Admissible Skein Modules and Ansular Functors: A Comparison

Lukas Müller; Lukas Woike

doi:10.5802/afst.1856

Admissible Skein Modules and Ansular Functors: A Comparison
[Modules d’écheveaux admissibles et foncteurs ansulaires : une comparaison]

Lukas Müller ¹ ; Lukas Woike ²

¹Perimeter Institute, N2L 2Y5 Waterloo, Canada
²Université Bourgogne Europe, CNRS, IMB UMR 5584, 21000 Dijon, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 517-542

Abstract (VO)
Résumé

Given a finite ribbon category, there are two possibilities for an extension defined on all three-dimensional handlebodies: on the one hand, one can use the admissible skein module construction of Costantino–Geer–Patureau-Mirand. On the other hand, by a construction of the authors using Costello’s modular envelope, one can build a so-called ansular functor, a handlebody version of the notion of a modular functor. Unlike the admissible skein modules with their construction through the Reshetikhin–Turaev graphical calculus, the ansular functor is defined purely through a universal property. In this note, we prove the widely held expectation that these constructions are related by giving an isomorphism between them, with the somewhat surprising subtlety that we need to include consistently on one of the sides an additional boundary component labeled by the distinguished invertible object of Etingof–Nikshych–Ostrik. Our comparison result includes the handlebody group action as well as the skein algebra action.

Étant donné une catégorie enrubannée finie, il existe deux possibilités d’extension définie sur tous les corps d’anse tridimensionnels : d’une part, on peut utiliser la construction des modules d’écheveaux admissibles de Costantino–Geer–Patureau-Mirand. D’autre part, par une construction des auteurs utilisant l’enveloppe modulaire de Costello, on peut construire un foncteur ansulaire, une version pour les corps d’anse de la notion de foncteur modulaire. Contrairement aux modules d’écheveaux admissibles construits par le calcul graphique de Reshetikhin–Turaev, le foncteur ansulaire est défini, à équivalence près, par une propriété universelle. Dans cette note, nous démontrons l’attente largement répandue selon laquelle ces constructions sont liées en donnant un isomorphisme entre elles, avec la subtilité un peu surprenante de devoir inclure systématiquement sur l’un des côtés une composante de bord supplémentaire étiquetée par l’objet inversible distingué d’Etingof–Nikshych–Ostrik. Notre résultat de comparaison inclut l’action du groupe de difféotopie du corps d’anse ainsi que l’action de l’algèbre d’écheveaux.

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Reçu le : 2025-04-01
Accepté le : 2025-10-27
Publié le : 2026-06-12

DOI : 10.5802/afst.1856

Classification : 18M20, 57R56, 57K31
Keywords: skein modules, skein algebras, factorization homology, mapping class groups
Mots-clés : modules d’écheveaux, algèbres d’écheveaux, homologie de factorisation, groupes de difféotopie

Affiliations des auteurs :

Lukas Müller ¹ ; Lukas Woike ²

¹ Perimeter Institute, N2L 2Y5 Waterloo, Canada
² Université Bourgogne Europe, CNRS, IMB UMR 5584, 21000 Dijon, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Lukas Müller; Lukas Woike. Admissible Skein Modules and Ansular Functors: A Comparison. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 517-542. doi: 10.5802/afst.1856

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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