Diagonals on the Permutahedra, Multiplihedra and Associahedra

2002/09/10 by Samson Saneblidze, Saneblidze, Samson, Ronald Umble +1
Mathematics · #55P35 #55U05 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55P35 #msc:55U05

paper · pdf · doi:10.48550/arxiv.math/0209109

45 pages, 13 figures. This (final) version is significantly more detailed than the previous

openalex publication_date 2002/09/10 · arxiv created 2004/08/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Ω²(C_*(X)) is a permutahedral set; consequently Δ_P lifts to a diagonal on Ω²(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_∞-(co)algebras in maximal generality.

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