Local Modules in Braided Monoidal 2-Categories
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910501443731456 |
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| author | Décoppet, Thibault D. Xu, Hao |
| author_facet | Décoppet, Thibault D. Xu, Hao |
| contents | Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3+1)d topological phases of matter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02843 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local Modules in Braided Monoidal 2-Categories Décoppet, Thibault D. Xu, Hao Category Theory Strongly Correlated Electrons High Energy Physics - Theory Quantum Algebra 18M15, 18M20, 18N10 Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3+1)d topological phases of matter. |
| title | Local Modules in Braided Monoidal 2-Categories |
| topic | Category Theory Strongly Correlated Electrons High Energy Physics - Theory Quantum Algebra 18M15, 18M20, 18N10 |
| url | https://arxiv.org/abs/2307.02843 |