Classifying submodules over monoidal categories

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.