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Example 1.4.7.10 (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

\[ \xymatrix { \operatorname{\partial \Delta }^1 \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \Delta ^0 \ar [r] & \Delta ^1 / \operatorname{\partial \Delta }^1. } \]

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

\[ \xymatrix { \bullet \ar@ (ur,ul)[] } \]

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 1.2.6.4) whose nerve is the simplicial set $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ of Example 1.2.4.3. Invoking Proposition 1.4.7.3 and Theorem 1.4.7.1, 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.