Definition 10.2.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čechnerve of $f$ if $C_{\bullet }$ is a Čechnerve (in the sense of Definition 10.2.5.1) and the face operator $d^{0}_{0}: C_{0} \rightarrow C_{-1}$ coincides with the morphism $f$ (so that $C_0 = X$ and $C_{-1} = Y$).
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