Kerodon

Example 6.2.5.11. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}= \operatorname{Path}[G]$ denote its path category (Construction 1.3.7.1). Let $S$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ which are either identity morphisms or are indecomposable. Then $S$ is a class of short morphisms for the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (the existence of $S$-optimal factorizations follows from Example 6.2.5.3, and the remaining requirements are immediate from the definitions). Moreover, the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\mathrm{short}}$ can be identified with the directed graph $G$ (regarded as a $1$-dimensional simplicial set; see §1.1.6). Applying Theorem 6.2.5.10 in this case, we recover the statement that the inclusion map $G \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is inner anodyne (Proposition 1.5.7.3).