Exercise 4.6.8.4 (01PZ)

Exercise 4.6.8.4. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X$ and $Y$, and let $\operatorname{\mathcal{C}}^{\Delta }$ denote the associated simplicial category (Construction 2.5.9.2). Show that the isomorphism $K( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) \xrightarrow {\sim } \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Y)$ of Example 4.6.5.15 factors as a composition

\[ K( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) = \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet } \xrightarrow {\theta } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } )}^{\mathrm{L}}( X, Y) \xrightarrow {\rho } \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X,Y ), \]

where $\theta $ is the comparison map of Construction 4.6.8.3 and $\rho $ is induced by the trivial Kan fibration $\mathfrak {Z}: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ of Proposition 2.5.9.10. Beware that $\theta $ and $\rho $ are generally not isomorphisms.