Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914817810366464 |
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| author | Fuchs, Jürgen Schaumann, Gregor Schweigert, Christoph Wood, Simon |
| author_facet | Fuchs, Jürgen Schaumann, Gregor Schweigert, Christoph Wood, Simon |
| contents | We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory.
In any Grothendieck-Verdier module category M we find two important subcategories M' and M''. The internal End of an object in M' that is a C-generator is an algebra such that its category of modules is equivalent to M as a module category. We also introduce a partially defined relative Serre functor S which furnishes an equivalence between M' and M''. Any isomorphism between an object m of M' and S(m) in M'' endows the internal End of m with the structure of a Grothendieck-Verdier Frobenius algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors Fuchs, Jürgen Schaumann, Gregor Schweigert, Christoph Wood, Simon Category Theory Quantum Algebra We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category M we find two important subcategories M' and M''. The internal End of an object in M' that is a C-generator is an algebra such that its category of modules is equivalent to M as a module category. We also introduce a partially defined relative Serre functor S which furnishes an equivalence between M' and M''. Any isomorphism between an object m of M' and S(m) in M'' endows the internal End of m with the structure of a Grothendieck-Verdier Frobenius algebra. |
| title | Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors |
| topic | Category Theory Quantum Algebra |
| url | https://arxiv.org/abs/2405.20811 |