Diagonals on the Permutahedra, Multiplihedra and Associahedra

Abstract

We construct an explicit diagonal Δ_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a permutahedral set Z and lift Δ_P to a diagonal on Z. We show that the double cobar construction Ω^2(C_*(X)) is a permutahedral set; consequently Δ_P lifts to a diagonal on Ω^2(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal generality.