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Construction (The Underlying Category of a Differential Graded Category). To every differential graded category $\operatorname{\mathcal{C}}$, we can associate an ordinary category $\operatorname{\mathcal{C}}^{\circ }$ as follows:

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

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\circ } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism from $X$ to $Y$ in $\operatorname{\mathcal{C}}^{\circ }$ is a $0$-cycle of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.

  • For each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\circ } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the identity morphism from $X$ to itself in $\operatorname{\mathcal{C}}^{\circ }$ is the identity morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{0}$ (which is a cycle by virtue of Remark

  • Composition of morphisms in $\operatorname{\mathcal{C}}^{\circ }$ is given by the composition law on $\operatorname{\mathcal{C}}$ (which preserves $0$-cycles by virtue of Remark

We will refer to $\operatorname{\mathcal{C}}^{\circ }$ as the underlying category of the differential graded category $\operatorname{\mathcal{C}}$ (note that $\operatorname{\mathcal{C}}^{\circ }$ can also be obtained by applying the general procedure described in Example