Reconstruction of module categories in the infinite and non-rigid settings

TL;DR

This paper extends the reconstruction theory of module categories to infinite and non-rigid settings using lax module monads, providing new categorical frameworks and concrete realizations in Hopf algebra contexts.

Contribution

It introduces a generalized reconstruction approach for module categories via lax module monads and applies it to non-rigid monoidal categories and Hopf algebra cases.

Findings

Reconstruction of module categories using lax module monads.

Realization of module categories as contramodules over Hopf trimodule algebras.

Categorical proof of the fundamental theorem of Hopf modules for Hopf trimodules.

Abstract

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 Brugui\`eres, Lack, and Virelizier.