Iterated Bar Complexes of E-infinity Algebras and Homology Theories

We proved in a previous article that the bar complex of an E∞algebra inherits a natural E∞-algebra structure. As a consequence, a welldefined iterated bar construction Bn(A) can be associated to any algebra over an E∞-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∞-algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over En-operads. Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an En-operad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ-homology with trivial coefficients.