Curved Koszul duality theory (Q695284)

The authors extend the bar-cobar adjunction to not necessarily augmented operads and properads. Due to the failure of augmentation, the objects of the dual category are endowed with curvature. As usual, the bar-cobar construction gives a cofibrant resolution for any properad. Applying this to the properad encoding unital and counital Frobenius algebras, it defines the associated notion up to homotopy. The authors define a curved Koszul duality theory for operads and properads. The obtained new theory is applied to the study of homotopy and cohomology theories of unital associated algebras.