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Construction 2.4.6.16 (The Homotopy $2$-Category of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We define a strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ are the objects of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the category $\underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ is the homotopy category of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition map

    \[ \circ : \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( X,Y) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( X, Z ) \]

    is given by the composition

    \begin{eqnarray*} \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( X,Y) & = & (\mathrm{h} \mathit{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }) \times (\mathrm{h} \mathit{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet }) \\ & \xleftarrow {\sim } & \mathrm{h} \mathit{( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )} \\ & \rightarrow & \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }} \\ & = & \operatorname{Hom}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}(X,Z), \end{eqnarray*}

    where the isomorphism is supplied by Corollary 1.4.3.6.

We will refer to $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy $2$-category of $\operatorname{\mathcal{C}}$.