Preface ....................................................... vii
Chapter 1. An Overview of Higher Category Theory ................ 1
1.1. Foundations for Higher Category Theory ..................... 1
1.2. The Language of Higher Category Theory .................... 26
Chapter 2. Fibrations of Simplicial Sets ....................... 53
2.1. Left Fibrations ........................................... 55
2.2. Simplicial Categories and oo-Categories ................... 72
2.3. Inner Fibrations .......................................... 95
2.4. Cartesian Fibrations ..................................... 114
Chapter 3. The ∞-Category of ∞-Categories ..................... 145
3.1. Marked Simplicial Sets ................................... 147
3.2. Straightening and Unstraightening ........................ 169
3.3. Applications ............................................. 204
Chapter 4. Limits and Colimits ............................... 223
4.1. Cofinality ............................................... 223
4.2. Techniques for Computing Colimits ........................ 240
4.3. Kan Extensions ........................................... 261
4.4. Examples of Colimits ..................................... 292
Chapter 5. Presentable and Accessible ∞-Categories ............ 311
5.1. ∞-Categories of Presheaves ............................... 312
5.2. Adjoint Functors ......................................... 331
5.3. ∞-Categories of Inductive Limits ......................... 377
5.4. Accessible ∞-Categories .................................. 414
5.5. Presentable ∞-Categories ................................. 455
Chapter 6. ∞-Topoi ............................................ 526
6.1. ∞-Topoi: Definitions and Characterizations ............... 527
6.2. Constructions of ∞-Topoi ................................. 569
6.3. The oo-Category of ∞-Topoi ............................... 593
6.4. n-Topoi .................................................. 632
6.5. Homotopy Theory in an ∞-Topos ............................ 651
6.1. Chapter 7. Higher Topos Theory in Topology ............... 682
7.1. Paracompact Spaces ....................................... 683
7.2. Dimension Theory ......................................... 711
7.3. The Proper Base Change Theorem ........................... 742
Appendix ...................................................... 781
A.l. Category Theory .......................................... 781
A.2. Model Categories ......................................... 803
A.3. Simplicial Categories .................................... 844
Bibliography .................................................. 909
General Index ................................................. 915
Index of Notation ............................................. 923
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