Module categories, internal bimodules and Tambara modules

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.