Diagonals on the permutahedra, multiplihedra and associahedra

Samson Saneblidze ; Ronald Umble

Homology, homotopy, and applications, Tome 6 (2004) no. 1, pp. 363-411.

Résumé

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 $P\rightarrow K$ \cite{tonks} 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.

Détail

DOI : 10.4310/HHA.2004.v6.n1.a20

Classification : 05A18, 05A19, 52B05, 55U05, 55P35
Keywords: diagonal, permutahedron, multiplihedron, associahedron