Curved cooperads and homotopy unital A-infty-algebras

Abstract: We provide bar and cobar constructions as functors acting between various categories of curved operads and curved cooperads. Cobar and bar constructions are adjoint to each other. Given a twisting cochain between a curved augmented cooperad C with an extra grading and a curved operad O we construct a couple of adjoint functors between the category of curved O-modules and the category of curved C-comodules. The important feature is that the curved operad O is not necessarily augmented.