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7.5 Homotopy Limits and Colimits

Let $\operatorname{\mathcal{C}}$ be a small category, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. Recall that the diagram $\mathscr {F}$ has a limit $\varprojlim ( \mathscr {F} )$ in the category of simplicial sets, given concretely by the formula

\[ \varprojlim ( \mathscr {F} )(C)_ n = \varprojlim _{C \in \operatorname{\mathcal{C}}} \mathscr {F}(C)_{n} \]

(Remark 1.1.1.13). However, from the perspective of homotopy theory, the construction $\mathscr {F} \mapsto \varprojlim (\mathscr {F})$ is poorly behaved:

  • Although each of the simplicial sets $\{ \mathscr {F}(C) \} _{C \in \operatorname{\mathcal{C}}}$ is assumed to be a Kan complex, the inverse limit $\varprojlim (\mathscr {F})$ need not be a Kan complex.

  • If $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a natural transformation between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ which is a levelwise homotopy equivalence (Remark 4.5.6.2), then the induced map $\varprojlim (\mathscr {F}) \rightarrow \varprojlim (\mathscr {G})$ need not be a (weak) homotopy equivalence (see Warning 3.4.0.1).

These deficiencies can be remedied by working in the framework of $\infty $-categories. By passing to the homotopy coherent nerve, every functor of ordinary categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ determines a functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}):\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}}$. By virtue of Corollary 7.4.5.6, the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ admits all (small) limits and colimits. In particular, there exists a Kan complex $X$ which is a limit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$. This construction has the advantage of being homotopy invariant: if $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a levelwise homotopy equivalence, then $X$ is also a limit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {G} )$ (see Remark 7.1.1.8). However, it has the disadvantage of being somewhat inexplicit: the Kan complex $X$ is a priori well-defined only up to homotopy equivalence, rather than up to isomorphism.

By combining the results of §7.4 and §5.3, we can obtain a more direct description of the Kan complex $X$. Let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (Definition 5.3.3.1). It follows from Example 5.6.5.6 that $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$ is a covariant transport representation for the left fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. By virtue of Corollary 7.4.5.2, the Kan complex $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) )$ is a limit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$. We will denote this Kan complex by $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ and refer to it as the homotopy limit of the diagram $\mathscr {F}$ (Construction 7.5.1.1). In §7.5.1, we give review some elementary properties of this construction (which goes back to the work of Bousfield and Kan; see [MR0365573]).

In §7.5.2, we extend the definition of the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ to the case where $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is a diagram of $\infty $-categories (rather than a diagram of Kan complexes). In this case, the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration (rather than a left fibration), and we define $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ to be the $\infty $-category of cocartesian sections of $U$ (that is, sections which carry each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$: see Construction 7.5.2.1). It follows from the results of §7.4 that $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a limit of the diagram of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ (Proposition 7.5.2.4).

In §7.5.3, we consider another perspective on the homotopy limit construction $\mathscr {F} \mapsto \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$: it can be viewed as a right derived functor of the usual inverse limit $\mathscr {F} \mapsto \varprojlim (\mathscr {F} )$. More precisely, for every diagram of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, there is a canonical isomorphism $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \varprojlim ( \mathscr {F}^{+} )$, where $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant replacement for the diagram $\mathscr {F}$ (see Construction 7.5.3.3 and Proposition 7.5.3.7). In particular, there is a tautological map $\varprojlim (\mathscr {F} ) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ (see Remark 7.5.2.10), which is an equivalence of $\infty $-categories when the diagram $\mathscr {F}$ is already isofibrant (Proposition 7.5.3.12). This condition is satisfied, for example, when the diagram $\mathscr {F}$ corresponds to a tower of $\infty $-categories

\[ \cdots \rightarrow \operatorname{\mathcal{E}}(3) \rightarrow \operatorname{\mathcal{E}}(2) \rightarrow \operatorname{\mathcal{E}}(1) \rightarrow \operatorname{\mathcal{E}}(0) \]

in which the transition functors are isofibrations (see Example 7.5.3.13).

Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ be an extension of $\mathscr {F}$, carrying the initial object of $\operatorname{\mathcal{C}}^{\triangleleft }$ to a Kan complex $X$. We say that $\overline{\mathscr {F}}$ is a categorical limit diagram if the composite map

\[ X \rightarrow \varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \]

is a homotopy equivalence (Definition 7.5.4.1). In §7.5.4, we show that this condition is equivalent to the requirement that $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \overline{\mathscr {F}} )$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ (Proposition 7.5.4.5). Moreover, we extend the definition of homotopy limit diagram to the case where $\overline{\mathscr {F}}$ is an arbitrary diagram of simplicial sets (Definition 7.5.4.8), and show that it generalizes the notion of homotopy pullback diagram introduced in §3.4.1 (Proposition 7.5.4.13). In §7.5.5, we introduce the parallel (and closely related) notion of categorical limit diagram (Definition 7.5.5.11), and show that it generalizes the notion of categorical pullback square introduced in §4.5.2 (Corollary 7.5.5.10).

There is a close relationship between the homotopy limit construction $\mathscr {F} \mapsto \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ of this section and the homotopy colimit construction $\mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ introduced in §5.3.2. If $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ is a diagram of simplicial sets and $X$ is a Kan complex, then there is a canonical isomorphism of simplicial sets

\[ \underset {\longleftarrow }{\mathrm{holim}}( X^{\mathscr {F}} )^{\operatorname{op}} \simeq \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{\operatorname{op}}), X^{\operatorname{op}} ), \]

where $X^{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ denotes the functor given by $C \mapsto \operatorname{Fun}( \mathscr {F}(C), X)$ (Example 7.5.1.7; see Example 7.5.2.9 for a generalization to the case where $X$ is an $\infty $-category). Just as the homotopy limit construction can be viewed as a right derived functor of the limit functor $\varprojlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$, the homotopy colimit construction can be viewed as a left derived functor of the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$. More precisely, we show in §7.5.6 that the homotopy colimit of a diagram $\mathscr {F}$ is isomorphic to the colimit $\varinjlim (\mathscr {G})$, where $\mathscr {G}$ is a projectively cofibrant diagram of simplicial sets equipped with a levelwise weak homotopy equivalence $\alpha : \mathscr {G} \rightarrow \mathscr {F}$ (Construction 7.5.6.8).

In §7.5.7, we show that the homotopy colimit construction has a close relationship with the formation of colimits in the $\infty $-category $\operatorname{\mathcal{S}}$. If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ is a diagram of Kan complexes, then a Kan complex is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$ if and only if it is weakly homotopy equivalent to $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ (Proposition 7.5.7.1). In fact, we can be more precise: if $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Kan}$ is a diagram extending $\mathscr {F}$ which carries the final object of $\operatorname{\mathcal{C}}^{\triangleright }$ to a Kan complex $X$, then $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$ is a colimit diagram if and only if the composite map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \twoheadrightarrow \varinjlim ( \mathscr {F} ) \rightarrow X$ is a weak homotopy equivalence (Corollary 7.5.7.7). If this condition is satisfied, we will say that $\overline{\mathscr {F}}$ is a homotopy colimit diagram (Definition 7.5.7.3). In §7.5.8, we introduce the parallel notion of categorical colimit diagram (Definition 7.5.8.2), which has a similar relationship with colimits in the $\infty $-category $\operatorname{\mathcal{QC}}$ (Corollary 7.5.8.9).

Structure

  • Subsection 7.5.1: Homotopy Limits of Kan Complexes
  • Subsection 7.5.2: Homotopy Limits of $\infty $-Categories
  • Subsection 7.5.3: The Homotopy Limit as a Derived Functor
  • Subsection 7.5.4: Homotopy Limit Diagrams
  • Subsection 7.5.5: Categorical Limit Diagrams
  • Subsection 7.5.6: The Homotopy Colimit as a Derived Functor
  • Subsection 7.5.7: Homotopy Colimit Diagrams
  • Subsection 7.5.8: Categorical Colimit Diagrams
  • Subsection 7.5.9: Application: Filtered Colimits of $\infty $-Categories