Exact dg categories II : The embedding theorem

TL;DR

This paper develops the theory of exact dg categories by introducing their bounded derived categories, establishing universal morphisms, and analyzing their structure under quotients and tensor products, with applications to algebraic conjectures.

Contribution

It introduces the bounded dg derived category for exact dg categories and proves universal properties and structural results, advancing the understanding of dg categorical frameworks.

Findings

Established the universal exact morphism from an exact dg category to its bounded derived category.

Proved that dg quotients by projective-injective subcategories have a canonical exact structure.

Confirmed a conjecture for algebraic 0-Auslander extriangulated categories.

Abstract

For an exact dg category $\mathcal A$, we introduce its bounded dg derived category $\mathcal{D}^b_{dg}(\mathcal A)$ and establish the universal exact morphism from $\mathcal A$ to $\mathcal{D}^b_{dg}(\mathcal A)$. We prove that the dg quotient of an exact dg category by a subcategory of projective-injectives carries a canonical exact structure. We show that exact dg categories reproduce under tensor products and functor dg categories. We apply our results to 0-Auslander extriangulated categories and confirm a conjecture by Fang-Gorsky-Palu-Plamondon-Pressland for the algebraic case.