Corollary 7.7.2.30. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks, let $K$ be a simplicial set, and let $\beta : F' \rightarrow F$ be a natural transformation between diagrams $F',F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Then $F'$ is a weak colimit diagram if and only if it satisfies the following condition:
- $(\ast )$
Let $\gamma : G \rightarrow F$ be a cartesian natural transformation in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Then the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})_{/F} }( F', G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})_{/F|_{K}} }( F'|_{K}, G|_{K} ) \]
is a homotopy equivalence of Kan complexes.
Proof.
Let $C \in \operatorname{\mathcal{C}}$ be the value of the diagram $F$ at the cone point of $K^{\triangleright }$, and let $\underline{C}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ denote the constant diagram taking the value $C$. By virtue of Variant 7.7.1.14, a natural transformation $\gamma : G \rightarrow F$ is cartesian if and only if there exists a morphism $u: D \rightarrow C$ of $\operatorname{\mathcal{C}}$ and a pullback diagram
\[ \xymatrix { G \ar [d]^{\gamma } \ar [r] & \underline{D} \ar [d]^{ \underline{u} } \\ F \ar [r] & \underline{C}. } \]
We may therefore assume without loss of generality that $F = \underline{C}$ is a constant diagram, and consider only constant natural transformations $\underline{u}: \underline{D} \rightarrow \underline{C}$ in the formulation of condition $(\ast )$.
Let $\widetilde{\operatorname{\mathcal{C}}}$ denote the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $. Then the pair $(F', \beta )$ can then be identified with a diagram $\widetilde{F}': K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. $\beta $ then determines a lift of $F'$ to a diagram $\widetilde{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. Let us abuse notation by writing $\underline{D}$ for the constant diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ taking the value $D$ (regarded as an obejct of $\operatorname{\mathcal{D}}$ via the morphism $u$). Recall that Corollary 4.6.4.18 supplies an equivalence of the slice $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})_{ / G }$ with the oriented fiber product
\[ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}) } \{ G\} \simeq \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{D}}) \]
and $\operatorname{Fun}( K, \operatorname{\mathcal{C}})_{ / G|_{K} }$ by $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$. We can therefore reformulate condition $(\ast )$ as follows:
- $(\ast ')$
For every object $D \in \operatorname{\mathcal{D}}$, the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( \widetilde{F}', \underline{D} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}})}( \widetilde{F}|_{K}, \underline{D}|_{K} ) \]
is a homotopy equivalence.
Using Corollary 7.1.6.14, we see that condition $(\ast ')$ is satisfied if and only if $\widetilde{F}'$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. By virtue of Corollary 7.7.2.28 (and Remark 7.7.2.29), this is equivalent to the condition that $F$ is a weak colimit diagram.
$\square$