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

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

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

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

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

Algèbre et opérade : cohomologie, homotopie et dualité de Koszul

Using the Koszul duality theory of operads, we make the André-Quillen cohomology of algebras over an operad explicit. This cohomology theory is represented by a chain complex: the cotangent complex. We provide criteria for the André-Quillen cohomology theory to be an Ext-functor. In particular, this is the case for algebras over cofibrant operads and this gives a new stable homotopy property for these algebras. Then we generalize the Koszul duality theory of associative algebras in two independant directions. On the one hand, we extend the Koszul duality theory to non necessarily augmented operads in order to treat algebras with unit. The notion of curvature appears to encode the default of augmentation. As a corollary, we obtain homotopical et cohomological theories for unital associative algebras or unital and counital Frobenius algebras. We make the case of unital associative algebras explicit. On the other hand, we generalize the Koszul duality theory to algebras over an operad. To do this, we show that the cotangent complex provides the good generalization of the Koszul complex.Nous explicitons la cohomologie d'André-Quillen des algèbres sur une opérade à l'aide de la dualité de Koszul des opérades. Cette cohomologie est représentée par le complexe cotangent. Nous donnons des critères assurant que cette cohomologie s'écrit en termes de foncteur Ext. En particulier, c'est le cas des algèbres sur des opérades cofibrantes, ce qui fournit une nouvelle propriété de stabilité homotopique de ces algèbres. Nous généralisons ensuite la dualité de Koszul des algèbres associatives dans deux directions indépendantes. D'un côté, nous étendons la dualité de Koszul aux opérades non nécessairement augmentées de façon à étudier les algèbres unitaires. La notion de courbure apparaît pour coder le défaut d'augmentation. Nous obtenons ainsi les théories homotopiques et cohomologiques des algèbres associatives unitaires ou des algèbres de Frobenius avec unité et counité. Nous détaillons le cas des algèbres associatives unitaires. D'un autre côté, nous généralisons la dualité de Koszul aux algèbres sur une opérade. Nous montrons pour cela que le complexe cotangent est la bonne généralisation du complexe de Koszul