Curved Koszul duality theory

We extend the bar–cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. As usual, the bar–cobar construction gives a cofibrant resolution for any properad. Applied to the properad encoding unital and counital Frobenius algebras, notion which appears in 2d-TQFT, it defines the associated notion up to homotopy. We further define a curved Koszul duality theory for operads or properads presented with quadratic, linear and constant relations. This provides smaller resolutions. We apply this new theory to study the homotopy theory and the cohomology theory of unital associative algebras.