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"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"-- Provided by publisher Cover 1 Title page 2 Acknowledgments 8 Introduction 10 I.1. Local topological field theory 10 I.2. Three-dimensional topology and three-dimensional algebra 11 I.2.1. From algebra to topology 11 I.2.2. From topology to algebra 12 I.3. Results 13 I.3.1. On 3-dualizability 13 I.3.2. On categorified 2-dimensional field theories 14 I.3.3. On quadruple duals 15 I.3.4. On tensor and bimodule categories 16 I.4. Outlook 18 I.5. Overview 19 Chapter 1. The algebra of 3-framed bordisms 22 1.1. n-framed manifolds and n-framed bordisms 23 1.1.1. n-framings from normally framed immersions 23 1.1.2. n-framings with boundary and corners 24 1.1.3. Low-dimensional examples of n-framed bordisms 25 1.2. Duality in the 2-framed bordism category 26 1.3. The Serre bordism and the Serre automorphism 28 1.4. The Radford bordism and the Radford equivalence 30 1.4.1. A decomposition of the Radford bordism 31 1.4.2. A categorical formula for the Radford equivalence 31 Chapter 2. Tensor categories 36 2.1. Conventions for duality 37 2.2. Tensor categories, bimodule categories, and the Deligne tensor product 39 2.2.1. Linear categories, finite categories, monoidal categories, rigid cats 39 2.2.2. Module categories, functors, and transformations 40 2.2.3. Balanced tensor products and the 3-category of finite tensor cats 42 2.3. Exact module categories 44 2.3.1. Properties of exact module categories over finite tensor categories 44 2.3.2. The tensor product of exact module categories is exact 45 2.4. Dual and functor bimodule categories 48 2.4.1. Flips and twists of bimodule categories 48 2.4.2. Duals of bimodule categories 49 2.4.3. The dual bimodule category is the functor dual 50 2.4.4. The relative Deligne tensor product as a functor category 51 2.4.5. Dual bimodule categories as modules over a double dual 53 2.5. Separable module categories and separable tensor categories 54 2.5.1. Separability and semisimplicity 55 2.5.2. Separable bimodules compose 56 2.6. Separability and global dimension 59 2.6.1. Global dimension via quantum trace 59 2.6.2. The algebra of enriched endomorphisms of the unit 61 2.6.3. Fusion categories are modules over a Frobenius algebra 62 2.6.4. The window element of the representing Frobenius algebra 63 Chapter 3. Dualizability 66 3.1. Duals of tensor categories and invariants of 1-framed bordisms 67 3.1.1. The dual tensor category is the monoidal opposite 67 3.1.2. The twice categorified 1-dim field theory associated to a tensor cat 69 3.2. Adjoints of bimodule categories and invariants of 2-framed bordisms 71 3.2.1. The adjoint bimodule category is the functor dual 71 3.2.2. The categorified 2-dim field theory associated to a finite tensor cat 72 3.3. The Radford adjoints and the quadruple dual 75 3.3.1. Finite tensor categories are Radford objects 76 3.3.2. A topological proof of the quadruple dual theorem 77 3.3.3. A computation of the Radford equivalence 78 3.4. Adjoints of bimodule functors: separable tensor cats are dualizable 81 3.5. Spherical structures and structured field theories 83 3.5.1. Pivotal structures and trivializations of the Serre automorphism 84 3.5.2. Spherical structures as square roots of the Radford equivalence 85 3.5.3. Semisimple sphericality as a trace condition 86 3.5.4. Oriented, combed, and spin field theory descent conjectures 87 Appendix A. The cobordism hypothesis 90 Bibliography 94 Back Cover 102