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Abstract: We proved in a previous article that the bar complex of an E-infinity algebrainherits a natural E-infinity algebra structure. As a consequence, awell-defined iterated bar construction B^nA can be associated to any algebraover an E-infinity operad. In the case of a commutative algebra A, our iteratedbar construction reduces to the standard iterated bar complex of A. The firstpurpose of this paper is to give a direct effective definition of the iteratedbar complexes of E-infinity algebras. We use this effective definition to provethat the n-fold bar complex B^nA admits an extension to categories ofalgebras over E n-operads. Then we prove that the n-fold bar complex determinesthe homology theory associated to a category of algebras over E n-operads. Forn infinite, we obtain an isomorphism between the homology of an infinite barconstruction and the usual Gamma-homology with trivial coefficients.



Author: Benoit Fresse

Source: https://arxiv.org/







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