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On dualizability of braided tensor categories

Part of: Modules, bimodules and ideals Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 March 2021

Adrien Brochier ,
David Jordan  and
Noah Snyder
Adrien Brochier
Affiliation:
Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75013Paris, Franceadrien.brochier@imj-prg.fr
David Jordan
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh, UKdjordan@ed.ac.uk
Noah Snyder
Affiliation:
Indiana University Bloomington, Bloomington, IN, USAnsnyder1@indiana.edu
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Abstract

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

Keywords

quantum groupsbraided tensor categorieshigher categoriestopological field theories

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality

Information

Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 3 , March 2021 , pp. 435 - 483
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Copyright
© The Author(s) 2021

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Footnotes

AB was supported during this work by RTG 1670 ‘Mathematics Inspired by String Theory and Quantum Field Theory’. DJ is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 637618). NS is supported by NSF grant DMS-1454767.

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