What is an exact dg category?

Abstract

Exact dg categories are a simultaneous generalisation of exact categories in the sense of Quillen and of pretriangulated dg categories in the sense of Bonda–Kapranov. The notion 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. In this paper, we present the definition and a number of fundamental results concerning the dg nerve, the dg derived category, tensor products and functor categories with exact dg target. We conclude with a generalisation of Rump's theorem on the existence of the greatest exact structure on an additive category.