Reflexive dg categories in algebra and topology

Reflexive dg categories were introduced by Kuznetsov and Shinder to abstract the duality between bounded and perfect derived categories. In particular this duality relates their Hochschild cohomologies, autoequivalence groups, and semiorthogonal decompositions. We establish reflexivity in a variety of settings including affine schemes, simple-minded collections, chain and cochain dg algebras of topological spaces, Ginzburg dg algebras, and Fukaya categories of cotangent bundles and surfaces as well as the closely related class of graded gentle algebras. Our proofs are based on the interplay of reflexivity with gluings, derived completions, and Koszul duality. In particular we show that for certain (co)connective dg algebras, reflexivity is equivalent to derived completeness.

💡 Research Summary

The paper “Reflexive dg categories in algebra and topology” develops a comprehensive theory of reflexive differential graded (dg) categories, a notion introduced by Kuznetsov and Shinder to capture the duality between bounded and perfect derived categories. The authors extend the class of known reflexive dg categories far beyond the previously studied smooth proper schemes, connective dg algebras, and homologically smooth dg categories. Their central methodological insight is the slogan: reflexivity = well‑behaved D_fd‑generators + derived completeness.

The core technical result is a “two‑out‑of‑three” theorem (Theorem H, 2.3.8): given a dg algebra A and a thick generator M of D_fd(A), any two of the following imply the third: (1) A is reflexive, (2) M is a thick generator of D_fd(REnd_A(M)^op), (3) the derived completion map A → A!!M is a quasi‑isomorphism. This reduces the problem of proving reflexivity to (i) identifying suitable generators and (ii) checking derived completeness. For connective algebras the standard t‑structure supplies generators; for coconnective algebras with semisimple H₀, co‑t‑structures (as in