A lifting theorem for Grothendieck-Verdier categories
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914269367369728 |
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| author | Demirdilek, Max |
| author_facet | Demirdilek, Max |
| contents | We identify additional structure on a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to $\mathcal{C}$ and the functor becomes Frobenius linearly distributive. As an application, we recover and extend conditions under which modules over Hopf monads and Hopf algebroids inherit Grothendieck-Verdier structures. We also characterize when categories of bimodules, modules, and local modules over (commutative) algebras internal to a Grothendieck-Verdier category admit such structures. Our results apply to quantales, smash product algebras, skew group algebras, and enveloping algebras of Lie-Rinehart algebras.
For applications of the lifting theorem, we construct a strict $2$-equivalence between a $2$-category of Grothendieck-Verdier categories and one of linearly distributive categories with negation, and extend this $2$-equivalence to the braided setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14812 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A lifting theorem for Grothendieck-Verdier categories Demirdilek, Max Category Theory Quantum Algebra Representation Theory 18M10 (Primary) 16T05, 16B50, 17B35 (Secondary) We identify additional structure on a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to $\mathcal{C}$ and the functor becomes Frobenius linearly distributive. As an application, we recover and extend conditions under which modules over Hopf monads and Hopf algebroids inherit Grothendieck-Verdier structures. We also characterize when categories of bimodules, modules, and local modules over (commutative) algebras internal to a Grothendieck-Verdier category admit such structures. Our results apply to quantales, smash product algebras, skew group algebras, and enveloping algebras of Lie-Rinehart algebras. For applications of the lifting theorem, we construct a strict $2$-equivalence between a $2$-category of Grothendieck-Verdier categories and one of linearly distributive categories with negation, and extend this $2$-equivalence to the braided setting. |
| title | A lifting theorem for Grothendieck-Verdier categories |
| topic | Category Theory Quantum Algebra Representation Theory 18M10 (Primary) 16T05, 16B50, 17B35 (Secondary) |
| url | https://arxiv.org/abs/2601.14812 |