Kerodon

Remark 2.5.9.5. Let $\operatorname{\mathcal{C}}$ be a differential graded category, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $f,g: X \rightarrow Y$ be morphisms from $X$ to $Y$ in the underlying category $\operatorname{\mathcal{C}}^{\circ }$ (that is, $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$). Then $f$ and $g$ are homotopic as morphisms of the differential graded category $\operatorname{\mathcal{C}}$ (in the sense of Definition 2.5.4.1) if and only if they are homotopic as morphisms of the simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ (Remark 2.4.1.9); see Example 2.5.6.6. It follows that the isomorphism of underlying categories $\operatorname{\mathcal{C}}^{\circ } \simeq \operatorname{\mathcal{C}}^{\Delta }_{0}$ of Remark 2.5.9.3 induces an isomorphism from the homotopy $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (given by Construction 2.5.4.6) to the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\Delta }$ (given by Construction 2.4.6.1).