Kerodon

Variant 7.1.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets carrying the cone point of $K^{\triangleright }$ to an object $Y \in \operatorname{\mathcal{C}}$. Set $u = \overline{u}|_{K}$, so that the diagram $\overline{u}$ can be identified with an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{u/}$. We will say that $\overline{u}$ is a colimit diagram if it is an initial object of $\operatorname{\mathcal{C}}_{u/}$. If this condition is satisfied, we say that $\overline{u}$ exhibits $Y$ as a colimit of the diagram $u$.