# Kerodon

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Definition 2.4.4.1. Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We will say that a morphism of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as a path category of $S_{\bullet }$ if, for every simplicial category $\operatorname{\mathcal{D}}_{\bullet }$, composition with $u$ induces a bijection

$\{ \text{Simplicial functors F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }} \} \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{D}}) ).$