Classifying submodules over monoidal categories

TL;DR

This paper develops a framework for classifying submodules of module categories over monoidal categories, extending existing ideas and applying to structures related to quantum symmetric pairs, with concrete classifications in skein categories.

Contribution

It introduces a new classification framework for submodules over monoidal categories with twisted structures, linking submodules to path-algebra modules and analyzing their properties.

Findings

Established a bijection between submodules of a module category and submodules of a path-algebra module.

Proved the correspondence respects idempotent completion and analyzed decategorification behavior.

Provided a complete classification of submodules in the disoriented skein category.

Abstract

We study the classification of submodules of module categories over monoidal categories, extending ideas of Coulembier on the classification of tensor ideals in monoidal categories. We develop a framework that applies to module categories equipped with a twisted cylinder twist, a structure closely related to the twisted reflection equation and quantum symmetric pairs. Under mild assumptions, we establish an order-preserving bijection between submodules of a module category $\mathcal{M}$ and submodules of the path-algebra module $\mathcal{M}(1,-)$. We show that this correspondence is compatible with idempotent completion and analyze its behavior under decategorification to the split Grothendieck group, giving criteria for classification in terms of indecomposable objects. As an application, we study the disoriented skein category as a module category over the oriented skein category, describe its indecomposable objects, and obtain a complete classification of its submodules.