Higher Topos Theory

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 1-Categories 72
2.3 Inner Fibrations 95
2.4 Cartesian Fibrations 114


Chapter 3. The 1-Category of 1-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 Co_nality 223
4.2 Techniques for Computing Colimits 240
4.3 Kan Extensions 261
4.4 Examples of Colimits 292


Chapter 5. Presentable and Accessible 1-Categories 311
5.1 1-Categories of Presheaves 312
5.2 Adjoint Functors 331
5.3 1-Categories of Inductive Limits 377
5.4 Accessible 1-Categories 414
5.5 Presentable 1-Categories 455


Chapter 6. 1-Topoi 526
6.1 1-Topoi: De_nitions and Characterizations 527
6.2 Constructions of 1-Topoi 569
6.3 The 1-Category of 1-Topoi 593
6.4 n-Topoi 632
6.5 Homotopy Theory in an 1-Topos 651


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. Appendix 781
A.1 Category Theory 781
A.2 Model Categories 803
A.3 Simplicial Categories 844


Bibliography 909
General Index 915
Index of Notation 923