18 research outputs found
The Koszul complex is the cotangent complex
23 pagesInternational audienceWe extend the Koszul duality theory of associative algebras to algebras over an operad. Recall that in the classical case, this Koszul duality theory relies on an important chain complex: the Koszul complex. We show that the cotangent complex, involved in the cohomology theory of algebras over an operad, generalizes the Koszul complex
Complex manifolds as families of homotopy algebras
International audienceWe prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex structure on a smooth manifold corresponds in this way to a family of algebras indexed by the points of the manifold
Corrigendum for the article ''Curved Koszul duality theory''
In this corrigendum, we explain and correct a mistake in our article ''Curved
Koszul duality theory''. Our definitions of morphisms between semi-augmented
properads and between curved coproperads have to be modified.Comment: Mathematische Annalen, 202
Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis
In 2007, Vallette built a bridge across posets and operads by proving that an
operad is Koszul if and only if the associated partition posets are
Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit
different refinements: our goal here is to link two of these refinements. We
more precisely prove that any (basic-set) operad whose associated posets admit
isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis.
Furthermore, we give counter-examples to the converse
Curved Koszul duality theory
38 pagesInternational audienceWe 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. We also define a curved Koszul duality theory for operads or properads presented with quadratic, linear and constant relations, which provides the possibility for smaller relations. We apply this new theory to study the homotopy theory and the cohomology theory of unital associative algebras