arXiv subjects

Benoit Fresse

Publications and source records attributed to Benoit Fresse.

27 records · Page 2Linked to original sources

arXiv2009

Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads

The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the operadic cobar construction to define explicit cyclinder objects in the category of operads. Then we apply this construction to prove that certain homotopy morphisms of algebras over operads are equivalent to left homotopies in the model category of operads.

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arXiv2009

Modules over operads and functors

In the theory of operads we consider functors of generalized symmetric powers defined by sums of coinvariant modules under actions of symmetric groups. One observes classically that the construction of symmetric functors provides an isomorphism from the category of symmetric modules to a subcategory of the category of functors on the base category. The purpose of this book is to obtain a similar relationship for functors on a category of algebras over an operad. We observe that right modules over operads, symmetric modules equipped with a right operad action, give rise to functors on categories of algebras and we prove that this construction yields an embedding of categories. Then we check that right modules over operads form a model category. In addition we prove that weak-equivalences of right modules correspond to pointwise weak-equivalences at the functor level. As a conclusion, we obtain that right modules over operads supply good models for the homotopy of associated functors on algebras over operads.

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arXiv2007

The universal Hopf operads of the bar construction

The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition that such a structure is homotopically unique provided that we consider unital operads which come equipped with a distinguished 0-ary operation that represents the natural unit of the bar complex. Our constructions rely on a Reedy model category for unital Hopf operads. For our purpose we define a unital Hopf endomorphism operad which operates functorially on the bar complex and which is universal with this property. Then we deduce our structure results from operadic lifting properties. To conclude this memoir we hint how to make our constructions effective and explicit.

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arXiv2003

The bar construction of an algebra as an E-infinite Hopf algebra

We prove that the bar construction of an $E_\infty$ algebra forms an $E_\infty$ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt-Eccles operad. (The surjection operad and the Barratt-Eccles operad are classical $E_\infty$ operads.)

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arXiv2003

Koszul duality of operads and homology of partition posets

We consider partitions of a set with $r$ elements ordered by refinement. We consider the simplicial complex $\bar{K}(r)$ formed by chains of partitions which starts at the smallest element and ends at the largest element of the partition poset. A classical theorem asserts that $\bar{K}(r)$ is equivalent to a wedge of $r-1$-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in $r$ letters. Consequently, the associated homology modules are representations of the symmetric groups. One observes that the $r-1$th homology modules of $\bar{K}(r)$, where $r = 1,2,...$, are dual to the Lie representation of the symmetric groups. In this article, we would like to point out that this theorem occurs a by-product of the theory of \emph{Koszul operads}. For that purpose, we improve results of V. Ginzburg and M. Kapranov in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov.

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arXiv2003

Derived division functors and mapping spaces

The normalized cochain complex of a simplicial set N^*(Y) is endowed with the structure of an E_{infinity} algebra. More specifically, we prove in a previous article that N^*(Y) is an algebra over the Barratt-Eccles operad. According to M. Mandell, under reasonable completeness assumptions, this algebra structure determines the homotopy type of Y. In this article, we construct a model of the mapping space Map(X,Y). For that purpose, we extend the formalism of Lannes' T functor in the framework of E_{infinity} algebras. Precisely, in the category of algebras over the Barratt-Eccles operad, we have a division functor -oslash N_(X) which is left adjoint to the functor Hom_F(N_*(X),-). We prove that the associated left derived functor -oslash^L N_*(X) is endowed with a quasi-isomorphism N^*(Y) oslash^L N_*(X) --> N^* Map(X,Y).

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arXiv2002

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.

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arXiv2002

A prismatic decomposition of the Barratt-Eccles operad

The Barratt-Eccles operad is a simplicial operad formed by the classical homogeneous bar construction of the symmetric groups. We prove that these simplicial sets decompose as unions of prisms indexed by surjections. We observe that the cellular complexes given by this prismatic structure are nothing but the components of the surjection operad (the operad introduced by J. McClure and J. Smith in their work on the Deligne conjecture).

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arXiv2002

Poisson structures over a complete intersection with isolated singularities

We study Poisson structures over singular varieties. In this purpose, we consider the Koszul complex associated to the equations of a complete intersection. This complex forms a differential graded algebra which is equivalent to the algebra of the variety. We show that a Poisson structure is equivalent to a sequence of multiderivations over the Koszul complex. If our variety has isolated singularities, then we can construct a sequence of multiderivations of reduced form.

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