Weak braiding for algebras in braided monoidal categories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914998294413312 |
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| author | Stockall, Devon |
| author_facet | Stockall, Devon |
| contents | Under appropriate conditions, if one picks a commutative algebra A with action of group G in braided monoidal category C, the category of A modules in C obtains a natural crossed G-braided structure. In the case of general commutative algebra object A in braided monoidal category C, one might ask what weakened notion of braiding one obtains on the category of A modules in C, and the relation that this braiding has to categorical symmetries acting on the associated quantum field theories, and on the algebra A itself.
In the following article, we present a definition of weak-braided monoidal category. It is proven that braided G-crossed categories, and categories of modules over commutative algebra objects in braided monoidal categories are weak-braided monoidal categories. Conversely, it is proven that, under reasonable assumptions, if a weak-braided category D is given by twisting the braiding by a collection of `twisting functors', then D is crossed-braided by a group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23027 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak braiding for algebras in braided monoidal categories Stockall, Devon Quantum Algebra Under appropriate conditions, if one picks a commutative algebra A with action of group G in braided monoidal category C, the category of A modules in C obtains a natural crossed G-braided structure. In the case of general commutative algebra object A in braided monoidal category C, one might ask what weakened notion of braiding one obtains on the category of A modules in C, and the relation that this braiding has to categorical symmetries acting on the associated quantum field theories, and on the algebra A itself. In the following article, we present a definition of weak-braided monoidal category. It is proven that braided G-crossed categories, and categories of modules over commutative algebra objects in braided monoidal categories are weak-braided monoidal categories. Conversely, it is proven that, under reasonable assumptions, if a weak-braided category D is given by twisting the braiding by a collection of `twisting functors', then D is crossed-braided by a group. |
| title | Weak braiding for algebras in braided monoidal categories |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2410.23027 |