Kerodon

Corollary 10.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. Then an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.2.1.3) if and only if it is a colimit of the underlying diagram $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow {X_{\bullet }} \operatorname{\mathcal{C}}$.