Global Koszul duality
Matt Booth, Andrey Lazarev
TL;DR
Global Koszul duality constructs a Quillen equivalence between curved algebras and curved coalgebras by replacing quasi-isomorphisms with Maurer–Cartan (MC) equivalences and employing the extended bar–cobar adjunction. The paper develops MC dg-categories, higher n-homotopies, and a monoidal model structure, proving that MC equivalences induce equivalences on derived categories of the second kind and on MC-dg-categories, thereby unifying local Koszul duality with a robust global deformation theory. A central achievement is the introduction of MC stacks, yielding homotopy-invariant moduli spaces for objects in dg-categories, flat connections, dg-modules, and twisted modules, all controlled by curved (co)algebras under Koszul duality. The framework recovers the classical conilpotent (local) Koszul duality upon restriction to coaugmented/dg contexts and extends to a categorical Koszul duality perspective, providing a comprehensive toolkit for global moduli problems and derived Morita-type phenomena in noncommutative settings.
Abstract
We construct a monoidal model structure on the category of all curved coalgebras and show that it is Quillen equivalent, via the extended bar-cobar adjunction, to another model structure we construct on the category of curved algebras. When the coalgebras under consideration are conilpotent and the algebras are dg, i.e. uncurved, this corresponds to the ordinary dg Koszul duality of Positselski and Keller-Lefèvre. As an application we construct global noncommutative moduli spaces for flat connections on vector bundles, holomorphic structures on almost complex vector bundles, dg modules over a dg algebra, objects in a dg category, and others.