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We proved in a previous article that the bar complex of an E∞–algebra inherits a
natural E∞–algebra structure. As a consequence, a well-defined 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.
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