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 7.3.8.20). Since $T$ is right cofinal (Proposition 7.2.1.10), 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 7.3.8.20.
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 7.3.8.20, 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.
$\square$