Entwined modules over representations of categories

TL;DR

This paper develops a new theory of modules over category representations valued in entwining structures over coalgebras, extending module category concepts to a broader categorical context.

Contribution

It introduces a novel framework for modules over representations of categories with entwining structures, generalizing existing theories and connecting to modules over small K-linear categories.

Findings

Established a theory of entwined modules over category representations.

Connected the new theory to modules over small K-linear categories via Frobenius and separable functors.

Extended the concept of module categories to entwining structures in categorical representations.

Abstract

We introduce a theory of modules over a representation of a small category taking values in entwining structures over a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to the same extent as that of module categories as well as the philosophy of Mitchell of working with rings with several objects. The representations are motivated by work of Estrada and Virili, who developed a theory of modules over a representation taking values in small preadditive categories, which were then studied in the same spirit as sheaves of modules over a scheme. We also describe, by means of Frobenius and separable functors, how our theory relates to that of modules over the underlying representation taking values in small $K$-linear categories.