Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

7.5.8 Categorical Colimit Diagrams

In ยง7.5.7, we introduced the notion of a homotopy colimit diagram (Definition 7.5.7.3), and showed that one can use homotopy colimit diagrams to compute colimits in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (Corollary 7.5.7.7). In this section, we introduce the closely related notion of categorical colimit diagram, which can be used to compute colimits in the larger $\infty $-category $\operatorname{\mathcal{QC}}\supset \operatorname{\mathcal{S}}$.

Proposition 7.5.8.1. 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 $\operatorname{\mathcal{C}}$, and let $W$ denote the collection of horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ (Definition 5.3.4.1). Then an $\infty $-category $\operatorname{\mathcal{D}}$ is a colimit of the diagram

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

if and only if it is a localization of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$, in the sense of Remark 6.3.2.2.

Proof. Let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the projection map of Definition 5.3.3.1 and let $W'$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. Choose a functor of $\infty $-categories $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ with respect to $W'$. Let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the taut scaffold of Construction 5.3.4.11. Then $\lambda _{t}$ is a categorical equivalence of simplicial sets (Corollary 5.3.5.9). Moreover, a morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ belongs to $W'$ if and only if it is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$) to an element of $\lambda _{t}(W)$ (see Proposition 5.3.3.15). It follows that the composite map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \xrightarrow { \lambda _{t} } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \xrightarrow {T} \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ with respect to $W$. We conclude by observing that $\operatorname{\mathcal{D}}$ is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ (Corollary 7.4.3.16). $\square$

Motivated by Proposition 7.5.8.1, we introduce the following variant of Definition 7.5.7.3:

Definition 7.5.8.2. 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. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, and let $W$ denote the collection of horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ (Definition 5.3.4.1). We will say that $\overline{ \mathscr {F} }$ is a categorical colimit diagram if the composite map

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

exhibits $\overline{\mathscr {F}}( {\bf 1} )$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$. (see Definition 6.3.1.9). 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.

Remark 7.5.8.3. Let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a categorical colimit diagram of simplicial sets. Then $\overline{\mathscr {F}}$ is also a homotopy colimit diagram of simplicial sets, in the sense of Definition 7.5.7.3. This follows from the observation that every localization of simplicial sets is a weak homotopy equivalence (Remark 6.3.1.16).

Proposition 7.5.8.4. 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. The following conditions are equivalent:

$(1)$

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

$(2)$

For every $\infty $-category $\operatorname{\mathcal{D}}$, the diagram of $\infty $-categories

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

is a categorical limit diagram (Definition 7.5.5.1).

$(3)$

For every $\infty $-category $\operatorname{\mathcal{D}}$, the diagram of Kan complexes

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

is a homotopy limit diagram (Definition 7.5.4.1).

Proof. The equivalence of $(1)$ and $(2)$ follows by combining Example 7.5.2.11 with Corollary 7.5.5.15. The equivalence with $(3)$ follows by combining the same results with Proposition 6.3.1.13 and Example 7.5.2.8. $\square$

Corollary 7.5.8.5. Suppose we are given a commutative diagram of simplicial sets

7.54
\begin{equation} \begin{gathered}\label{equation:categorical-pushout-vs-categorical-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 (7.54) is a categorical pushout square (in the sense of Definition 4.5.4.1) if and only if $\mathscr {F}$ is a categorical colimit diagram (in the sense of Definition 7.5.8.2).

Corollary 7.5.8.6 (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 categorical equivalence of simplicial sets. Then any two of the following conditions imply the third:

$(1)$

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

$(2)$

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

$(3)$

The natural transformation $\alpha $ induces a categorical 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. By virtue of Proposition 7.5.8.4 (and Proposition 4.5.3.8), it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{D}}$, any two of the following conditions imply the third:

$(1_{\operatorname{\mathcal{D}}} )$

The functor

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

is a categorical limit diagram.

$(2_{\operatorname{\mathcal{D}}})$

The functor

\[ (\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {G}}(C), \operatorname{\mathcal{D}}) \]

is a categorical limit diagram.

