On fusion 2-categories
<p>The theory of fusion 1-categories has received much attention, and is therefore well established. Fusion 2-categories were recently introduced as a categorification of the notion of a fusion 1-category. The present thesis is devoted to the study of the general properties of fusion 2-categor...
| المؤلف الرئيسي: | |
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| التنسيق: | أطروحة |
| اللغة: | English |
| منشور في: |
2023
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| الموضوعات: |
| _version_ | 1826313352944549888 |
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| author | Décoppet, TD |
| author_facet | Décoppet, TD |
| author_sort | Décoppet, TD |
| collection | OXFORD |
| description | <p>The theory of fusion 1-categories has received much attention, and is therefore well established. Fusion 2-categories were recently introduced as a categorification of the notion of a fusion 1-category. The present thesis is devoted to the study of the general properties of fusion 2-categories.</p>
<p>At the heart of the definition of a fusion 2-category is the concept of finite semisimple 2-category over an algebraically closed field of characteristic zero. We generalize this concept by presenting the notion of a compact semisimple 2-category, which is sensible over an arbitrary field. Further, over an algebraically closed field or a real closed field, we prove that compact semisimple 2-categories are always finite.</p>
<p>Algebras in fusion 1-categories play a key role in the theory of fusion 1-categories. Accordingly, it is natural to study algebras in fusion 2-categories. We focus our attention on rigid algebras in fusion 2-categories, a notion which generalizes the definition of a fusion 1-category. In particular, we study the 2-categories of bimodules over rigid algebras. We also consider separable algebras, which are particularly well-behaved rigid algebras, and prove that a rigid algebra in a fusion 2-category is separable if and only if the associated 2-category of bimodules is finite semisimple.</p>
<p>Categorifying the notion of Morita equivalence between fusion 1-categories, we set up the Morita theory of fusion 2-categories. In particular, we give three equivalent characterizations of Morita equivalence between fusion 2-categories based on the 3-category of separable module 2-categories, the dual fusion 2-category, and the fusion 2-category of bimodules.</p>
<p>A crucial property of fusion 1-categories over an algebraically closed field of characteristic zero is that their Drinfeld centers are finite semisimple 1-categories. Categorifying the aforementioned result, we show that the Drinfeld center of any fusion 2-category over an algebraically closed field of characteristic zero is a finite semisimple 2-category. Our proof consists in a careful analysis of the Morita equivalence classes of fusion 2-categories.</p> |
| first_indexed | 2024-09-25T04:11:47Z |
| format | Thesis |
| id | oxford-uuid:301debf5-e28f-4955-a23c-f4fb47fb9457 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:11:47Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | oxford-uuid:301debf5-e28f-4955-a23c-f4fb47fb94572024-06-27T08:18:13ZOn fusion 2-categoriesThesishttp://purl.org/coar/resource_type/c_db06uuid:301debf5-e28f-4955-a23c-f4fb47fb9457MathematicsQuantum AlgebraCategory TheoryEnglishHyrax Deposit2023Décoppet, TD<p>The theory of fusion 1-categories has received much attention, and is therefore well established. Fusion 2-categories were recently introduced as a categorification of the notion of a fusion 1-category. The present thesis is devoted to the study of the general properties of fusion 2-categories.</p> <p>At the heart of the definition of a fusion 2-category is the concept of finite semisimple 2-category over an algebraically closed field of characteristic zero. We generalize this concept by presenting the notion of a compact semisimple 2-category, which is sensible over an arbitrary field. Further, over an algebraically closed field or a real closed field, we prove that compact semisimple 2-categories are always finite.</p> <p>Algebras in fusion 1-categories play a key role in the theory of fusion 1-categories. Accordingly, it is natural to study algebras in fusion 2-categories. We focus our attention on rigid algebras in fusion 2-categories, a notion which generalizes the definition of a fusion 1-category. In particular, we study the 2-categories of bimodules over rigid algebras. We also consider separable algebras, which are particularly well-behaved rigid algebras, and prove that a rigid algebra in a fusion 2-category is separable if and only if the associated 2-category of bimodules is finite semisimple.</p> <p>Categorifying the notion of Morita equivalence between fusion 1-categories, we set up the Morita theory of fusion 2-categories. In particular, we give three equivalent characterizations of Morita equivalence between fusion 2-categories based on the 3-category of separable module 2-categories, the dual fusion 2-category, and the fusion 2-category of bimodules.</p> <p>A crucial property of fusion 1-categories over an algebraically closed field of characteristic zero is that their Drinfeld centers are finite semisimple 1-categories. Categorifying the aforementioned result, we show that the Drinfeld center of any fusion 2-category over an algebraically closed field of characteristic zero is a finite semisimple 2-category. Our proof consists in a careful analysis of the Morita equivalence classes of fusion 2-categories.</p> |
| spellingShingle | Mathematics Quantum Algebra Category Theory Décoppet, TD On fusion 2-categories |
| title | On fusion 2-categories |
| title_full | On fusion 2-categories |
| title_fullStr | On fusion 2-categories |
| title_full_unstemmed | On fusion 2-categories |
| title_short | On fusion 2-categories |
| title_sort | on fusion 2 categories |
| topic | Mathematics Quantum Algebra Category Theory |
| work_keys_str_mv | AT decoppettd onfusion2categories |