On exact dg categories

TL;DR

This paper introduces the concept of exact dg categories, generalizing several existing notions, and explores their fundamental properties and relationships with dg nerves, derived categories, and exact structures.

Contribution

It defines exact dg categories in analogy with Quillen's exact categories, linking them to dg nerves and proving key properties and existence results.

Findings

Bijective correspondence between exact structures on dg categories and their dg nerves.

Existence of the greatest exact structure on small dg categories with additive homotopy.

Generalization of Rump's theorem to 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.