Duality in Monoidal Categories
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915776181567488 |
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| author | Halbig, Sebastian Zorman, Tony |
| author_facet | Halbig, Sebastian Zorman, Tony |
| contents | We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal hom is tensor-representable? We provide a counterexample in terms of the category of sl2-crystals. As a byproduct, we obtain characterisations of the Grothendieck-Verdier duality and rigidity of functor categories endowed with Day convolution as their tensor product. This has various applications, three of which we study in detail: generalisations of quasi-Frobenius algebras, called QF-2 algebras; Mackey functors, where we prove that, as expected due to work of Bouc, an object being rigidly dualisable is equivalent to it being finitely-generated projective; and crossed modules of finite groups, where we associate to each of these objects a Grothendieck-Verdier category of group-graded representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03545 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Duality in Monoidal Categories Halbig, Sebastian Zorman, Tony Category Theory Quantum Algebra 18D15, 18M10 We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal hom is tensor-representable? We provide a counterexample in terms of the category of sl2-crystals. As a byproduct, we obtain characterisations of the Grothendieck-Verdier duality and rigidity of functor categories endowed with Day convolution as their tensor product. This has various applications, three of which we study in detail: generalisations of quasi-Frobenius algebras, called QF-2 algebras; Mackey functors, where we prove that, as expected due to work of Bouc, an object being rigidly dualisable is equivalent to it being finitely-generated projective; and crossed modules of finite groups, where we associate to each of these objects a Grothendieck-Verdier category of group-graded representations. |
| title | Duality in Monoidal Categories |
| topic | Category Theory Quantum Algebra 18D15, 18M10 |
| url | https://arxiv.org/abs/2301.03545 |