Combinatorial operad actions on cochains

By C. Berger, Benoit Fresse. Mathematical Proceedings of the Cambridge Philosophical Society, 2001

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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 paper 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a3697b750ce474b620243e1b8bd287d19460e301

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Citations in the corpus

Notes on factorization algebras, factorization homology and applications (2013, background): “ion bigger than 2 64other possible models for the symmetric monoidal ¥-category (En-Alg; ) are given by algebraic Hopf operads such as those arising from the ﬁltration of the Barratt-Eccles operad in [BF] 65An ¥-operad O is an ¥-category together with a functor O !N(Fin) satisfying a list of axioms,”
Operadic Twisting -- with an application to Deligne's conjecture (2012, background): “ete and locally bounded, we deduce Theorem 9.3 from [8, Lemma A.3]. 9.3. The dg operad Br is a homotopy xed point of Tw. Various solutions of the Deligne conjecture on Hochschild cochain complex [1], [2], [9], [26], [27], [21], [31], [32] imply that the dg operad Br is quasiisomorphic to the dg oper”
Higher Hochschild Homology, Topological Chiral Homology and Factorization Algebras (2010, background): “1]) of algebras over the usual operad of singular chains on the little n-dimensional disk operad or as algebras over the Barratt-Eccles operad (which is an Hopf operad) [4].”
On differential graded categories (2006, background): “Deligne [29] which has been obtained, for example, in [105], [92], [14].”
A bordism approach to string topology (2003, unknown): “Both examples are related by Deligne’s conjecture proved in many different ways [2, 28, 29, 33, 40, 41].”
Constructive Homological Algebra and Applications (2012, background): “ C∗(B) not only with the coalgebra structure, but with some E∞-coalgebra structure, E∞ being an appropriate algebraic operad, which can be understood as the completion of the Steenrod operations. See [7, 38] for this essential point. 126 It is a ﬁrst quadrant bicomplex, the horizontal degree is pand”