Kerodon

Notation 10.2.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. It follows from Remarks 10.2.5.3 and 7.3.6.6 that if $f$ admits a Čechnerve $C_{\bullet }$, then the augmented simplicial object $C_{\bullet }$ is determined up to isomorphism and depends functorially on $f$. To emphasize this dependence, we will denote $C_{\bullet }$ by $\operatorname{\check{C}}_{\bullet }(X/Y)$ and refer to it as the Čechnerve of the morphism $f: X \rightarrow Y$. Alternatively, we can identify $\operatorname{\check{C}}_{\bullet }(X/Y)$ with the simplicial object of $\operatorname{\mathcal{C}}_{/Y}$ given by the Čechnerve of $f$ (in the sense of Notation 10.2.4.4).