Reflexive dg categories in algebra and topology

TL;DR

This paper explores the concept of reflexive dg categories across various algebraic and topological contexts, revealing their properties and relationships with duality, Hochschild cohomology, and derived structures.

Contribution

It establishes reflexivity in multiple settings, linking it to derived completeness and providing new insights into duality and categorical structures in algebra and topology.

Findings

Reflexivity holds for affine schemes and simple-minded collections.

Reflexivity relates to derived completeness for certain dg algebras.

The work connects reflexivity with gluings, derived completions, and Koszul duality.

Abstract

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.