Curved Koszul duality theory

Understand

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.
• We handle the lack of augmentation by extending the category of coproperads to include objects endowed with a curvature.
• As usual, the bar-cobar construction gives a (large) cofibrant resolution for any properad, such as the properad encoding unital and counital Frobenius algebras, a notion which appears in 2d-TQFT.