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Global Koszul duality

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.