Reconstruction of module categories in the infinite and non-rigid settings
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| Format: | Preprint |
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2024
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| _version_ | 1866917850271186944 |
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| author | Stroiński, Mateusz Zorman, Tony |
| author_facet | Stroiński, Mateusz Zorman, Tony |
| contents | By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More explicitly, instead of algebra objects in finite tensor categories, we consider quasi-finite coalgebra objects in locally finite tensor categories. Moreover, we show that module categories over non-rigid monoidal categories can be reconstructed via lax module monads, which generalize algebra objects. For the monoidal category of finite-dimensional comodules over a (non-Hopf) bialgebra, we give this result a more concrete form, realizing module categories as categories of contramodules over Hopf trimodule algebras -- this specializes to our tensor-categorical results in the Hopf case. In this context, we also give a precise Morita theorem, as well as an analogue of the Eilenberg--Watts theorem for lax module monads and, as a consequence, for Hopf trimodule algebras. Using lax module functors we give a categorical proof of the variant of the fundamental theorem of Hopf modules which applies to Hopf trimodules. We also give a characterization of fusion operators for a Hopf monad as coherence cells for a module functor structure, using which we similarly reinterpret and reprove the Hopf-monadic fundamental theorem of Hopf modules due to Bruguières, Lack, and Virelizier. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconstruction of module categories in the infinite and non-rigid settings Stroiński, Mateusz Zorman, Tony Quantum Algebra Category Theory Representation Theory By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More explicitly, instead of algebra objects in finite tensor categories, we consider quasi-finite coalgebra objects in locally finite tensor categories. Moreover, we show that module categories over non-rigid monoidal categories can be reconstructed via lax module monads, which generalize algebra objects. For the monoidal category of finite-dimensional comodules over a (non-Hopf) bialgebra, we give this result a more concrete form, realizing module categories as categories of contramodules over Hopf trimodule algebras -- this specializes to our tensor-categorical results in the Hopf case. In this context, we also give a precise Morita theorem, as well as an analogue of the Eilenberg--Watts theorem for lax module monads and, as a consequence, for Hopf trimodule algebras. Using lax module functors we give a categorical proof of the variant of the fundamental theorem of Hopf modules which applies to Hopf trimodules. We also give a characterization of fusion operators for a Hopf monad as coherence cells for a module functor structure, using which we similarly reinterpret and reprove the Hopf-monadic fundamental theorem of Hopf modules due to Bruguières, Lack, and Virelizier. |
| title | Reconstruction of module categories in the infinite and non-rigid settings |
| topic | Quantum Algebra Category Theory Representation Theory |
| url | https://arxiv.org/abs/2409.00793 |