Example 1.5.7.12 (Free Monoids). Let $M$ be the free monoid generated by a set $E$. Then we can identify $BM$ with the path category $\operatorname{Path}[G]$ of a directed graph $G$ satisfying
see Example 1.3.7.3. Invoking Proposition 1.5.7.3 and Theorem 1.5.7.1, we obtain the following:
- $(a)$
The inclusion of simplicial sets $G_{\bullet } \hookrightarrow B_{\bullet }M$ is inner anodyne.
- $(b)$
For any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( B_{\bullet }M , \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.
Note that if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then a map of simplicial sets $\sigma _0: G_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ can be identified with a choice of object $X \in \operatorname{\mathcal{C}}$ together with a collection of morphisms $\{ f_{e}: X \rightarrow X \} _{e \in E}$ indexed by $E$. It follows from $(b)$ that any such map admits an (essentially unique) extension to a functor $\sigma : B_{\bullet } M \rightarrow \operatorname{\mathcal{C}}$, which we can interpret as an action of the monoid $M$ on the object $X \in \operatorname{\mathcal{C}}$.