The universal Hopf operads of the bar construction

TL;DR

This paper proves that the bar complex of an E-infinity algebra naturally carries a Hopf E-infinity algebra structure, which is homotopically unique under certain conditions, using operadic and model category techniques.

Contribution

It establishes the functorial Hopf E-infinity algebra structure on bar complexes and demonstrates its homotopical uniqueness with a new operadic framework.

Findings

Bar complex of E-infinity algebra has a Hopf E-infinity structure

The structure is homotopically unique under unital operads with a distinguished 0-ary operation

Operadic lifting properties are used to derive the main results

Abstract

The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition that such a structure is homotopically unique provided that we consider unital operads which come equipped with a distinguished 0-ary operation that represents the natural unit of the bar complex. Our constructions rely on a Reedy model category for unital Hopf operads. For our purpose we define a unital Hopf endomorphism operad which operates functorially on the bar complex and which is universal with this property. Then we deduce our structure results from operadic lifting properties. To conclude this memoir we hint how to make our constructions effective and explicit.