Definition 10.2.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. We will say that $X_{\bullet }$ is a Čechnerve if, for every integer $n \geq 0$, the following condition is satisfied:
- $(\ast _ n)$
For $0 \leq i \leq n$, let $\nu _{i}: X_{n} \rightarrow X_{0}$ be the morphism of $\operatorname{\mathcal{C}}$ induced by the inclusion $[0] \simeq \{ i \} \subseteq [n]$. Then the morphisms $\{ \nu _{i} \} _{0 \leq i \leq n}$ exhibit $X_{n}$ as a product of $(n+1)$-copies of $X_0$.