Benoit Fresse

Modules over Operads and Functors

Categorical and operadic background.- Symmetric monoidal categories for operads.- Symmetric objects and functors.- Operads and algebras in symmetric monoidal categories.- Miscellaneous structures associated to algebras over operads.- The category of right modules over operads and functors.- Definitions and basic constructions.- Tensor products.- Universal constructions on right modules over operads.- Adjunction and embedding properties.- Algebras in right modules over operads.- Miscellaneous examples.- Homotopical background.- Symmetric monoidal model categories for operads.- The homotopy of algebras over operads.- The (co)homology of algebras over operads.- The homotopy of modules over operads and functors.- The model category of right modules.- Modules and homotopy invariance of functors.- Extension and restriction functors and model structures.- Miscellaneous applications.- Appendix: technical verifications.- Shifted modules over operads and functors.- Shifted functors and pushout-products.- Applications of pushout-products of shifted functors.

Combinatorial operad actions on cochains

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.

On the homotopy of simplicial algebras over an operad

According to a result of H. Cartan (cf. [5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad. To be precise, we work over a fixed field F, and we consider operads in the category of F-modules. An operad is an algebraic device that specifies a type of algebras. There are operads Com, As, Lie, and Pois, whose algebras are respectively commutative algebras, associative algebras, Lie algebras and Poisson algebras. In general, if P denotes an operad, then we call P-algebras the associated algebras. First, we generalize the notion of a divided power in the context of algebras over an operad. This is done as follows. Recall that the free commutative algebra is given by the formula T (Com, V ) = ⊕n(V )Sn , for V ∈ModF .

Koszul duality of En-operads

The goal of this paper is to prove a Koszul duality result for En-operads in differential graded modules over a ring. The case of an E1-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an En-operad is identified with the n-Gerstenhaber operad and forms another well-known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an En-operad

Une décomposition prismatique de l'opérade de Barratt-Eccles

{\mathtt{E}_n}

The cotriple resolution of differential graded algebras

defines a cofibrant model of

Koszul duality complexes for the cohomology of iterated loop spaces of spheres

{\mathtt{E}_n}