Skein Categories in Non-semisimple Settings
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908793592348672 |
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| author | Brown, Jennifer Haïoun, Benjamin |
| author_facet | Brown, Jennifer Haïoun, Benjamin |
| contents | We introduce a version of skein categories of surfaces which depends on a tensor ideal in a linear ribbon category, thereby extending the existing theory to the setting of non-semisimple TQFTs. We obtain modified notions of skein algebras of surfaces and skein modules of 3-cobordisms for non-semisimple ribbon categories.
We prove that these skein categories built from ideals coincide with factorization homology, shedding new light on the similarities and differences between the semisimple and non-semisimple settings. The essential difference is the need to work with profunctors in the non-semisimple setting. Doing so produces a ``distinguished presheaf'' which plays the role of the distinguished object in skein categories in semisimple settings. As a consequence, we get a skein-theoretic description of factorization homology for a large class of balanced braided presentable categories, precisely all those which are expected to induce oriented categorified 3-TQFTs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Skein Categories in Non-semisimple Settings Brown, Jennifer Haïoun, Benjamin Quantum Algebra Category Theory Geometric Topology 18M15, 57K31 We introduce a version of skein categories of surfaces which depends on a tensor ideal in a linear ribbon category, thereby extending the existing theory to the setting of non-semisimple TQFTs. We obtain modified notions of skein algebras of surfaces and skein modules of 3-cobordisms for non-semisimple ribbon categories. We prove that these skein categories built from ideals coincide with factorization homology, shedding new light on the similarities and differences between the semisimple and non-semisimple settings. The essential difference is the need to work with profunctors in the non-semisimple setting. Doing so produces a ``distinguished presheaf'' which plays the role of the distinguished object in skein categories in semisimple settings. As a consequence, we get a skein-theoretic description of factorization homology for a large class of balanced braided presentable categories, precisely all those which are expected to induce oriented categorified 3-TQFTs. |
| title | Skein Categories in Non-semisimple Settings |
| topic | Quantum Algebra Category Theory Geometric Topology 18M15, 57K31 |
| url | https://arxiv.org/abs/2406.08956 |