Curved Cooperads and Homotopy
The notions of operads and cooperads are recalled. In particular, free operads and cofree cooperads are constructed via planar rooted trees with inputs. We discuss also φ-derivations, where φ is a morphism of operads, and, dually, φ-coderivations, where φ is a morphism of cooperads. Derivations are easily described, when the source operad is free, and similarly for coderivations, when the target cooperad is cofree'. Attention is paid to (co)operads with values in graded modules and their generalizations. Such are curved (co)operads, defined as above plus extra data including id-(co)derivations of degree 1. 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 C , we construct a couple of adjoint functors between the category of curved C -modules and the category of curved C-comodules. The important feature is that the curved operad C is not necessarily augmented. Mathematical Studies Monograph Series Vol 17.
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