Complete Segal animae

Constructing \(\infty \)-categories is more subtle than constructing classical categories: it does not suffice to define composition of morphisms by hand and check unitality and associativity, due to the fact that there are higher coherences that should be encoded. In most cases, the \(\infty \)-categories one is interested in can be built up from a few elementary constructions, such as products, pullbacks, and functor categories. However, in certain cases, like that for \(\infty \)-categories of spans (see Chapter 13), this is not immediately possible.

In this chapter, we will introduce an alternative method for constructing \(\infty \)-categories, originally due to Rezk (2001): by writing down a simplicial anima \(\simp \catop \to \An \) satisfying certain Segal and Rezk conditions. Given an \(\infty \)-category \(C\), the assignment \([n] \mapsto \Map ([n],C)\) can be made functorial in \([n] \in \simp \catop \), resulting in a simplicial anima \(N(C) \colon \simp \catop \to \An \) called the nerve of \(C\). It turns out that this assignment defines a fully faithful functor \(N\colon \Cat _{\infty } \to \sAn \) from the \(\infty \)-category of small \(\infty \)-categories to the \(\infty \)-category of simplicial animae. The image of this functor consists of those simplicial animae that satisfy the Segal and Rezk conditions, which are called complete Segal animae. A precise formulation of this result is given in Theorem 24.1.10 below.

This theorem is powerful since it lets us manipulate or construct \(\infty \)-categories at the level of simplicial animae. We recall the definition of complete Segal animae in Section 24.1, and use the theorem in Section 24.2 to draw some consequences regarding filtered colimits and geometric realizations of \(\infty \)-categories.

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