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Dualizable tensor categories
About this Title
Christopher L. Douglas, Christopher Schommer-Pries and Noah Snyder
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 268, Number 1308
ISBNs: 978-1-4704-4361-0 (print); 978-1-4704-6347-2 (online)
DOI: https://doi.org/10.1090/memo/1308
Published electronically: May 6, 2021
Keywords: Tensor category,
fusion category,
bimodule category,
dualizable,
topological field theory,
local field theory,
pivotal,
spherical,
framing,
combing,
3-manifold,
Serre automorphism,
Radford equivalence
Table of Contents
Chapters
- Acknowledgments
- Introduction
- 1. The algebra of 3-framed bordisms
- 2. Tensor categories
- 3. Dualizability
- A. The cobordism hypothesis
Abstract
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-dimensional 3-framed local field theories. We also show that all finite tensor categories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds determine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach produces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expected connection between pivotal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures.- N. Andruskiewitsch, I. Angiono, A. García Iglesias, B. Torrecillas, and C. Vay, From Hopf algebras to tensor categories, Conformal field theories and tensor categories, Math. Lect. Peking Univ., Springer, Heidelberg, 2014, pp. 1–31. MR 3585364
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