Unitary pseudonatural transformations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866914015972687872 |
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| author | Verdon, Dominic |
| author_facet | Verdon, Dominic |
| contents | We suggest two approaches to a definition of unitarity for pseudonatural transformations between unitary pseudofunctors on pivotal dagger 2-categories. The first is to require that the 2-morphism components of the transformation be unitary. The second is to require that the dagger of the transformation be equal to its inverse. We show that the `inverse' making these definitions equivalent is the right dual of the transformation in the 2-category Fun(C,D) of pseudofunctors C -> D, pseudonatural transformations, and modifications. We show that the subcategory Fun_u(C,D) $\subset$ Fun(C,D) whose objects are unitary pseudofunctors and whose 1-morphisms are unitary pseudonatural transformations is a pivotal dagger 2-category. We apply these results to obtain a Morita-theoretical classification of unitary pseudonatural transformations between fibre functors on the category of representations of a compact quantum group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_12760 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Unitary pseudonatural transformations Verdon, Dominic Category Theory Quantum Algebra We suggest two approaches to a definition of unitarity for pseudonatural transformations between unitary pseudofunctors on pivotal dagger 2-categories. The first is to require that the 2-morphism components of the transformation be unitary. The second is to require that the dagger of the transformation be equal to its inverse. We show that the `inverse' making these definitions equivalent is the right dual of the transformation in the 2-category Fun(C,D) of pseudofunctors C -> D, pseudonatural transformations, and modifications. We show that the subcategory Fun_u(C,D) $\subset$ Fun(C,D) whose objects are unitary pseudofunctors and whose 1-morphisms are unitary pseudonatural transformations is a pivotal dagger 2-category. We apply these results to obtain a Morita-theoretical classification of unitary pseudonatural transformations between fibre functors on the category of representations of a compact quantum group. |
| title | Unitary pseudonatural transformations |
| topic | Category Theory Quantum Algebra |
| url | https://arxiv.org/abs/2004.12760 |