Diagonals on the permutahedra, multiplihedra and associahedra

Abstract

We construct an explicit diagonal $\Delta_{P}$ on the permutahedra $P.$ Related diagonals on the multiplihedra $J$ and the associahedra $K$ are induced by Tonks' projection $P\rightarrow K$ [19] and its factorization through $J.$ We introduce the notion of a permutahedral set $% \mathcal{Z}$ and lift $\Delta_{P}$ to a diagonal on $\mathcal{Z}$. We show that the double cobar construction $\Omega^{2}C_{\ast}(X)$ is a permutahedral set; consequently $\Delta_{P}$ lifts to a diagonal on $% \Omega^{2}C_{\ast}(X)$. Finally, we apply the diagonal on $K$ to define the tensor product of $A_{\infty}$-(co)algebras in maximal generality.