A duality between standard simplices and Stasheff polytopes

Abstract

We show that the family of chain modules over the standard simplices can be equipped with an operad structure. Similarly, the family of cochain modules of the Stasheff polytopes can be equipped with an operad structure. We first show that these operads are Koszul dual to each other, and second that they are both Koszul operads. The algebras over the standard simplices operad, called associative trialgebras, are defined by 3 operations and 11 relations. The algebras over the Stasheff polytopes operad, called dendriform trialgebras, are defined by 3 operations and 7 relations.