Kerodon

Remark 10.2.5.3. In the augmented simplex category $\operatorname{{\bf \Delta }}_{+}$, there is unique morphism $\delta ^{0}_{0}: [-1] \rightarrow [0]$. This morphism determines a fully faithful functor $[1] \rightarrow \operatorname{{\bf \Delta }}_{+}$, whose image is the full subcategory $\operatorname{{\bf \Delta }}_{+}^{\leq 0} \subseteq \operatorname{{\bf \Delta }}_{+}$ spanned by the objects $[0]$ and $[-1]$. Combining Remarks 10.2.4.2 and 7.3.2.4, we see that an augmented simplicial object $X_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ is a Čechnerve if and only if it is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq 0} )^{\operatorname{op}} \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} )^{\operatorname{op}}$.