Kerodon

Notation 2.4.4.2 (The Path Category of a Simplicial Set). Let $S_{\bullet }$ be a simplicial set. It follows immediately from the definitions that if there exists a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as the path category of $S_{\bullet }$, then the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ (and the morphism $u$) are uniquely determined up to isomorphism and depend functorially on $S_{\bullet }$. We will emphasize this dependence by denoting $\operatorname{\mathcal{C}}_{\bullet }$ by $\operatorname{Path}[ S ]_{\bullet }$ and referring to it as the path category of the simplicial set $S_{\bullet }$.