Definition 7.7.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$. Suppose that $\mathscr {F}$ admits a colimit: that is, there exists an object $C \in \operatorname{\mathcal{C}}$ and a natural transformation $\beta : \mathscr {F} \rightarrow \underline{C}$ which exhibits $C$ as a colimit of $\mathscr {F}$. We say that the colimit of $\mathscr {F}$ is universal if the following condition is satisfied:
- $(\ast )$
For every morphism $f: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$ and every levelwise pullback square
\[ \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{C}' \ar [d]^{ \underline{f} } \\ \mathscr {F} \ar [r]^{\beta } & \underline{C} } \]in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, the natural transformation $\beta '$ exhibits $C'$ as a colimit of $\mathscr {F}'$.
We say that $K$-indexed colimits in $\operatorname{\mathcal{C}}$ are universal if, for every diagram $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ which admits a colimit, the colimit of $\mathscr {F}$ is universal.