Example 2.4.4.6 (The Path Category of a Simplex). Let $n \geq 0$ be a nonnegative integer and let $\operatorname{Path}[n]_{\bullet }$ denote the simplicial category of Notation 2.4.3.1. For any simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, we have canonical bijections
\[ \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[n]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) \simeq \operatorname{N}^{\operatorname{hc}}_{n}(\operatorname{\mathcal{C}}) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{N}^{\operatorname{hc}}_{\bullet }(\operatorname{\mathcal{C}}) ). \]
It follows that $\operatorname{Path}[n]_{\bullet }$ is a path category for the standard simplex $\Delta ^ n$, in the sense of Definition 2.4.4.1.