We use the theory of Tambara modules to extend and generalize the reconstruction theorem for module categories over a rigid monoidal category to the nonrigid 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 nonclosed 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.

We develop a theory of adjunctions in semigroup categories, that is, monoidal categories without a unit object. We show that a rigid semigroup category is promonoidal, and thus one can naturally adjoin a unit object to it. This extends the previous results of Houston in the symmetric case, and addresses a question of his. It also extends the results in the non-symmetric case with additional finiteness assumptions, obtained by Benson-Etingof-Ostrik, Coulembier, and Ko-Mazorchuk-Zhang. We give an interpretation of these results using comonad cohomology, and, in the absence of finiteness conditions, using enriched traces of monoidal categories. As an application of our results, we give a characterization of finite tensor categories in terms of the finitary 2-representation theory of Mazorchuk–Miemietz.

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.