Samson Saneblidze and Ronald Umble
نویسنده
چکیده
An associahedral set is a combinatorial object generated by Stasheff associahedra {Kn} and equipped with appropriate face and degeneracy operators. Associahedral sets are similar in many ways to simplicial or cubical sets. In this paper we give a formal definition of an associahedral set, discuss some naturally occurring examples and construct an explicit geometric diagonal ∆ : C∗(Kn) −→ C∗(Kn)⊗C∗(Kn) on the cellular chains C∗ (Kn) . The diagonal ∆, which is analogous to the AlexanderWhitney diagonal on the simplices, gives rise to a diagonal on any associahedral set and leads immediately to an explicit diagonal on the A∞-operad. As an application of this, we use the diagonal ∆ to define a tensor product in the A∞ category. This tensor product will play a central role in our discussion of “A∞-Hopf algebras” to appear in the sequel. We mention that Chapoton [1], [2] constructed a diagonal of the form ∆ : C∗ (Kn) → ∑ i+j=n C∗ (Ki) ⊗ C∗ (Kj) on the direct sum ∑ n≥2 C∗ (Kn) , which coincides with the diagonal of Loday and Ronco [8] in dimension zero. Whereas Chapoton’s diagonal is formally defined to be primitive on generators, our diagonal is obtained by a purely geometrical decomposition of the generators and is totally different from his. The second author wishes to thank Millersville University for its generous financial support and the University of North Carolina at Chapel Hill for its hospitality during the final stages of this project.
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