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Construction Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. For every nonnegative integer $n \geq 0$, we define an ordinary category $\operatorname{\mathcal{C}}_{n}$ as follows:

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

  • Let $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_ n) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$ be objects of $\operatorname{\mathcal{C}}_ n$. A morphism from $X$ to $Y$ in the category $\operatorname{\mathcal{C}}_ n$ is an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. In other words, we have an equality of sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}_ n}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$.

  • For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}_ n$, the composition $g \circ f: X \rightarrow Z$ is given by the image of the $n$-simplex $(g,f)$ under the map of simplicial sets

    \[ c_{Z,Y,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }. \]
  • For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}_ n) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the identity morphism from $X$ to itself in the category $\operatorname{\mathcal{C}}_ n$ is the $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet }$ which corresponds to the composite map

    \[ \Delta ^ n \rightarrow \Delta ^{0} \xrightarrow { \operatorname{id}_ X } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \]