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7.5.7 Homotopy Colimit Diagrams

Let $\operatorname{\mathcal{C}}$ be a (small) category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a (strictly commutative) diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. Passing to the homotopy coherent nerve, we obtain 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, this functor admits a colimit in the $\infty $-category $\operatorname{\mathcal{S}}$. This colimit admits a classical description, using the homotopy colimit of Construction 5.3.2.1.

Proposition 7.5.7.1. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. Then a Kan complex $X$ is a colimit of the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ if and only if it is weakly homotopy equivalent to the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$.

Proof. Let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the taut scaffold of Construction 5.3.4.11. Then $\lambda _{t}$ is a categorical equivalence of simplicial sets (Corollary 5.3.5.9), and therefore a weak homotopy equivalence (Remark 4.5.3.4). The desired result now follows from Corollary 7.4.5.7. $\square$

Example 7.5.7.2. Let $\operatorname{\mathcal{C}}$ be a groupoid and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is a Kan complex (Corollary 5.3.4.23). In this case, Proposition 7.5.7.1 guarantees that $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ in the $\infty $-category $\operatorname{\mathcal{S}}$. For example, if $X$ is a Kan complex equipped with an action of a group $G$, then the homotopy quotient $X_{\mathrm{h}G}$ is a colimit of the associated diagram $B_{\bullet }G \rightarrow \operatorname{\mathcal{S}}$ (Example 5.3.4.24).

Our goal in this section is to formulate a companion to Proposition 7.5.7.1, which provides concrete models for colimit diagrams in the $\infty $-category $\operatorname{\mathcal{S}}$ (rather than colimits in the abstract).

Definition 7.5.7.3. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets restriction $\mathscr {F} = \overline{ \mathscr {F} }|_{\operatorname{\mathcal{C}}}$. We will say that $\overline{ \mathscr {F} }$ is a homotopy colimit diagram if the composite map

\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} ) \rightarrow \overline{\mathscr {F}}( {\bf 1} ) \]

is a weak homotopy equivalence of simplicial sets. Here ${\bf 1}$ denotes the final object of the cone $\operatorname{\mathcal{C}}^{\triangleright } \simeq \operatorname{\mathcal{C}}\star \{ {\bf 1}\} $, and the morphism on the left is the comparison map of Remark 5.3.2.9.

Example 7.5.7.4. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a colimit diagram in the category of simplicial sets. If the diagram $\mathscr {F} = \overline{ \mathscr {F} }|_{ \operatorname{\mathcal{C}}}$ is projectively cofibrant, then $\overline{\mathscr {F}}$ is a homotopy colimit diagram: this is a reformulation of Corollary 7.5.6.14 (for a stronger statement, see Corollary 7.5.8.7).

Proposition 7.5.7.5 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ be a natural transformation between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {G}}(C)$ is a weak homotopy equivalence of simplicial sets. Then any two of the following conditions imply the third:

$(1)$

The functor $\overline{\mathscr {F}}$ is a homotopy colimit diagram.

$(2)$

The functor $\overline{\mathscr {G}}$ is a homotopy colimit diagram.

$(3)$

The natural transformation $\alpha $ induces a weak homotopy equivalence $\overline{\mathscr {F}}( {\bf 1} ) \rightarrow \overline{\mathscr {G}}( {\bf 1} )$, where ${\bf 1}$ denotes the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$.

Proof. Setting $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ and $\mathscr {G} = \overline{\mathscr {G}}|_{\operatorname{\mathcal{C}}}$, we observe that $\alpha $ determines a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r] \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \ar [d] \\ \overline{\mathscr {F}}( {\bf 1} ) \ar [r] & \overline{\mathscr {G}}( {\bf 1} ) } \]

where the upper horizontal map is a weak homotopy equivalence (Proposition 5.3.2.18). The desired result now follows from the two-out-of-three property (Remark 3.1.6.16). $\square$

There is a close relationship between homotopy colimit diagrams (in the sense of Definition 7.5.7.3) and homotopy limit diagrams (in the sense of Definition 7.5.4.1).

Proposition 7.5.7.6. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then $\overline{\mathscr {F}}$ is a homotopy colimit diagram if and only if, for every Kan complex $X$, the functor

\[ X^{\overline{\mathscr {F}}}: ( \operatorname{\mathcal{C}}^{\triangleright } )^{\operatorname{op}} \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {F}}(C), X) \]

is a homotopy limit diagram.

Proof. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, let ${\bf 1}$ denote the final object of $\operatorname{\mathcal{C}}^{\triangleright }$, and let $\theta : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \overline{\mathscr {F}}({\bf 1})$ be the map appearing in Definition 7.5.7.3. Then $\overline{\mathscr {F}}$ is a homotopy colimit diagram if and only if, for every Kan complex $X$, precomposition with $\theta $ induces a homotopy equivalence of Kan complexes

\[ \theta ^{\ast }: \operatorname{Fun}( \overline{\mathscr {F}}({\bf 1}), X) \rightarrow \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), X). \]

Setting $\mathscr {G} = \mathscr {F}^{\operatorname{op}}$, $\overline{\mathscr {G}} = \overline{\mathscr {F}}^{\operatorname{op}}$, and $Y = X^{\operatorname{op}}$, Example 7.5.1.7 identifies $\theta ^{\ast }$ with the opposite of the restriction map $Y^{ \overline{\mathscr {G}} }( {\bf 1} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( Y^{\mathscr {G}} )$ appearing in Definition 7.5.4.1. In particular, $\theta ^{\ast }$ is a homotopy equivalence if and only if $Y^{ \overline{\mathscr {G}} }$ is a homotopy limit diagram of Kan complexes. By virtue of Corollary 7.5.4.12, this is equivalent to the requirement that $X^{\overline{\mathscr {F}}}$ is a homotopy limit diagram. $\square$

Corollary 7.5.7.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then $\overline{\mathscr {F}}$ is a homotopy colimit diagram (in the sense of Definition 7.5.7.3) if and only if the induced functor of $\infty $-categories

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

is a colimit diagram (in the sense of Variant 7.1.2.5).

Proof. Combine Proposition 7.5.7.6 with Corollary 7.5.4.6 (applied to the simplicial category $\operatorname{Kan}^{\operatorname{op}}$). $\square$

Corollary 7.5.7.8. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a functor. Let $\overline{ \mathscr {F} }^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be the functor given on objects by $\overline{ \mathscr {F} }^{\operatorname{op}}(C) = \overline{\mathscr {F}}(C)^{\operatorname{op}}$. Then $\overline{\mathscr {F}}$ is a homotopy colimit diagram if and only if $\overline{\mathscr {F}}^{\operatorname{op}}$ is a homotopy colimit diagram.

Proof. Combine Proposition 7.5.7.6 with Corollary 7.5.7.8. $\square$

Corollary 7.5.7.9. Suppose we are given a commutative diagram of simplicial sets

7.52
\begin{equation} \begin{gathered}\label{equation:homotopy-pushout-as-homotopy-colimit} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_{0} \ar [d] \\ A_{1} \ar [r] & A_{01}, } \end{gathered} \end{equation}

which we identify with a functor $\mathscr {F}: [1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$. Then (7.52) is a homotopy pushout square (in the sense of Definition 3.4.2.1) if and only if $\mathscr {F}$ is a homotopy colimit diagram (in the sense of Definition 7.5.7.3).