Exact dg structures are classified intrinsically by bi-Serre subcategories

Abstract: The aim of this article is to leverage Enomoto's classification of exact structures to the dg level. Let $\mathscr{A}$ be a connective additive idempotent complete dg category. We establish an intrinsic Enomoto-type classification of exact dg structures on $\mathscr{A}$ in terms of bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Our proof is a direct dg argument based on dg totalizations of $3$-term complexes and dg duality, rather than a reduction to the greatest exact dg structure and the classification of its substructures. As a consequence, we establish a bijection between exact dg structures on $\mathscr{A}$ and bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Moreover, this bijection is an isomorphism of posets and yields a complete lattice structure on the class of exact dg structures on $\mathscr{A}$. In particular, we provide another construction of Rump--Chen's greatest exact dg structure on $\mathscr{A}$.