Mixed Hodge structures and formality of symmetric monoidal functors

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor to a functor defined via the first term of the weight spectral sequence.Comment: 26 page

Sullivan minimal models of operad algebras

preprintWe prove the existence of Sullivan minimal models of operad algebras, for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan’s original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass and Lie, as well as over their minimal models Com8, Ass8 and Lie8. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.Preprin

Mixed Hodge structures and formality of symmetric monoidal functors

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex varieties whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex varieties with non-pure weight filtration, we relate the singular chains functor to a functor defined via the first term of the weight spectral sequence. Résumé. Nous utilisons la théorie de Hodge mixte pour montrer que le foncteur des chaînes singulières à coefficients rationnels est formel, comme foncteur symétrique monoïdal lax, lorsqu'on le restreint aux variétés complexes dont la filtration par le poids en cohomologie satisfait une certaine propriété de pureté. Ce résultat a des applications directes à la formalité d'opérades ou plus généralement à des structures algébriques encodées par une opérade colorée. Nous prouvons aussi le résultat dual, avec des applications à la formalité dans le contexte de la théorie de l'homotopie rationnelle. Dans le cas général d'une variété dont la filtration par le poids n'est pas pure, nous relions le foncteur des chaînes singulières à un foncteur défini par la première page de la suite spectrale des poids

Sullivan minimal models of operad algebras

We prove the existence of Sullivan minimal models of operad algebras for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan's original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass, and Lie, as well as over their minimal models Com∞, Ass∞, and Lie∞. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study