Proposition 4.6.9.21 (03E0)

Proposition 4.6.9.21. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing objects $X$, $Y$, and $Z$, so that the composition law

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \otimes \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } \]

induces a bilinear map of simplicial abelian groups

\[ \mu : \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \rightarrow \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } ) \]

(see Proposition 2.5.9.1). Then the diagram of Kan complexes

4.74

\begin{equation} \begin{gathered}\label{equation:composition-law-on-dg-nerve} \xymatrix { \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \ar [r]^-{\mu } \ar [d]^{ \rho _{Z,Y} \times \rho _{Y,X} } & \mathrm{K}(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }) \ar [d]^{ \rho _{Z,X} } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }(Y,Z) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X,Y) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z) } \end{gathered} \end{equation}

commutes up to homotopy, where the bottom horizontal map is the composition law of Construction 4.6.9.9.

Proof of Proposition 4.6.9.21. Let $\operatorname{\mathcal{C}}^{\Delta }$ denote the underlying simplicial category of the differential graded category $\operatorname{\mathcal{C}}$ (Construction 2.5.9.2). By virtue of Exercise 4.6.8.4, we can identify (4.74) with the outer rectangle of a larger diagram

\[ \xymatrix { \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \ar [r]^-{\mu } \ar [d] & \mathrm{K}(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }) \ar [d] \\ \operatorname{Hom}_{ \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )}( Y,Z) \times \operatorname{Hom}_{ \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )}( X,Y) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta })}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }(Y,Z) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X,Y) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z), } \]

where middle horizontal map is given by the composition law of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } )$. We now observe that the upper square commutes up to homotopy by virtue of Proposition 4.6.9.19, and the lower square commutes up to homotopy by the functoriality of Construction 4.6.9.9. $\square$