Iterated bar complexes of E-infinity algebras and homology theories

TL;DR

This paper develops a direct definition of iterated bar complexes for E-infinity algebras, extending their applicability and linking them to Gamma-homology, thereby advancing algebraic homology theories.

Contribution

It provides a new effective definition of iterated bar complexes for E-infinity algebras and connects these complexes to E_n-operad homology theories.

Findings

Iterated bar complexes extend to E_n-operad algebras.

The n-fold bar complex determines associated homology theories.

Infinite bar complex homology matches Gamma-homology.

Abstract

We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity 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-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) 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 a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients.