2023/06/14 by Chen, Xiaofa · 1 citation
#13F60 #16D90 #16G20 #18G10 #18G25 #18G35 (Primary) #18G65 #18G80 (Secondary) #18N60 #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2306.08231
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 ∞-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 A with additive homotopy category H0(A), there is a bijection between exact structures on A and exact structures (in the sense of Barwick) on the dg nerve of 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.