$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.6.2.20. Let $\operatorname{\mathcal{C}}$ be a differential graded category, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and suppose we are given a morphism of simplicial sets $e_0: S \rightarrow \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$, which we identify with a morphism of chain complexes $f: \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. Let $e: S \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Y)$ denote the composition of $e_0$ with the morphism $\rho _{X,Y}$. The following conditions are equivalent:
- $(1)$
The morphism $e$ exhibits $Y$ as a tensor product of $X$ by $S$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.
- $(2)$
Let $Z$ be an object of $\operatorname{\mathcal{C}}$, so that $f$ induces a morphism of chain complexes
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } )_{\ast }. \]
Then $\theta $ is an isomorphism on homology in degrees $\geq 0$.
Proof of Proposition 7.6.2.20.
Fix an object $Z \in \operatorname{\mathcal{C}}$. Using Proposition 4.6.9.21, we see that the diagram of Kan complexes
\[ \xymatrix@C =50pt@R=50pt{ \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \ar [r]^-{ \mathrm{K}(\theta ) } \ar [dd]^{ \rho _{Y,Z} } & \mathrm{K}( \operatorname{Hom}_{\operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } )_{\ast } ) \ar [d]^-{\psi } \\ & \operatorname{Fun}(S, \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } ) ) \ar [d]^-{ \rho _{X,Z} \circ } \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(Y,Z) \ar [r] & \operatorname{Fun}(S, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z))} \]
commutes up to homotopy, where $\psi $ is the homotopy equivalence of Example 3.1.6.11 and the bottom horizontal map is given by combining $e$ with the composition law on the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$. Note that condition $(1)$ is equivalent to the requirement that the bottom horizontal map is a homotopy equivalence (for each object $Z \in \operatorname{\mathcal{C}}$). Since the map $\rho _{Y,Z}$ and $\rho _{X,Z}$ are also homotopy equivalences (Proposition 4.6.5.10), this is equivalent to the requirement that $\mathrm{K}( \theta )$ is a homotopy equivalence (for each object $Z \in \operatorname{\mathcal{C}}$). The equivalence of $(1)$ and $(2)$ now follows from the criterion of Corollary 3.2.7.4.
$\square$