Curved operadic calculus

Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational arti cle is to introduce the notion of a curved operad, and to develop the operadic calculus at this new level. The algebraic side of the curved operadic calculus provides us with universal constructions: using a new notion of curved operadic bimodules, we construct curved universal enveloping algebras. Since there is no notion of quasi-isomorphism in the curved context, we develop the homotopy theory of curved operads using new methods. This approach leads us to introduce the new notion of a curved absolute operad, which is the notion Koszul dual to counital cooperads non-necessarily conilpo tent, and we construct a complete bar-cobar adjunction between them. We endow curved absolute operads with a suitable model category structure. We establish a du ality square of duality functors that intertwines this complete bar-cobar construction with the bar-cobar adjunction between unital operads and conilpotent curved coop erads. This allows us to compute minimal cofibrant resolutions for various curved absolute operads. Using the complete bar construction, we show a general homotopy transfer theorem for curved algebras. Along the way, we construct the non-necessarily conilpotent cofree cooperad.