Proposition 2.4.4.3. Let $S_{\bullet }$ be a simplicial set. Then there exists a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ and a morphism of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as a path category of $S_{\bullet }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. This is a special case of Proposition 1.2.3.15, since the category $\operatorname{Cat_{\Delta }}$ admits small colimits (Proposition 2.4.1.13). Explicitly, the simplicial path category of a simplicial set $S_{\bullet }$ is given by the generalized geometric realization
\[ | S_{\bullet } |^{ \operatorname{Path}[-]_{\bullet } } = \varinjlim _{ \Delta ^ n \rightarrow S_{\bullet } } \operatorname{Path}[n]_{\bullet }, \]
where $\operatorname{Path}[-]_{\bullet }$ denotes the cosimplicial object of $\operatorname{Cat_{\Delta }}$ defined in Notation 2.4.3.1. $\square$