Module categories, internal bimodules and Tambara modules

TL;DR

This paper extends the theory of module categories over monoidal categories using Tambara modules, providing new reconstruction theorems and embedding results for non-rigid cases.

Contribution

It generalizes the reconstruction theorem to non-rigid monoidal categories via Tambara modules and establishes a biequivalence with algebra and bimodule objects.

Findings

Biequivalence between cyclic module categories and algebra/bimodule objects in Tambara modules

Reconstruction of cyclic module categories as free module objects in Tambara modules

Embedding of module categories into categories enriched in Tambara modules

Abstract

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 $\mathscr{C}$ and the bicategory of algebra and bimodule objects in the category of Tambara modules on $\mathscr{C}$. 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 $\mathscr{C}$, and give a sufficient condition for its reconstructability as module objects in $\mathscr{C}$. 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 $\mathscr{C}$-module categories, in which $\mathscr{C}$-module functors are characterized as $1$-morphisms admitting a right adjoint. Finally, we show that the $2$-category of all $\mathscr{C}$-module categories embeds into the $2$-category of categories enriched in Tambara modules on $\mathscr{C}$, giving an ''action via enrichment'' result.