Example 1.5.7.11 (The Simplicial Circle). Let $\Delta ^1 / \operatorname{\partial \Delta }^1$ denote the simplicial set obtained from $\Delta ^1$ by collapsing the boundary $\operatorname{\partial \Delta }^1$ to a point, so that we have a pushout diagram of simplicial sets
We will refer to $\Delta ^1 / \operatorname{\partial \Delta }^1$ as the simplicial circle; note that the geometric realization $| \Delta ^1 / \operatorname{\partial \Delta }^1 |$ is isomorphic to the standard circle $S^1$ as a topological space. The simplicial set $\Delta ^1 / \operatorname{\partial \Delta }^1$ has dimension $\leq 1$, and can therefore be identified with the directed graph $G$ depicted in the diagram
Note that the path category $\operatorname{Path}[G]$ can be identified with the category $B\operatorname{\mathbf{Z}}_{\geq 0}$ associated to the monoid $\operatorname{\mathbf{Z}}_{\geq 0}$ of nonnegative numbers under addition (Example 1.3.7.4) whose nerve is the simplicial set $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ of Construction 1.3.2.5. Invoking Proposition 1.5.7.3 and Theorem 1.5.7.1, we obtain the following:
- $(a)$
The inclusion of simplicial sets $\Delta ^1 / \operatorname{\partial \Delta }^1 \hookrightarrow B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ is inner anodyne.
- $(b)$
For any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.
If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then a morphism of simplicial sets $\Delta ^1 / \operatorname{\partial \Delta }^1 \rightarrow \operatorname{\mathcal{C}}$ can be identified with a pair $(X,f)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $f: X \rightarrow X$ is an endomorphism of $X$ (Definition 1.4.1.5). Theorem 1.5.7.1 then guarantees that the pair $(X,f)$ can be extended to a functor of $\infty $-categories $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathcal{C}}$.