$\otimes$-Frobenius functors and exact module categories
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| Format: | Preprint |
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2025
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| _version_ | 1866917284544512000 |
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| author | Jaklitsch, David Yadav, Harshit |
| author_facet | Jaklitsch, David Yadav, Harshit |
| contents | We call a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between finite tensor categories $\otimes$-Frobenius if its left and right adjoints are isomorphic as $\mathcal{C}$-bimodule functors. We give several characterizations of this notion -- most notably, $F$ is $\otimes$-Frobenius if and only if the centralizer $Z({}_{F}\!{\mathcal{D}}_{\!F})$ is unimodular. We use them to analyze how actions on module categories behave under pullback along $F$. For perfect functors, we show that twisting a $\mathcal{D}$-module category $\mathcal{M}$ along $F$ preserves exactness, and that pivotality, unimodularity, and sphericality are preserved whenever $F$ is $\otimes$-Frobenius (or, more generally, Frobenius with respect to $\mathcal{M}$).
Applications include: (i) explicit criteria for $\otimes$-Frobenius functors arising from bialgebra maps $f\!:\!H'\!\to\!H$ between finite-dimensional Hopf algebras; and (ii) criteria ensuring that objects of internal natural transformations are (symmetric) Frobenius algebras in $Z(\mathcal{C})$. Along the way we show that central tensor functors are Frobenius iff they are $\otimes$-Frobenius and that any tensor functor between separable fusion categories is $\otimes$-Frobenius, answering questions of Flake-Laugwitz-Posur. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_16978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\otimes$-Frobenius functors and exact module categories Jaklitsch, David Yadav, Harshit Quantum Algebra Category Theory Representation Theory 16T05, 18M15, 18M20 We call a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between finite tensor categories $\otimes$-Frobenius if its left and right adjoints are isomorphic as $\mathcal{C}$-bimodule functors. We give several characterizations of this notion -- most notably, $F$ is $\otimes$-Frobenius if and only if the centralizer $Z({}_{F}\!{\mathcal{D}}_{\!F})$ is unimodular. We use them to analyze how actions on module categories behave under pullback along $F$. For perfect functors, we show that twisting a $\mathcal{D}$-module category $\mathcal{M}$ along $F$ preserves exactness, and that pivotality, unimodularity, and sphericality are preserved whenever $F$ is $\otimes$-Frobenius (or, more generally, Frobenius with respect to $\mathcal{M}$). Applications include: (i) explicit criteria for $\otimes$-Frobenius functors arising from bialgebra maps $f\!:\!H'\!\to\!H$ between finite-dimensional Hopf algebras; and (ii) criteria ensuring that objects of internal natural transformations are (symmetric) Frobenius algebras in $Z(\mathcal{C})$. Along the way we show that central tensor functors are Frobenius iff they are $\otimes$-Frobenius and that any tensor functor between separable fusion categories is $\otimes$-Frobenius, answering questions of Flake-Laugwitz-Posur. |
| title | $\otimes$-Frobenius functors and exact module categories |
| topic | Quantum Algebra Category Theory Representation Theory 16T05, 18M15, 18M20 |
| url | https://arxiv.org/abs/2501.16978 |