$(3_{\operatorname{\mathcal{D}}})$

The natural transformation $\alpha $ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \overline{\mathscr {G}}( {\bf 1} ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \overline{\mathscr {F}}( {\bf 1} ), \operatorname{\mathcal{D}})$.

This follows from Remark 7.5.5.6. $\square$

Corollary 7.5.8.7. 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 categorical colimit diagram.

Proof. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and define $\overline{\mathscr {G}}: (\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}$ by the formula $\overline{\mathscr {G}}(C) = \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}})$. By virtue of Proposition 7.5.8.4, it will suffice to show that the diagram of Kan complexes $\overline{\mathscr {G}}^{\simeq }$ is a homotopy limit diagram. Setting $\mathscr {G} = \overline{\mathscr {G}}|_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }$, our assumption that $\mathscr {F}$ is projectively cofibrant guarantees that the diagram $\mathscr {G}$ is isofibrant (Remark 7.5.6.6). It follows that the diagram of Kan complexes $\mathscr {G}^{\simeq }$ is also isofibrant, and that $\overline{\mathscr {G}}^{\simeq }$ is a limit diagram (Corollary 4.5.6.19). The desired result now follows from Example 7.5.4.2. $\square$

Corollary 7.5.8.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\theta : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \varinjlim (\mathscr {F})$ be the comparison map of Remark 5.3.2.9, and let $W$ denote the collection of all horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ (Definition 5.3.4.1). If $\mathscr {F}$ is projectively cofibrant (Definition 7.5.6.1), then $\theta $ exhibits $\varinjlim ( \mathscr {F} )$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$.

Proof. This is a restatement of Corollary 7.5.8.7. $\square$

Corollary 7.5.8.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. Then $\overline{\mathscr {F}}$ is a categorical colimit diagram (in the sense of Definition 7.5.7.3) if and only if the induced functor of $\infty $-categories

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

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

Proof. Combine Proposition 7.5.8.4 with Corollary 7.5.4.6 (applied to the simplicial category $\operatorname{QCat}^{\operatorname{op}}$). $\square$

Corollary 7.5.8.10. 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 categorical colimit diagram if and only if it is a homotopy colimit diagram.

Corollary 7.5.8.11. 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 categorical colimit diagram if and only if $\overline{\mathscr {F}}^{\operatorname{op}}$ is a categorical colimit diagram.

We close this section with an application of the formalism of categorical colimit diagrams.

Proposition 7.5.8.12 (Rewriting Colimits). Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a categorical colimit diagram which carries the final object of $\operatorname{\mathcal{C}}^{\triangleright }$ to a simplicial set $K$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category equipped with a diagram $q: K \rightarrow \operatorname{\mathcal{D}}$ satisfying the following condition:

$(\ast )$

For each object $C \in \operatorname{\mathcal{C}}$, the composite map

\[ q_{C}: \overline{\mathscr {F}}(C) \rightarrow K \xrightarrow {q} \operatorname{\mathcal{D}} \]

admits a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$.

Then there exists a functor $Q: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ with the following properties:

$(1)$

For each object $C \in \operatorname{\mathcal{C}}$, the object $Q(C) \in \operatorname{\mathcal{D}}$ is a colimit of the diagram $q_{C}$.

$(2)$

An object $X \in \operatorname{\mathcal{D}}$ is a colimit of the diagram $q$ if and only if it is a colimit of $Q$. In particular, the diagram $q$ has a colimit in $\operatorname{\mathcal{D}}$ if and only if the diagram $Q$ has a colimit in $\operatorname{\mathcal{D}}$.

$(3)$

Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories which preserves the colimit of each of the diagrams $q_{C}$, and suppose that the diagrams $q$ and $Q$ admit colimits in $\operatorname{\mathcal{D}}$. Then $G$ preserves the colimit of $q$ if and only if it preserves the colimit of $Q$.

