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Example 2.4.2.3 (Ordinary Categories as Simplicial Categories). Let $\operatorname{\mathcal{C}}$ be an ordinary category. We define a simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ as follows:

  • The objects of $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \underline{\operatorname{\mathcal{C}}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, $\operatorname{Hom}_{ \underline{\operatorname{\mathcal{C}}} }( X, Y )_{\bullet }$ is the constant simplicial set associated to the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Construction 1.1.4.2).

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition law

    \[ c_{Z,Y,X}: \operatorname{Hom}_{\underline{\operatorname{\mathcal{C}}}}( Y,Z)_{\bullet } \times \operatorname{Hom}_{ \underline{\operatorname{\mathcal{C}}} }( X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \underline{\operatorname{\mathcal{C}}}}( X, Z )_{\bullet } \]

    on $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ is determined by the composition law $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ on $\operatorname{\mathcal{C}}$.

We will refer to $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ as the constant simplicial category associated to $\operatorname{\mathcal{C}}$. Under the fully faithful embedding $\operatorname{Cat_{\Delta }}\hookrightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ of Remark 2.4.1.12, it corresponds to the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \{ \operatorname{\mathcal{C}}\} \hookrightarrow \operatorname{Cat}$ (see Construction 1.1.4.2). In particular, the underlying category of $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ (in the sense of Example 2.4.1.4) is the ordinary category $\operatorname{\mathcal{C}}$.