Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908415616352256 |
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| author | Décoppet, Thibault D. |
| author_facet | Décoppet, Thibault D. |
| contents | Fix a finite symmetric tensor category $\mathcal{E}$ over an algebraically closed field. We derive an $\mathcal{E}$-enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. In order to do so, we consider, associated to any $\mathcal{E}$-enriched finite braided tensor category $\mathcal{A}$ satisfying a mild technical assumption, a Hopf algebra $\mathbb{F}_{\mathcal{E}/\mathcal{A}}$ in $\mathcal{A}$. This is a generalization of Lyubashenko's universal Hopf algebra $\mathbb{F}_{\mathcal{A}}$ in $\mathcal{A}$. In fact, we show that there is a short exact sequence $\mathbb{F}_{\mathcal{E}}\rightarrow\mathbb{F}_{\mathcal{A}}\rightarrow\mathbb{F}_{\mathcal{E}/\mathcal{A}}$ of Hopf algebras in $\mathcal{A}$, and that the canonical pairing on $\mathbb{F}_{\mathcal{A}}$ descends to a pairing $ω_{\mathcal{E}/\mathcal{A}}$ on $\mathbb{F}_{\mathcal{E}/\mathcal{A}}$. We prove that $\mathcal{A}$ is $\mathcal{E}$-non-degenerate, i.e.\ its symmetric center is exactly $\mathcal{E}$, if and only if the pairing $ω_{\mathcal{E}/\mathcal{A}}$ is non-degenerate.
We then use the above characterization to show that an $\mathcal{E}$-enriched finite braided tensor category is invertible in the Morita 4-category of $\mathcal{E}$-enriched pre-tensor categories if and only if it is $\mathcal{E}$-non-degenerate. As an application of our relative invertibility criterion, we extend the full dualizability result of Brochier-Jordan-Snyder by showing that a finite braided tensor category is fully dualizable as an object of the Morita 4-category of braided pre-tensor categories if its symmetric center is separable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories Décoppet, Thibault D. Quantum Algebra Category Theory 16D90, 16T05, 18M15, 55U30 Fix a finite symmetric tensor category $\mathcal{E}$ over an algebraically closed field. We derive an $\mathcal{E}$-enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. In order to do so, we consider, associated to any $\mathcal{E}$-enriched finite braided tensor category $\mathcal{A}$ satisfying a mild technical assumption, a Hopf algebra $\mathbb{F}_{\mathcal{E}/\mathcal{A}}$ in $\mathcal{A}$. This is a generalization of Lyubashenko's universal Hopf algebra $\mathbb{F}_{\mathcal{A}}$ in $\mathcal{A}$. In fact, we show that there is a short exact sequence $\mathbb{F}_{\mathcal{E}}\rightarrow\mathbb{F}_{\mathcal{A}}\rightarrow\mathbb{F}_{\mathcal{E}/\mathcal{A}}$ of Hopf algebras in $\mathcal{A}$, and that the canonical pairing on $\mathbb{F}_{\mathcal{A}}$ descends to a pairing $ω_{\mathcal{E}/\mathcal{A}}$ on $\mathbb{F}_{\mathcal{E}/\mathcal{A}}$. We prove that $\mathcal{A}$ is $\mathcal{E}$-non-degenerate, i.e.\ its symmetric center is exactly $\mathcal{E}$, if and only if the pairing $ω_{\mathcal{E}/\mathcal{A}}$ is non-degenerate. We then use the above characterization to show that an $\mathcal{E}$-enriched finite braided tensor category is invertible in the Morita 4-category of $\mathcal{E}$-enriched pre-tensor categories if and only if it is $\mathcal{E}$-non-degenerate. As an application of our relative invertibility criterion, we extend the full dualizability result of Brochier-Jordan-Snyder by showing that a finite braided tensor category is fully dualizable as an object of the Morita 4-category of braided pre-tensor categories if its symmetric center is separable. |
| title | Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories |
| topic | Quantum Algebra Category Theory 16D90, 16T05, 18M15, 55U30 |
| url | https://arxiv.org/abs/2506.16241 |