The Universal Hopf Operads of the Bar Construction

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. Date: 9 January 2007. 2000 Mathematics Subject Classification. Primary: 55P48; Secondary: 57T30, 16W30. Research supported in part by the ANR grant JCJC06-143080. The author enjoyed a stay at the Institut Mittag Leffler (Sweden) during the preparation of this memoir. 1