# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 2.5.9.4. Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then the underlying simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ is locally Kan (Definition 2.4.1.8). This follows from the observation that each of the simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet } = \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ has the structure of a simplicial abelian group, and is therefore automatically a Kan complex (Proposition 1.1.9.9).