Volume 21, issue 4 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
Invertible braided tensor categories

Adrien Brochier, David Jordan, Pavel Safronov and Noah Snyder

Algebraic & Geometric Topology 21 (2021) 2107–2140
Abstract

We prove that a finite braided tensor category 𝒜 is invertible in the Morita 4–category BrTens of braided tensor categories if and only if it is nondegenerate. This includes the case of semisimple modular tensor categories, but also nonsemisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4–dimensional framed topological field theories, which we regard as a nonsemisimple framed version of the Crane–Yetter–Kauffman invariants, after the Freed–Teleman and Walker constructions in the semisimple case. More generally, we characterize invertibility for E1– and E2–algebras in an arbitrary symmetric monoidal –category, and we conjecture a similar characterization of invertible En–algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of nondegenerate braided fusion categories, and pose a number of open questions about it.

Keywords
braided tensor category, topological field theory, higher categories
Mathematical Subject Classification
Primary: 18M20, 57R56
References
Publication
Received: 8 June 2020
Revised: 13 July 2020
Accepted: 27 July 2020
Published: 18 August 2021
Authors
Adrien Brochier
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Université de Paris
Sorbonne Université, CNRS
Paris
France
David Jordan
School of Mathematics
University of Edinburgh
Edinburgh
United Kingdom
Pavel Safronov
School of Mathematics
University of Edinburgh
Edinburgh
United Kingdom
Noah Snyder
Mathematics Department
Indiana University
Bloomington, IN
United States