ 383
10.5 Fibre and Cofibre Sequences ........................... 391
10.6 Postnikov Sections and Slice Filtrations .............. 405
10.7 T-Complexes ........................................... 412
11 Symmetric T-spectra ........................................ 431
11.1 Symmetric Spaces ...................................... 436
11.2 First Model Structures ................................ 441
11.3 Localized Model Structures ............................ 447
11.4 Stable Homotopy Theory of Symmetric Spectra ........... 451
11.5 Equivalence of Stable Categories ...................... 461
11.6 The Smash Product ..................................... 472
11.7 Symmetric T-complexes ................................. 483
References .................................................... 499
Index ......................................................... 505
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