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Benoit Fresse

Publications and source records attributed to Benoit Fresse.

At least 19 recordsLinked to original sources
arXiv2020

On a notion of homotopy Segal $ E_\infty $-Hopf cooperad

We define a notion of homotopy Segal cooperad in the category of $ E_\infty $-algebras. This model of Segal cooperad that we define in the paper, which we call homotopy Segal $ E_\infty $-Hopf cooperad, covers examples given by the cochain complex of topological operads and provides a framework for the study of the homotopy of such objects. In a first step, we consider a category of Segal $ E_\infty $-Hopf cooperads, which consists of collections of $ E_\infty $-algebras indexed by trees and equipped with coproduct operators, corresponding to tree morphisms, together with facet operators, corresponding to subtree inclusions. The coproduct operators model coproducts of operations inside a tree. The facet operators are assumed to satisfy a Segal condition. The homotopy Segal cooperads that we aim to define are formed by integrating homotopies in the composition schemes of the coproduct operators. For this purpose, we replace the functorial structure that governs the composition of the coproduct operators by the structure of a homotopy functor which we shape on a cubical enrichment of the category of $ E_\infty $-algebras. We prove that every homotopy Segal $ E_\infty $-Hopf cooperad in our sense is weakly-equivalent to a strict Segal $ E_\infty $-Hopf cooperad. We also define a notion of homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads. We prove that every homotopy Segal $ E_\infty $-Hopf cooperad admits a cobar construction and that every homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads induces a morphism on this cobar construction, so that our approach provides a lifting to the context of $ E_\infty $-algebras of classical homotopy cooperad structures that are modeled on the bar duality of operads when we work in a category of differential graded modules.

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arXiv2020

On the rational homotopy type of embedding spaces of manifolds in $R^n$

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined $L_\infty$-algebras of diagrams.

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arXiv2020

Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of $E_n$-operads

We define a simplicial enrichment on the category of differential graded Hopf cooperads (the category of dg Hopf cooperads for short). We prove that our simplicial enrichment satisfies, in part, the axioms of a simplicial model category structure on the category of dg Hopf cooperads. We use this simplicial model structure to define a model of mapping spaces in the category of dg Hopf cooperads and to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. The rational homotopy theory of operads implies that the homotopy automorphism spaces of dg Hopf cooperads can be regarded as models for the homotopy automorphism spaces of the rationalization of operads in topological spaces (or in simplicial sets). We prove, as a main application, that the spaces of Maurer--Cartan forms on the Kontsevich graph complex Lie algebras are homotopy equivalent, in the category of simplicial monoids, to the homotopy automorphism spaces of the rationalization of the operads of little discs.

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arXiv2019

Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.

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arXiv2018

The extended rational homotopy theory of operads

In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and Grothendieck-Teichm\"uller groups". In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite type rational cohomology groups).

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arXiv2017

The rational homotopy of mapping spaces of E${}_n$ operads

We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the rational homotopy type of the deloopings of spaces of long embeddings. Some of the results hold also for mapping spaces $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$, $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n)$, $n-m\geq 2$, of the truncated little discs operads, which allows one to determine rationally the delooping of the Goodwillie-Weiss tower for the spaces of long embeddings.

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arXiv2017

The homotopy theory of operad subcategories

We study the subcategory of topological operads $P$ such that $P(0) = *$ (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of $k$-truncated unitary operads within the model category of all $k$-truncated operads, for any fixed arity bound $k\geq 1$, where a $k$-truncated operad denotes an operad that is defined up to arity $k$.

