Remark 8.5.3.3 (The Structure of $\operatorname{N}_{\bullet }( \operatorname{Idem})$). For every integer $n \geq 0$, the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ contains a unique nondegenerate $n$-simplex $\sigma _{n}$, given by the diagram
\[ \widetilde{X} \xrightarrow { \widetilde{e}} \widetilde{X} \xrightarrow {\widetilde{e}} \widetilde{X} \xrightarrow {\widetilde{e}} \cdots \rightarrow \widetilde{X} \xrightarrow { \widetilde{e}} \widetilde{X}. \]
Moreover, the face morphisms of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ satisfy $d^{n}_{i}( \sigma _ n) = \sigma _{n-1}$ for $0 \leq i \leq n$. Applying Corollary 3.3.1.8, we obtain an isomorphism of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ with the simplicial set $(\Delta ^0)^{+}$ introduced in Construction 3.3.1.6. Here we abuse notation by identifying $\Delta ^0$ with its underlying semisimplicial set.