We proved in a previous article that the bar complex of an –algebra inherits a natural –algebra structure. As a consequence, a well-defined iterated bar construction can be associated to any algebra over an –operad. In the case of a commutative algebra , our iterated bar construction reduces to the standard iterated bar complex of .
The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of –algebras. We use this effective definition to prove that the –fold bar construction admits an extension to categories of algebras over –operads.
Then we prove that the –fold bar complex determines the homology theory associated to the category of algebras over an –operad. In the case , we obtain an isomorphism between the homology of an infinite bar construction and the usual –homology with trivial coefficients.
"Iterated bar complexes of $E$–infinity algebras and homology theories." Algebr. Geom. Topol. 11 (2) 747 - 838, 2011. https://doi.org/10.2140/agt.2011.11.747