Dualizable tensor categories
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 th...
| Main Authors: | , , |
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| Format: | Working paper |
| Language: | English |
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University of Oxford
2020
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| _version_ | 1797082097971625984 |
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| author | Douglas, CL Schommer-Pries, C Snyder, N |
| author_facet | Douglas, CL Schommer-Pries, C Snyder, N |
| author_sort | Douglas, CL |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T01:23:24Z |
| format | Working paper |
| id | oxford-uuid:911f94a4-dafc-4110-a0ef-7fd4004d1de1 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:23:24Z |
| publishDate | 2020 |
| publisher | University of Oxford |
| record_format | dspace |
| spelling | oxford-uuid:911f94a4-dafc-4110-a0ef-7fd4004d1de12022-03-26T23:16:36ZDualizable tensor categoriesWorking paperhttp://purl.org/coar/resource_type/c_8042uuid:911f94a4-dafc-4110-a0ef-7fd4004d1de1EnglishSymplectic ElementsUniversity of Oxford2020Douglas, CLSchommer-Pries, CSnyder, NWe 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. |
| spellingShingle | Douglas, CL Schommer-Pries, C Snyder, N Dualizable tensor categories |
| title | Dualizable tensor categories |
| title_full | Dualizable tensor categories |
| title_fullStr | Dualizable tensor categories |
| title_full_unstemmed | Dualizable tensor categories |
| title_short | Dualizable tensor categories |
| title_sort | dualizable tensor categories |
| work_keys_str_mv | AT douglascl dualizabletensorcategories AT schommerpriesc dualizabletensorcategories AT snydern dualizabletensorcategories |