Kerodon

Remark 8.5.4.13. Since the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is $1$-dimensional, the morphism $Q$ of Notation 8.5.4.12 factors (uniquely) through the $1$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$, which we can identify with the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$. Under this identification, $Q$ corresponds to a morphism of simplicial sets $q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \Delta ^1 / \operatorname{\partial \Delta }^1$. This is a covering map (see Definition 3.1.4.1), which exhibits the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$ as the quotient of $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ by a free action of the group $(\operatorname{\mathbf{Z}},+)$ by translations. The induced map of geometric realizations $| \operatorname{Spine}[\operatorname{\mathbf{Z}}] | \rightarrow | \Delta ^1 / \operatorname{\partial \Delta }^1 |$ can be identified with the standard covering map $\mathbf{R} \rightarrow S^1$ in the category of topological spaces.