Opérades cellulaires et espaces de lacets itérés

Berger, Clemens

doi:10.5802/aif.1543

Cited by 21 publications

(83 citation statements)

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“…As pointed out in [1], [2], the above constructions unify in the language of operads. Let L be the category of (non-empty) finite sets with injective maps as morphisms.…”

Section: Operadic Descriptionmentioning

confidence: 79%

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Braids and crossed modules

Huebschmann¹

2012

Journal of Group Theory

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“…As pointed out in [1], [2], the above constructions unify in the language of operads. Let L be the category of (non-empty) finite sets with injective maps as morphisms.…”

Section: Operadic Descriptionmentioning

confidence: 79%

“…Likewise the construction J ðkÞ (k d 1) [25] yields approximations to the functors W k S k and, as noted in [1], [2], each J ðkÞ is a preoperad; the preoperad J ðkÞ is defined to be the collection…”

Section: Operadic Descriptionmentioning

confidence: 99%

Braids and crossed modules

Huebschmann¹

2012

Journal of Group Theory

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“…The main purpose of this section is to revisit definitions of [8] in order to give an abstract formalization of the complete graph cell decompositions of E 1 -operads and to extend the applications of these cell structures to iterated bar modules.…”

Section: Interlude: Operads Shaped On Complete Graph Posetsmentioning

confidence: 99%

“…Hence, we have a nested sequence of operads E n .r / D colim Ä2K n .r / E.Ä/, formed from the Barratt-Eccles operad E, and weakly-equivalent to the nested sequence of the chain operads of little n-cubes (see [8]). …”

Section: 7mentioning

confidence: 99%

Iterated bar complexes ofE–infinity algebras and homology theories

Fresse¹

2011

Algebr. Geom. Topol.

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We proved in a previous article that the bar complex of an E 1 -algebra inherits a natural E 1 -algebra structure. As a consequence, a well-defined iterated bar construction B n .A/ can be associated to any algebra over an E 1 -operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A.The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E 1 -algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over E n -operads.Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an E n -operad. In the case n D 1, we obtain an isomorphism between the homology of an infinite bar construction and the usual -homology with trivial coefficients. 57T30; 55P48, 18G55, 55P35 IntroductionThe standard reduced bar complex B.A/ is basically defined as a functor from the category of associative differential graded algebras (associative dg-algebras for short) to the category of differential graded modules (dg-modules for short). In the case of a commutative dg-algebra, the bar complex B.A/ inherits a natural multiplicative structure and still forms a commutative dg-algebra. This observation implies that an iterated bar complex B n .A/ is naturally associated to any commutative dg-algebra A, for every n 2 N . In this paper, we use techniques of modules over operads to study extensions of iterated bar complexes to algebras over E n -operads. Our main result asserts that the n-fold bar complex B n .A/ determines the homology theory associated to an E n -operad.For the purpose of this work, we take the category of dg-modules as a base category and we assume that all operads belong to this category. An E n -operad refers to a dg-operad equivalent to the chain operad of little n-cubes. Many models of E n -operads come in nested sequences ( )such that E D colim n E n is an E 1 -operad, an operad equivalent in the homotopy category of dg-operads to the operad of commutative algebras C. Recall that an E 1 -operad is equivalent to the operad of associative algebras A and forms, in another usual terminology, an A 1 -operad. The structure of an algebra over an A 1 -operad includes a product W A˝A ! A and a full set of homotopies that make this product associative. The structure of an algebra over an E 1 -operad includes a product W A˝A ! A and a full set of homotopies that make this product associative and commutative. The intermediate structure of an algebra over an E n -operad includes a product W A˝A ! A which is fully homotopy associative, but homotopy commutative up to some degree only.Throughout the paper, we use the letter C to denote the base category of dg-modules and the notation P C , where P is any operad, refers to the category of P-algebras in dg-modules. The category of commutative dg-algebras, identified with the category of algebras over the commutative operad C, is denoted by ...

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“…Berger conjectured in [3] and [4] an En operad structure of the permutohedra, whose associated monad is the Milgram construction J n .…”

mentioning

confidence: 99%