Proof. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, let $U: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the projection map, and let $W$ be the collection of all horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$. The diagram $\overline{\mathscr {F}}$ then determines a morphism of simplicial sets $T: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow K$ which exhibits $K$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$. It follows from assumption $(\ast )$ that for each object $C \in \operatorname{\mathcal{C}}$, the composite map

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \hookrightarrow \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \xrightarrow {T} K \xrightarrow {q} \operatorname{\mathcal{D}} \]

admits a colimit in $\operatorname{\mathcal{D}}$. Applying Corollary 7.3.5.3, we conclude that there is a functor $Q: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : T \circ q \rightarrow Q \circ U$ which exhibits $Q$ as a left Kan extension of $T \circ q$ along $U$. We will complete the proof by showing that $Q$ satisfies conditions $(1)$, $(2)$, and $(3)$ of Proposition 7.5.8.12. Condition $(1)$ follows immediately from Remark 7.3.5.4.

We now prove $(2)$. Assume first that $X \in \operatorname{\mathcal{D}}$ is a colimit of the diagram $Q$. For every simplicial set $S$, we let $\underline{X}_{S}$ denote the image of $X$ in the $\infty $-category $\operatorname{Fun}(S, \operatorname{\mathcal{D}})$. Choose a natural transformation $\alpha : Q \rightarrow \underline{X}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$ which exhibits $X \in \operatorname{\mathcal{D}}$ as a colimit of the diagram $Q$, let $\widetilde{\alpha }: Q \circ U \rightarrow \underline{X}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})}$ denote the image of $\alpha $ in $\operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}), \operatorname{\mathcal{D}})$, and let $\widetilde{\gamma }: q \circ T \rightarrow \underline{X}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})} = \underline{X}_{K} \circ T$ be a composition of $\beta $ with $\widetilde{\alpha }$ in $\operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}), \operatorname{\mathcal{D}})$. Since precomposition with $T$ induces a fully faithful functor $\operatorname{Fun}(K, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}), \operatorname{\mathcal{D}})$, we may assume without loss of generality that $\widetilde{\gamma }$ is the image of a natural transformation $\gamma : q \rightarrow \underline{X}_{K}$. Note that $\widetilde{\gamma }$ exhibits $X$ as a colimit of the diagram $q \circ T$ (Corollary 7.3.7.20). Since $T$ is right cofinal (Proposition 7.2.1.9), it follows that $\gamma $ exhibits $X$ as a colimit of the diagram $q$ (Corollary 7.2.2.7).

To prove the reverse implication, it will suffice to show that if the diagram $q: K \rightarrow \operatorname{\mathcal{D}}$ admits a colimit, then $Q$ also admits a colimit. Since $T$ is right cofinal, the diagram $q \circ T$ also admits a colimit in $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.11), so the desired result is immediate from Corollary 7.3.7.20.

We now prove $(3)$. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories which preserves the colimit of the diagram $q_{C}$, for each object $C \in \operatorname{\mathcal{C}}$. Let $\alpha : Q \rightarrow \underline{X}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$ and $\gamma : q \rightarrow \overline{X}_{K}$ be defined as above; we wish to show that $G(\alpha )$ exhibits $G(X)$ as a colimit of the diagram $G \circ Q$ if and only if $G(\gamma )$ exhibits $G(X)$ as a colimit of the diagram $G \circ q$. Using Corollary 7.2.2.7, we see that latter condition is equivalent to the requirement that $G( \widetilde{\gamma } )$ exhibits $G(X)$ as a colimit of the diagram $G \circ q \circ T$. By virtue of Corollary 7.3.7.20, we are reduced to showing that the natural transformation $G(\beta )$ exhibits $G \circ Q$ as a left Kan extension of $G \circ q \circ T$ along $U$. This follows from the criterion of Remark 7.3.5.4. $\square$

Corollary 7.5.8.13. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a categorical colimit diagram carrying the final object of $\operatorname{\mathcal{C}}^{\triangleright }$ to a simplicial set $K$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and $\overline{\mathscr {F}}(C)$-indexed colimits, for each object $C \in \operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{D}}$ also admits $K$-indexed colimits. Moreover, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor of $\infty $-categories which preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and $\overline{\mathscr {F}}(C)$-indexed colimits for each $C \in \operatorname{\mathcal{C}}$, then $G$ also preserves $K$-indexed colimits.