Background on ∞-category theory

This part collects the \(\infty \)-categorical background used throughout the book. The material is standard, but developing it at the point where it is first needed would repeatedly interrupt the main line of argument. These chapters are therefore meant to be consulted rather than read in order: whenever the main text appeals to one of their results, it forward-references the precise statement it uses.

Chapter 21 treats adjunctions between \(\infty \)-categories, their equivalent characterizations, their interaction with (co)limits and Kan extensions, and Bousfield localizations. Chapter 22 develops presentable \(\infty \)-categories through presentations by generators and relations, which is the form in which we use them to construct adjoints and localizations; it also treats compact generation and tensor products of categories with prescribed colimits. Chapter 23 treats cartesian and cocartesian fibrations together with the straightening/unstraightening correspondence, our main device for constructing functors into \(\Cat _{\infty }\), such as the Hom functor \(\Hom _C\colon C\catop \times C \to \An \). Finally, Chapter 24 presents \(\infty \)-categories as complete Segal animae, which is how the span categories of Part II are constructed.