Subsection 7.5.1 (039Z): Homotopy Limits of Kan Complexes

7.5.1 Homotopy Limits of Kan Complexes

In this section, we introduce the homotopy limit of a diagram of Kan complexes, following Bousfield and Kan (see [MR0365573]).

Construction 7.5.1.1 (Homotopy Limits of Kan Complexes). Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$, and $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of $\mathscr {F}$ (Definition 5.3.3.1). We define

\[ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} ) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \]

to be the simplicial set which parametrizes sections of the projection map $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. We will refer to $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ as the homotopy limit of the diagram $\mathscr {F}$.

Proposition 7.5.1.2. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ is a Kan complex.

Proof. This is a special case of Corollary 4.4.2.5, since the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration (Corollary 5.3.3.19). $\square$

Remark 7.5.1.3 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$. Then $\alpha $ induces a morphism of weighted nerves $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$, and therefore a morphism of Kan complexes $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ): \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G} )$. If $\alpha $ is a levelwise homotopy equivalence, then $T$ is an equivalence of left fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Corollary 5.3.3.20), so $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha )$ is a homotopy equivalence.

Warning 7.5.1.4. In [MR0365573], Bousfield and Kan define the homotopy limit of an arbitrary diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ to be the simplicial set $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}))$ appearing in Construction 7.5.1.1. We will avoid this convention for two reasons:

Many important features of the Bousfield-Kan construction (such as the homotopy invariance property of Remark 7.5.1.3) are true for diagrams of Kan complexes, but not for general diagrams of simplicial sets.
In the case where $\mathscr {F}$ is a diagram of $\infty $-categories, it will be convenient to adopt a slightly different definition of homotopy limit (Construction 7.5.2.1), which generally does not agree with the Bousfield-Kan construction.

Note that every (strictly commutative) diagram of Kan complexes $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ determines a diagram

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}} \]

in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$.

Proposition 7.5.1.5. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then the Kan complex $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a limit of the diagram

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}} \]

in the $\infty $-category $\operatorname{\mathcal{S}}$.

Proof. This is a special case of Corollary 7.4.5.2, since the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F} )$ is a covariant transport representation for the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Example 5.6.5.6). $\square$

We now give a more concrete description of the homotopy limit.

Remark 7.5.1.6. Let $\operatorname{\mathcal{C}}$ be a category. For each object $C \in \operatorname{\mathcal{C}}$, let $\mathscr {E}(C)$ denote the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ / C} )$. The construction $C \mapsto \mathscr {E}(C)$ determines a functor $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, which we view as an object of the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$. For every diagram of Kan complexes $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$, Proposition 5.3.3.24 supplies a canonical isomorphism

\[ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) = \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F} )_{\bullet }, \]

where the right hand side is defined using the simplicial enrichment of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ described in Example 2.4.2.2.

Stated more concretely, we can identify $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F})$ with a simplicial subset of the product ${\prod }_{C \in \operatorname{\mathcal{C}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ), \mathscr {F}(C) )$, whose $n$-simplices are collections of maps $\{ \sigma _{C}: \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \rightarrow \mathscr {F}(C) \} $ which satisfy the following condition:

$(\ast )$

For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \ar [r]^-{\circ f} \ar [d]^{ \sigma _{C} } & \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \ar [d]^{ \sigma _ D } \\ \mathscr {F}(C) \ar [r]^-{ \mathscr {F}(f) } & \mathscr {F}(D) } \]

is commutative.

In particular, we have an equalizer diagram of simplicial sets

\[ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow {\prod }_{C} \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{/C} ), \mathscr {F}(C) ) \rightrightarrows {\prod }_{f: C \rightarrow D} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ), \mathscr {F}(D) ). \]

Example 7.5.1.7 (Duality with Homotopy Colimits). Let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $X$ be a Kan complex, and let $X^{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be the diagram of Kan complexes given by the formula $X^{\mathscr {F}}(C) = \operatorname{Fun}( \mathscr {F}(C), X)$. Let us write $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ for the functor given by the formula $\mathscr {F}^{\operatorname{op}}(C) = \mathscr {F}(C)^{\operatorname{op}}$, and let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by $\mathscr {E}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$. Combining Remark 7.5.1.6 with Proposition 5.3.2.21, we obtain canonical isomorphisms

\begin{eqnarray*} \underset {\longleftarrow }{\mathrm{holim}}( X^{\mathscr {F}} )^{\operatorname{op}} & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( \mathscr {E}, X^{\mathscr {F}} )^{\operatorname{op}}_{\bullet } \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, X^{\mathscr {E}} )^{\operatorname{op}}_{\bullet } \\ & \simeq & \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{\operatorname{op}} ), X^{\operatorname{op}} ). \end{eqnarray*}