## 7.2 Cofinality

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$ indexed by a simplicial set $B$. In §7.1, we introduced the definition of a *limit* $\varprojlim (f)$ and *colimit* $\varinjlim (f)$ of the diagram $f$ (Definition 7.1.1.11). In practice, it is often convenient to replace $f$ by a simpler diagram having the same limit (or colimit). The primary goal of this section is to introduce a general formalism which will allow us to make replacements of this sort.

We begin in §7.2.1 by introducing the notions of *left cofinal* and *right cofinal* morphisms of simplicial sets (Definition 7.2.1.1). Roughly speaking, one can regard left cofinality as a homotopy-invariant replacement for the notion of left anodyne morphism introduced in Definition 4.2.4.1. More precisely, the collection of left cofinal morphisms of simplicial sets is uniquely determined by the following assertions:

A monomorphism of simplicial sets $f: A \hookrightarrow B$ is left cofinal if and only if it is left anodyne (Proposition 7.2.1.3).

Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ A' \ar [r]^-{f'} & B', } \]where the vertical maps are categorical equivalences. Then $f$ is left cofinal if and only if $f'$ is left cofinal (Corollary 7.2.1.22).

In §7.2.2, we connect the notion of cofinality with the theory of limits and colimits developed in §7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $g: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. We will show that if $f: A \rightarrow B$ is a left cofinal morphism of simplicial sets, then the limit of the diagram $g$ (if it exists) can be identified with with the limit of the composite diagram $(g \circ f): A \rightarrow \operatorname{\mathcal{C}}$ (Corollary 7.2.2.11). Similarly, if $f$ is right cofinal, then the colimit of $g$ can be identified with the colimit of $g \circ f$. Consequently, cofinality is a very useful tool for computing (or verifying the existence of) limits and colimits.

In §7.2.3, we specialize to the study of cofinal functors between $\infty $-categories. Our main result asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is right cofinal if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is weakly contractible (Theorem 7.2.3.1). In particular, the weak contractibility of each slice $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ guarantees that $F$ is a weak homotopy equivalence of simplicial sets: this is an $\infty $-categorical generalization of Quillen's “Theorem A” (see Example 7.2.3.3). We will deduce Theorem 7.2.3.1 from a general fact about the stability of right cofinality with respect to pullback along cocartesian fibrations (Proposition 7.2.3.12), which is of independent interest.