The Tangle Hypothesis: Dimension 1
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916495800401920 |
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| author | Ayala, David Francis, John |
| author_facet | Ayala, David Francis, John |
| contents | We introduce an $(\infty,1)$-category ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$, the morphisms in which are framed tangles in $\mathbb{R}^n\times \mathbb{D}^1$. We prove that ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$ has the universal mapping out property of the 1-dimensional Tangle Hypothesis of Baez--Dolan and Hopkins--Lurie: it is the rigid $\mathcal{E}_n$-monoidal $(\infty,1)$-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal $(\infty,1)$-category gives link invariants, generalizing the Reshetikhin--Turaev invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Tangle Hypothesis: Dimension 1 Ayala, David Francis, John Algebraic Topology Category Theory Geometric Topology Quantum Algebra Primary 57R56. Secondary 57R90, 18B30, 18D10 We introduce an $(\infty,1)$-category ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$, the morphisms in which are framed tangles in $\mathbb{R}^n\times \mathbb{D}^1$. We prove that ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$ has the universal mapping out property of the 1-dimensional Tangle Hypothesis of Baez--Dolan and Hopkins--Lurie: it is the rigid $\mathcal{E}_n$-monoidal $(\infty,1)$-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal $(\infty,1)$-category gives link invariants, generalizing the Reshetikhin--Turaev invariants. |
| title | The Tangle Hypothesis: Dimension 1 |
| topic | Algebraic Topology Category Theory Geometric Topology Quantum Algebra Primary 57R56. Secondary 57R90, 18B30, 18D10 |
| url | https://arxiv.org/abs/2410.23965 |