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

Example (Edges of the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then $\operatorname{N}_{1}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ can be identified with the collection of all triples $(X_0, X_1, f)$ where $X_0$ and $X_1$ are objects of $\operatorname{\mathcal{C}}$ and $f$ is a $0$-cycle in the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_1)_{\bullet }$. In other words, $\operatorname{N}_{1}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is the collection of all morphisms in the underlying category $\operatorname{\mathcal{C}}^{\circ }$ of Construction