Remark 2.4.1.12 (00K2)

Remark 2.4.1.12. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a (small) simplicial category. Then the construction $[n] \mapsto \operatorname{\mathcal{C}}_{n}$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ (Definition 1.1.0.2) to the category $\operatorname{Cat}$ of (small) categories. Allowing $\operatorname{\mathcal{C}}_{\bullet }$ to vary, we obtain a functor $\operatorname{Cat_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$, which fits into a pullback diagram of categories

\[ \xymatrix@C =80pt@R=50pt{ \operatorname{Cat_{\Delta }}\ar [r]^-{ \operatorname{\mathcal{C}}_{\bullet } \mapsto ([n] \mapsto \operatorname{\mathcal{C}}_ n)} \ar [d]^{\operatorname{Ob}} & \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat}) \ar [d]^{ \operatorname{Ob}} \\ \operatorname{Set}\ar [r] & \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set}), } \]

where the lower horizontal map carries each set $S$ to the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ taking the value $S$.

Phrased more informally: simplicial categories can be identified with simplicial objects $\operatorname{\mathcal{C}}_{\bullet }$ of $\operatorname{Cat}$ for which the underlying simplicial set of objects $[n] \mapsto \operatorname{Ob}( \operatorname{\mathcal{C}}_{n} )$ is constant. In particular, the functor $\operatorname{Cat_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ is a fully faithful embedding.