Kerodon

Definition 10.2.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a semisimplicial object of $\operatorname{\mathcal{C}}$. We say that an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ if it is a colimit of the diagram $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}} ) \rightarrow \operatorname{\mathcal{C}}$. If $X^{\bullet }$ is a cosemisimplicial object of $\operatorname{\mathcal{C}}$, we say that an object $X \in \operatorname{\mathcal{C}}$ is a totalization of $X^{\bullet }$ if it is a limit of the diagram $X^{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} ) \rightarrow \operatorname{\mathcal{C}}$.