Title: Diagonals on the Permutahedra, Multiplihedra and Associahedra
Authors: Samson Saneblidze, Ronald Umble
MSC: 55P35, 55U05
ArXive: math.AT/0209109

Abstract: We construct an explicit diagonal on the permutahedra {P_n}. Related diagonals on the multiplihedra {J_n} and the associahedra {K_n} are induced by Tonks' projection P_n --> K_{n+1} and its factorization through J_n. We use the diagonal on {K_n} to define the tensor product of A_infty-(co)algebras. We introduce the notion of a permutahedral set Z, observe that the double cobar construction Omega^{2}C_*(X) is a naturally occurring example and lift the diagonal on {P_n} to a diagonal on Z.