Lifting Quillen exact structures to exact dg structures

Determine which Quillen exact structures on the homotopy category H^0(A) of a connective additive dg category A lift to exact dg structures on A, and characterize the relationship between the poset Ex(A) of exact dg structures on A and the poset Ex_Q(H^0(A)) of Quillen exact structures on H^0(A), both ordered by inclusion.

Background

The paper classifies exact dg structures on a connective, additive, idempotent-complete dg category A by bi-Serre subcategories of the intrinsically defined category C_2(A). It also compares this classification with Quillen’s classification of exact structures on an additive category via Serre-type subcategories of module categories.

The concluding question asks whether an arbitrary Quillen exact structure on H0(A) can be lifted through the homotopy category projection to an exact dg structure on A. More broadly, it seeks a description of how the inclusion-based poset of exact dg structures on A relates to the inclusion-based poset of Quillen exact structures on H0(A).

References

We conclude with a natural question suggested by the preceding classification. Which Quillen exact structures on $H0(A)$ lift to exact dg structures on $A$? More generally, how can one describe the relationship between the posets

\Ex_{}(A) \qquad\text{and}\qquad \Ex_{\mathrm Q}\bigl(H0(A)\bigr)?

Here $\Ex_{\mathrm Q}\bigl(H0(A)\bigr)$ denotes the poset of Quillen exact structures on $H0(A)$, ordered by inclusion.

— Exact dg structures are classified intrinsically by bi-Serre subcategories  (2609.25583 - Ogawa, 22 Sep 2026) in Section 2, subsection 'The lattice of exact dg structures', concluding Question