7 Limits and Colimits

In this chapter, we extend the classical theory of limits and colimits to the setting of higher category theory. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that an object $Y \in \operatorname{\mathcal{D}}$ is a limit of $F$ if there exists a natural transformation $\alpha : \underline{Y} \rightarrow F$ having the following universal property: for every object $X \in \operatorname{\mathcal{C}}$, composition with $\alpha $ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X , Y ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( \underline{X}, F ); \]

here $\underline{X}, \underline{Y} \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the constant functors taking the values $X$ and $Y$, respectively. In this case, the object $Y$ is uniquely determined up to isomorphism; to emphasize this, we often denote $Y$ by $\varprojlim (F)$, or by $\varprojlim _{C \in \operatorname{\mathcal{C}}}( F(C) )$. In §7.1, we summarize the formal properties of this notion (as well as the dual notion of colimit, which plays an equally essential role in the theory).

Throughout this book, we will often be faced with the problem of computing (or describing) the limit of a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. In such situations, it is useful to have some flexibility to modify the $\infty $-category $\operatorname{\mathcal{C}}$. In §7.2, we introduce the notion of a left cofinal morphism of simplicial sets $e: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ (Definition 7.2.1.1). If $e: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is left cofinal, then an object of $\operatorname{\mathcal{D}}$ is a limit of $F$ if and only if it is a limit of the composite map $F' = F \circ e$ (see Corollary 7.2.2.11, and Corollary 7.4.3.12 for a converse). When $\operatorname{\mathcal{C}}$ is an $\infty $-category, cofinality admits a simple characterization: a morphism $e: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is left cofinal if and only if, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is weakly contractible (Theorem 7.2.3.1). We will encounter many situations where this criterion is easy to verify. In such cases, it is harmless to replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ for the purpose of calculating the limit of a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

In §7.3, we consider another important technique for computing limits. Suppose we are given a cartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Under some mild assumptions, one can show that the limit of a diagram $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ obeys a transitivity formula, which we can write informally as

\[ \varprojlim _{X \in \operatorname{\mathcal{E}}}( F(X) ) \simeq \varprojlim _{C \in \operatorname{\mathcal{C}}} ( \varprojlim _{ X \in \operatorname{\mathcal{E}}_ C } F(X) ). \]

More precisely, suppose that for every object $C \in \operatorname{\mathcal{C}}$, the diagram $F_{C} = F|_{ \operatorname{\mathcal{E}}_{C} }$ admits a limit in the $\infty $-category $\operatorname{\mathcal{D}}$. Then one can construct a new functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, given on objects by the formula $G(C) = \varprojlim (F_ C)$; we refer to $G$ as a right Kan extension of $F$ along $U$ (see Definition 7.3.1.2 and Proposition 7.3.4.4). Moreover, an object of the $\infty $-category $\operatorname{\mathcal{D}}$ is a limit of the functor $F$ if and only if it is a limit of the functor $G$ (Corollary 7.3.8.20).

The remainder of this chapter is devoted to studying limits and colimits in special situations. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1). For any $\infty $-category $\operatorname{\mathcal{C}}$, Corollary 5.6.0.6 supplies a bijection from the set of isomorphism classes of functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and the set of equivalence classes of (essentially small) left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. In §7.4, we use this identification to give an explicit description of limits and colimits in $\operatorname{\mathcal{S}}$:

$(1)$: The Kan complex $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ parametrizing sections of $U$ is a limit of the diagram $\mathscr {F}$ (Proposition 7.4.1.6),.
$(2)$: A Kan complex $X$ is a colimit of $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$ (Proposition 7.4.3.1).

These assertions are special cases of more general results which apply to diagrams taking values in the $\infty $-category $\operatorname{\mathcal{QC}}\supset \operatorname{\mathcal{S}}$; see Theorems 7.4.4.6 and 7.4.5.11.

Recall that the $\infty $-category $\operatorname{\mathcal{S}}$ is defined as the homotopy coherent nerve of the ordinary category of Kan complexes $\operatorname{Kan}$. In particular, if $\mathscr {F}_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}$ is a functor between ordinary categories, then passing to the homotopy coherent nerve gives a functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$. In this case, there is a natural candidate for the corresponding left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, obtained by taking $\operatorname{\mathcal{E}}$ to be the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0)$ of Definition 5.3.3.1. In §7.5, we combine this observation with assertions $(1)$ and $(2)$ to compare limits and colimits in the $\infty $-category $\operatorname{\mathcal{S}}$ with the classical theory of homotopy limits and colimits introduced by Bousfield and Kan in [MR0365573].

In §7.6, we provide a detailed discussion of some special classes of limits which arise frequently in practice, such as products (Definition 7.6.1.3), powers (Definition 7.1.2.1), pullbacks (Definition 7.6.2.1), equalizers (Definition 7.6.4.4), and sequential limits (Definition 7.6.5.1). From these primitives, many other examples can be constructed: for example, arbitrary limits in an $\infty $-category $\operatorname{\mathcal{D}}$ can be built by combining products and equalizers (see Corollary 7.6.4.25 and Proposition 7.6.6.9).

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7 Limits and Colimits

Structure