Diagonals on the permutahedra, multiplihedra and associahedra

Abstract

We construct an explicit diagonal ΔP on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P → K [19] and its factorization through J. We introduce the notion of a permutahedral set Z and lift ΔP to a diagonal on Z. We show that the double cobar construction Ω 2C*( X) is a permutahedral set; consequently ΔP lifts to a diagonal on Ω2C *( X). Finally, we apply the diagonal on K to define the tensor product of A∞-(co)algebras in maximal generality. © 2004, Samson Saneblidze and Ronald Umble.