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arXiv2015

The intrinsic formality of $E_n$-operads

We establish that $E_n$-operads satisfy a rational intrinsic formality theorem for $n\geq 3$. We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in spaces. We consider, in this context, the dual cooperad of the $n$-Poisson operad $\mathsf{Pois}_n^c$, which represents the cohomology of the operad of little $n$-discs $\mathsf{D}_n$. We assume $n\geq 3$. We explicitly prove that a Hopf cooperad in cochain graded dg-modules $\mathsf{K}$ is weakly-equivalent (quasi-isomorphic) to $\mathsf{Pois}_n^c$ as a Hopf cooperad as soon as we have an isomorphism at the cohomology level $H^*(\mathsf{K})\simeq\mathsf{Pois}_n^c$ when $4\nmid n$. We just need the extra assumption that $\mathsf{K}$ is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little $n$-discs operad in order to extend this formality statement in the case $4\mid n$. We deduce from these results that any operad in simplicial sets $\mathsf{P}$ which satisfies the relation $H^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c$ in rational cohomology (and an analogue of our extra involution requirement in the case $4\mid n$) is rationally weakly equivalent to an operad in simplicial sets $LG_{\bullet}(\mathsf{Pois}_n^c)$ which we determine from the $n$-Poisson cooperad $\mathsf{Pois}_n^c$. We also prove that the morphisms $\iota: \mathsf{D}_m\rightarrow\mathsf{D}_n$, which link the little discs operads together, are rationally formal as soon as $n-m\geq 2$. These results enable us to retrieve the (real) formality theorems of Kontsevich by a new approach, and to sort out the question of the existence of formality quasi-isomorphisms defined over the rationals (and not only over the reals) in the case of the little discs operads of dimension $n\geq 3$.

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arXiv2015

The cotriple resolution of differential graded algebras

We consider the cotriple resolution of algebras over operads in differential graded modules. We focus, to be more precise, on the example of algebras over the differential graded Barratt-Eccles operad and on the example of commutative alegbras. We prove that the geometric realization of the cotriple resolution (in the sense of model categories) gives a cofibrant resolution functor on these categories of differential graded algebras.

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arXiv2014

Iterated bar complexes and E_n-homology with coefficients

The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized to E_n-algebras, and that one can calculate E_n-homology with trivial coefficients via this iterated bar construction. We extend this result to E_n-homology and E_n-cohomology of a commutative algebra A with coefficients in a symmetric A-bimodule.

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arXiv2010

Props in model categories and homotopy invariance of structures

We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object X homotopy equivalent to an algebra A over a cofibrant prop P inherits a P-algebra structure so that X defines a model of A in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of A-infinity algebras.

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arXiv2010

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

The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connection between the cohomology of iterated loop spaces and the cohomology of algebras over E_n-operads; and a Koszul duality result for E_n-operads.

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arXiv2009

Batanin's category of pruned trees is Koszul

The category of pruned trees has been defined by M. Batanin with the aim of understanding the cell structure of certain E_n-operads in categorical terms. The objects of this category are planar trees with n levels so that all leaves are at the top level of the tree. The goal of this article is to prove that the category of pruned trees is Koszul. This result gives us a minimal differential graded model of this category, small complexes to compute Tor and Ext functors in associated categories of diagrams, and allows us to generalize a recent result of M. Livernet and B. Richter about the interpretation of E_n-homology in terms of categorical Tor functors.

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arXiv2009

On mapping spaces of differential graded operads with the commutative operad as target

The category of differential graded operads is a cofibrantly generated model category and as such inherits simplicial mapping spaces. The vertices of an operad mapping space are just operad morphisms. The 1-simplices represent homotopies between morphisms in the category of operads. The goal of this paper is to determine the homotopy of the operadic mapping spaces Map(E_n,C) with a cofibrant E_n-operad on the source and the commutative operad on the target. First, we prove that the homotopy class of a morphism phi: E_n -> C is uniquely determined by a multiplicative constant which gives the action of phi on generating operations in homology. From this result, we deduce that the connected components of Map(E_n,C) are in bijection with the ground ring. Then we prove that each of these connected components is contractible. In the case where n is infinite, we deduce from our results that the space of homotopy self-equivalences of an E-infinity-operad in differential graded modules has contractible connected components indexed by invertible elements of the ground ring.

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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

Koszul duality of E_n-operads

The goal of this paper is to prove a Koszul duality result for E_n-operads in differential graded modules over a ring. The case of an E_1-operad, which is equivalent to the associative operad, is classical. For n>1, the homology of an E_n-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 E_n-operad defines a cofibrant model of E_n. This cofibrant model gives a realization at the chain level of the minimal model of the n-Gerstenhaber operad arising from Koszul duality. Most models of E_n-operads in differential graded modules come in nested sequences of operads homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings E_n-1 --> E_n at the level of cobar constructions.

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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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