The diagonal of the associahedra

Masuda, Naruki; Thomas, Hugh; Tonks, Andy; Vallette, Bruno

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 121-146.

Cet article introduit pour la première fois une méthode générale permettant de résoudre le problème de l’approximation de la diagonale de familles de polytopes satisfaisant à une propriété de cohérence par faces. On retrouve les cas classiques des simplexes et des cubes et on résout celui des associaèdres, appelés aussi polytopes de Stasheff. On montre que ce dernier cas vérifie une formule cellulaire facile à énoncer. Pour la première fois, nous munissons une famille de réalisations des associaèdres (celle de Loday) d’une structure d’opérade topologique cellulaire, dont nous montrons qu’elle est compatible avec les diagonales.

This paper introduces the first general method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.

Classification : 52B11, 18M75, 18M70, 06A07
Keywords: Associahedra, approximation of the diagonal, operads, fiber polytopes, $\mathrm{A}_\infty $-algebras
Mot clés : Associaèdres, approximation de la diagonale, opérades, polytopes fibrés, $A_\infty $-algèbres

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