Construction 2.5.9.2. Let $\operatorname{\mathcal{C}}$ be a differential graded category. We define a simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ as follows:
The objects of $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\Delta }_{\bullet }) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet }$ is the generalized Eilenberg-MacLane space $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$. More concretely, the $n$-simplices of $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet }$ are chain maps $\sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.
For every triple of objects $X,Y,Z \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\Delta }_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$ and every nonnegative integer $n \geq 0$, the composition law
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(Y,Z)_{n} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}( X,Y)_{n} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}( X,Z)_{n} \]carries a pair $(\sigma , \tau )$ to the $n$-simplex of $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } )$ given by the composite map
\begin{eqnarray*} \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) & \hookrightarrow & \mathrm{N}_{\ast }( \Delta ^ n \times \Delta ^ n; \operatorname{\mathbf{Z}}) \\ & \xrightarrow { \mathrm{AW} } & \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \\ & \xrightarrow {\sigma \boxtimes \tau } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \boxtimes \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \\ & \xrightarrow {\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }. \end{eqnarray*}
We will refer to $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ as the underlying simplicial category of the differential graded category $\operatorname{\mathcal{C}}$.