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Abstract: We extend the bar-cobar adjunction to operads and properads, not necessarilyaugmented. Due to the default of augmentation, the objects of the dual categoryare endowed with a curvature. We handle the lack of augmentation by extendingthe category of coproperads to include objects endowed with a curvature. Asusual, the bar-cobar construction gives a large cofibrant resolution for anyproperad, such as the properad encoding unital and counital Frobenius algebras,a notion which appears in 2d-TQFT. We also define a curved Koszul dualitytheory for operads or properads presented with quadratic, linear and constantrelations, which provides the possibility for smaller relations. We apply thisnew theory to study the homotopy theory and the cohomology theory of unitalassociative algebras.

Autor: Joseph Hirsh, Joan Millès


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