$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.7.2.28. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $F$ is a weak colimit diagram.
- $(2)$
For any object $C \in \operatorname{\mathcal{C}}$, any lift of $F$ to a morphism $\widetilde{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/C}$ is a colimit diagram in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/C}$.
- $(3)$
There exists an object $C \in \operatorname{\mathcal{C}}$ and a lift of $F$ to a colimit diagram $\widetilde{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/C}$.
Proof.
The implication $(2) \Rightarrow (1) \Rightarrow (3)$ are immediate from the definitions. To prove the reverse implications, we note that if $\widetilde{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/C}$ is a lift of $F$, then $\widetilde{F}$ is a weak colimit diagram in $\operatorname{\mathcal{C}}_{/C}$ if and only if $F$ is a weak colimit diagram in $\operatorname{\mathcal{C}}$. It will therefore suffice to show that this condition is satisfied if and only if $\widetilde{F}$ is a colimit diagram in $\operatorname{\mathcal{C}}_{/C}$. This follows from Proposition 7.7.2.26, since the $\infty $-category $\operatorname{\mathcal{C}}_{/C}$ admits finite products (Corollary 7.6.2.17).
$\square$