Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
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| Format: | Preprint |
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2023
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| _version_ | 1866915163637022720 |
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| author | Décoppet, Thibault D. Yu, Matthew |
| author_facet | Décoppet, Thibault D. Yu, Matthew |
| contents | We introduce group-theoretical fusion 2-categories, a strong categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories, and investigate the group gradings of such fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara-Yamagami defect, that is $\mathbb{Z}/2$-graded fusion 2-categories whose non-trivially graded factor is $\mathbf{2Vect}$. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara-Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08117 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories Décoppet, Thibault D. Yu, Matthew Category Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra 16D90, 18M20, 18M25, 18N10 We introduce group-theoretical fusion 2-categories, a strong categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories, and investigate the group gradings of such fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara-Yamagami defect, that is $\mathbb{Z}/2$-graded fusion 2-categories whose non-trivially graded factor is $\mathbf{2Vect}$. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara-Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results. |
| title | Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories |
| topic | Category Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra 16D90, 18M20, 18M25, 18N10 |
| url | https://arxiv.org/abs/2306.08117 |