Diagonals on the Permutahedra, Multiplihedra and Associahedra
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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.
45 pages, 13 figures. This (final) version is significantly more detailed than the previous
45 pages, 13 figures. This (final) version is significantly more detailed than the previous