Kerodon

Example 10.3.1.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a sieve, and let $X$ be an object of $\operatorname{\mathcal{C}}^{0}$. If $C_{\bullet }$ is any simplicial object of $\operatorname{\mathcal{C}}$ satisfying $C_{0} = X$, then $C_{\bullet }$ can also be regarded as a simplicial object of $\operatorname{\mathcal{C}}^{0}$. Applying Remark 10.3.1.9, we deduce that $C_{\bullet }$ is a Čechnerve of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}^{0}$ if and only if it is a Čechnerve of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Definition 10.2.4.1).