# Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?

Let $$\mathcal{C}$$ be an additive category and let $$N_{dg}(\mathcal{C})$$ be the differential graded nerve of the differential graded category $$Ch(\mathcal{C})$$. This is a stable $$\infty$$-category.

Note that $$Ind(\mathcal{C})$$ is naturally an additive category. Then is it true that $$Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))?$$

In general, when does the nerve functor $$N_{dg}$$ commute with "categorical" constructions that can be performed in both the setting of $$dg$$-categories and the setting of $$\infty$$-categories?

• I don't think it's true the way it's written for $C$ the category of finite-dimensional vector spaces. The RHS is the $\infty$-category of unbounded complexes which is $Ind$ of the $\infty$-category of bounded complexes of finite-dimensional vector spaces. – Pavel Safronov Oct 15 '18 at 7:31
• Also, if you take $C$ the category of finitely-generated projective $R$-modules, the RHS is the $\infty$-category of complexes of $R$-modules whose homotopy category is $K(R)$ which is usually not compactly generated. – Pavel Safronov Oct 15 '18 at 7:55
• @PavelSafronov Your comment suggests there could still be some kind of relationship between the two indizations. In general I'd like to know how much computation can be done in the dg setting before passing to the infinity setting – leibnewtz Oct 15 '18 at 14:54