We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with cer-tain 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 cat-egories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds deter-mine 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 pro-duces 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 piv-otal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures.
Author(s): Christopher L. Douglas; Christopher Schommer-Pries; Noah Snyder
Series: Memoirs of the American Mathematical Society Volume 268, (Number 1308)
Publisher: American Mathematical Soc.
Year: 2021
Language: English
Commentary: decrypted from 618B4B0EDBEED0AEC7C366B8E43C0031 source file
Pages: 88
City: Providence, RI
Cover
Title page
Acknowledgments
Introduction
I.1. Local topological field theory
I.2. Three-dimensional topology and three-dimensional algebra
I.2.1. From algebra to topology
I.2.2. From topology to algebra
I.3. Results
I.3.1. On 3-dualizability
I.3.2. On categorified 2-dimensional field theories
I.3.3. On quadruple duals
I.3.4. On tensor and bimodule categories
I.4. Outlook
I.5. Overview
Chapter 1. The algebra of 3-framed bordisms
1.1. ?-framed manifolds and ?-framed bordisms
1.1.1. ?-framings from normally framed immersions
1.1.2. ?-framings with boundary and corners
1.1.3. Low-dimensional examples of ?-framed bordisms
1.2. Duality in the 2-framed bordism category
1.3. The Serre bordism and the Serre automorphism
1.4. The Radford bordism and the Radford equivalence
1.4.1. A decomposition of the Radford bordism
1.4.2. A categorical formula for the Radford equivalence
Chapter 2. Tensor categories
2.1. Conventions for duality
2.2. Tensor categories, bimodule categories, and the Deligne tensor product
2.2.1. Linear categories, finite categories, monoidal categories, rigid cats
2.2.2. Module categories, functors, and transformations
2.2.3. Balanced tensor products and the 3-category of finite tensor cats
2.3. Exact module categories
2.3.1. Properties of exact module categories over finite tensor categories
2.3.2. The tensor product of exact module categories is exact
2.4. Dual and functor bimodule categories
2.4.1. Flips and twists of bimodule categories
2.4.2. Duals of bimodule categories
2.4.3. The dual bimodule category is the functor dual
2.4.4. The relative Deligne tensor product as a functor category
2.4.5. Dual bimodule categories as modules over a double dual
2.5. Separable module categories and separable tensor categories
2.5.1. Separability and semisimplicity
2.5.2. Separable bimodules compose
2.6. Separability and global dimension
2.6.1. Global dimension via quantum trace
2.6.2. The algebra of enriched endomorphisms of the unit
2.6.3. Fusion categories are modules over a Frobenius algebra
2.6.4. The window element of the representing Frobenius algebra
Chapter 3. Dualizability
3.1. Duals of tensor categories and invariants of 1-framed bordisms
3.1.1. The dual tensor category is the monoidal opposite
3.1.2. The twice categorified 1-dim field theory associated to a tensor cat
3.2. Adjoints of bimodule categories and invariants of 2-framed bordisms
3.2.1. The adjoint bimodule category is the functor dual
3.2.2. The categorified 2-dim field theory associated to a finite tensor cat
3.3. The Radford adjoints and the quadruple dual
3.3.1. Finite tensor categories are Radford objects
3.3.2. A topological proof of the quadruple dual theorem
3.3.3. A computation of the Radford equivalence
3.4. Adjoints of bimodule functors: separable tensor cats are dualizable
3.5. Spherical structures and structured field theories
3.5.1. Pivotal structures and trivializations of the Serre automorphism
3.5.2. Spherical structures as square roots of the Radford equivalence
3.5.3. Semisimple sphericality as a trace condition
3.5.4. Oriented, combed, and spin field theory descent conjectures
Appendix A. The cobordism hypothesis
Bibliography
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