Notation 8.5.4.14. Let $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ denote the $1$-dimensional simplicial set associated to the directed graph
\[ \cdots \rightarrow -2 \rightarrow -1 \rightarrow 0 \rightarrow 1 \rightarrow 2 \rightarrow \cdots \]
We let $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ be the morphism of simplicial sets corresponding to the diagram
\[ \cdots \rightarrow \widetilde{X} \xrightarrow { \widetilde{e} } \widetilde{X} \xrightarrow { \widetilde{e} } \widetilde{X} \xrightarrow { \widetilde{e} } \widetilde{X} \xrightarrow {\widetilde{e}} \widetilde{X} \rightarrow \cdots \]
in the category $\operatorname{Idem}$.