Compact Semisimple Tensor 2-Categories are Morita Connected
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912391038500864 |
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| author | Décoppet, Thibault D. Sanford, Sean |
| author_facet | Décoppet, Thibault D. Sanford, Sean |
| contents | In arXiv:2211.04917, it was shown that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. This result has recently allowed for a complete classification of fusion 2-categories. Here we establish that compact semisimple tensor 2-categories, which generalize fusion 2-categories to an arbitrary field of characteristic zero, also enjoy this ``Morita connectedness'' property. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compact Semisimple Tensor 2-Categories are Morita Connected Décoppet, Thibault D. Sanford, Sean Quantum Algebra Category Theory 18M15, 18M20 In arXiv:2211.04917, it was shown that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. This result has recently allowed for a complete classification of fusion 2-categories. Here we establish that compact semisimple tensor 2-categories, which generalize fusion 2-categories to an arbitrary field of characteristic zero, also enjoy this ``Morita connectedness'' property. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups. |
| title | Compact Semisimple Tensor 2-Categories are Morita Connected |
| topic | Quantum Algebra Category Theory 18M15, 18M20 |
| url | https://arxiv.org/abs/2412.15019 |