  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-Lef\`evre. 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.
