Proposition 7.7.1.19 (05T7)

Proposition 7.7.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism, and define $\beta : \mathscr {F} \rightarrow \underline{C}$ as above. The following conditions are equivalent:

$(1)$

The morphism $\overline{ \mathscr {F} }$ is a universal colimit diagram, in the sense of Definition 7.7.1.15.

$(2)$

For every morphism $u: C' \rightarrow C$ in $\operatorname{\mathcal{C}}$ and every levelwise pullback diagram

7.82

\begin{equation} \begin{gathered}\label{equation:universal-colimit-diagram-reformulated} \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{C}' \ar [d]^{ \underline{u} } \\ \mathscr {F} \ar [r]^{\beta } & \underline{C} } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, the natural transformation $\beta '$ exhibits $C'$ as a colimit of $\mathscr {F}'$

Proof. The implication $(2) \Rightarrow (1)$ follows from Remark 7.1.3.6 and Proposition 7.7.1.13. To prove the converse, it suffices to observe that every diagram of the form (7.82) is isomorphic to one which is obtained from a natural transformation $\overline{\mathscr {F}}' \rightarrow \overline{\mathscr {F}}$ of $K^{\triangleright }$-indexed diagrams in $\operatorname{\mathcal{C}}$. This follows from the fact that the comparison map $K \diamond \Delta ^0 \twoheadrightarrow K \star \Delta ^0$ of Theorem 4.5.8.8 is a categorical equivalence of simplicial sets. $\square$