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Remark Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\operatorname{\mathcal{C}}^{\circ }$ denote its underlying category (in the sense of Construction Then $\operatorname{\mathcal{C}}^{\circ }$ is isomorphic to the underlying ordinary category $\operatorname{\mathcal{C}}^{\Delta }_{0}$ of the simplicial category $\operatorname{\mathcal{C}}^{\Delta }$ (in the sense of Example Both of these categories can be described concretely as follows:

  • The objects of $\operatorname{\mathcal{C}}^{\circ } \simeq \operatorname{\mathcal{C}}^{\Delta }_0$ are the objects of $\operatorname{\mathcal{C}}$.

  • For objects $X,Y \in \operatorname{\mathcal{C}}$, the morphisms from $X$ to $Y$ in the category $\operatorname{\mathcal{C}}^{\circ } \simeq \operatorname{\mathcal{C}}^{\Delta }_0$ are given by $0$-cycles in the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.