On exact dg categories

Abstract

We introduce the notion of an exact dg category, which is a simultaneous generalization of the notions of exact category in the sense of Quillen and of pretriangulated dg category in the sense of Bondal--Kapranov. It is also a differential graded analogue of Barwick's notion of exact $\infty$-category and a differential graded enhancement of Nakaoka--Palu's notion of extriangulated category. It is completely different from Positselski's notion of exact DG-category. Our motivations come for example from the categories appearing in the additive categorification of cluster algebras with coefficients. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We obtain a number of fundamental results concerning the dg nerve, the dg derived category, tensor products and functor categories with exact dg target. For example, we show that for a given dg category $\mathcal{A}$ with additive homotopy category $H^0(\mathcal{A})$, there is a bijection between exact structures on $\mathcal{A}$ and exact structures (in the sense of Barwick) on the dg nerve of $\mathcal{A}$. We also show the existence of the greatest exact structure on a (small) dg category with additive homotopy category. This generalizes a theorem of Rump for Quillen exact categories.