Affinization of monoidal categories

Youssef Mousaaid; Alistair Savage

doi:10.5802/jep.158

Affinization of monoidal categories
[Affinisation de catégories monoïdales]

Youssef Mousaaid ¹ ; Alistair Savage ¹

¹ Department of Mathematics and Statistics, University of Ottawa Ottawa, ON K1N 6N5, Canada

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 791-829.

Résumé
Abstract

Nous définissons l’affinisation d’une catégorie monoïdale $𝒞$ arbitraire, correspondant à la catégorie des $𝒞$ -diagrammes sur le cylindre. Nous donnons aussi une autre caractérisation en termes de l’adjonction à $𝒞$ de générateurs pointés. L’affinisation formalise et unifie plusieurs constructions qui existent dans la littérature. En particulier, nous décrivons un grand nombre d’exemples provenant d’algèbres de type de Hecke, tresses, enchevêtrements, et invariants de nœuds. Lorsque $𝒞$ est rigide, son affinisation est isomorphe à sa trace horizontale, bien que les deux définitions paraissent assez différentes. En général, l’affinisation et la trace horizontale ne sont pas isomorphes.

We define the affinization of an arbitrary monoidal category $𝒞$ , corresponding to the category of $𝒞$ -diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to $𝒞$ . The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When $𝒞$ is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In general, the affinization and the horizontal trace are not isomorphic.

Reçu le : 2020-11-01
Accepté le : 2021-03-12
Publié le : 2021-03-22

DOI : 10.5802/jep.158

Classification : 18M15, 18M30, 57K14, 57K31
Keywords: Monoidal category, affinization, string diagram, annulus, cylinder, braid, tangle, skein theory
Mot clés : Catégorie monoïdale, affinisation, diagramme de cordes, anneau, cylindre, tresse, enchevêtrement, relation d’écheveaux

Affiliations des auteurs :

Youssef Mousaaid ¹ ; Alistair Savage ¹

¹ Department of Mathematics and Statistics, University of Ottawa Ottawa, ON K1N 6N5, Canada

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Affinization of monoidal categories},
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Youssef Mousaaid; Alistair Savage. Affinization of monoidal categories. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 791-829. doi : 10.5802/jep.158. https://jep.centre-mersenne.org/articles/10.5802/jep.158/

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