The diagonal of the multiplihedra and the tensor product of $\mathrm{A}_\infty $-morphisms

Guillaume Laplante-Anfossi; Thibaut Mazuir

doi:10.5802/jep.221

The diagonal of the multiplihedra and the tensor product of

A_{\infty}

-morphisms
[La diagonale des multiplièdres et le produit tensoriel de morphismes A-infini]

Guillaume Laplante-Anfossi ¹ ; Thibaut Mazuir ²

¹ School of Mathematics and Statistics, University of Melbourne Parkville, VIC, 3010, Australia
² Institut für Mathematik, Humboldt Universität zu Berlin Rudower Chaussee 25, 12489 Berlin, Germany

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 405-446.

Résumé
Abstract

On définit une approximation cellulaire de la diagonale des réalisations de Forcey–Loday des multiplièdres, et on les munit d’une structure de bimodule opéradique topologique et cellulaire compatible sur les réalisations de Loday des associaèdres. On obtient ainsi un modèle algébrique et topologique pour les morphismes A-infini, de même qu’une formule universelle explicite pour leur produit tensoriel. On étudie la monoïdalité de ce nouveau produit tensoriel et on conclut en esquissant plusieurs applications en topologie algébrique et en topologie symplectique.

We define a cellular approximation for the diagonal of the Forcey–Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic $A_{\infty}$ -morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology.

Reçu le : 2022-07-19
Accepté le : 2023-02-14
Publié le : 2023-02-22

DOI : 10.5802/jep.221

Classification : 52B11, 18M70
Keywords: Multiplihedra, approximation of the diagonal, associahedra, operads, tensor product, A-infinity algebras, A-infinity morphisms, A-infinity categories
Mot clés : Multiplièdres, approximation de la diagonale, associaèdres, opérades, produit tensoriel, algèbres A-infini, morphismes A-infini, catégories A-infini

Affiliations des auteurs :

Guillaume Laplante-Anfossi ¹ ; Thibaut Mazuir ²

¹ School of Mathematics and Statistics, University of Melbourne Parkville, VIC, 3010, Australia
² Institut für Mathematik, Humboldt Universität zu Berlin Rudower Chaussee 25, 12489 Berlin, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The diagonal of the multiplihedra and the tensor product of $\mathrm{A}_\infty $-morphisms},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Guillaume Laplante-Anfossi; Thibaut Mazuir. The diagonal of the multiplihedra and the tensor product of $\mathrm{A}_\infty $-morphisms. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 405-446. doi : 10.5802/jep.221. https://jep.centre-mersenne.org/articles/10.5802/jep.221/

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