We use the theory of Tambara modules to extend and generalize the reconstruction theorem for module categories over a rigid monoidal category to the non-rigid case. We show a biequivalence between the 2-category of cyclic module categories over a monoidal category C and the bicategory of algebra and bimodule objects in the category of Tambara modules on C . Using it, we prove that a cyclic module category can be reconstructed as the category of certain free module objects in the category of Tambara modules on C , and give a sufficient condition for its reconstructability as module objects in C . To that end, we extend the definition of the Cayley functor to the non-closed case, and show that Tambara modules give a proarrow equipment for C-module categories, in which C-module functors are characterized as 1-morphisms admitting a right adjoint. Finally, we show that the 2-category of all C -module categories embeds into the 2-category of categories enriched in Tambara modules on C , giving an “action via enrichment” result.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-489853 |
| Date | January 2022 |
| Creators | Stroinski, Mateusz |
| Publisher | Uppsala universitet, Algebra, logik och representationsteori, Uppsala |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, comprehensive summary, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | U.U.D.M. report / Uppsala University, Department of Mathematics, 1101-3591 |
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