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

Warning The differential graded nerve construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ can be used to produce many interesting examples of $\infty $-categories. However, not every $\infty $-category can be obtained in this way (even up to equivalence). Put differently, $\infty $-categories of the form $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ have some special features, which are not shared by general $\infty $-categories. For example, if $\operatorname{\mathcal{C}}$ is a pretriangulated differential graded category (Definition ), then the differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$ is a stable $\infty $-category (see Proposition ).