Iterated bar complexes of $E$–infinity algebras and homology theories

Abstract

We proved in a previous article that the bar complex of an $E_{\infty }$–algebra inherits a natural $E_{\infty }$–algebra structure. As a consequence, a well-defined iterated bar construction ${\mathtt{B}}^n\left(A\right)$ can be associated to any algebra over an $E_{\infty }$–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 $E_{\infty }$–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 $n=\infty$, we obtain an isomorphism between the homology of an infinite bar construction and the usual $\Gamma$–homology with trivial coefficients.