Title: The bar construction of an $E$-infinity algebra
Author: Benoit Fresse
E-mail: Benoit.Fresse@math.univ-lille1.fr
AMS classification number: Primary: 57T30; Secondary: 55P48, 18G55, 55P35.
ArXiv reference: math.AT/0601085
Abstract: We consider the classical reduced bar construction of
associative algebras B(A). If the product of A is commutative, then B(A)
can be equipped with the classical shuffle product, so that B(A) is
still a commutative algebra. This assertion can be generalized for
algebras which are commutative up to homotopy. Namely, one observes
that the bar construction of an E-infinite algebra B(A) can be endowed
with the structure of an E-infinite algebra.
The purpose of this article is to give an existence and uniqueness
theorem for this claim. We would like to insist on the uniqueness
property: our statement makes the construction of $E$-infinite
structures easier and more flexible. Therefore, the proof of our
existence theorem differs from other constructions of the literature.
In addition, the uniqueness property allows to give easily a homotopy
interpretation of the bar construction.