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Remark (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 maps of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ satisfy $d_{i}( \sigma _ n) = \sigma _{n-1}$ for $0 \leq i \leq n$. Applying Corollary, we obtain an isomorphism of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ with the simplicial set $(\Delta ^0)^{+}$ introduced in Construction Here we abuse notation by identifying $\Delta ^0$ with its underlying semisimplicial set.