Higher Topos Theory

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.


The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

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