# Entwined modules over linear categories and Galois extensions

## TL;DR

This paper explores modules over categorified fiber bundles using entwining structures involving linear categories and coalgebras, establishing Frobenius and separability conditions, and introducing Galois extensions of categories.

## Contribution

It introduces the concept of C-Galois extensions of categories and characterizes entwined modules over these extensions as modules over subcategories of coinvariants.

## Key findings

- Frobenius and separability conditions for functors on entwined modules
- Definition of C-Galois extensions of categories
- Equivalence of entwined modules and modules over coinvariant subcategories

## Abstract

In this paper, we study modules over quotient spaces of certain categorified fiber bundles. These are understood as modules over entwining structures involving a small $K$-linear category $\mathcal D$ and a $K$-coalgebra $C$. We obtain Frobenius and separability conditions for functors on entwined modules. We also introduce the notion of a $C$-Galois extension $\mathcal E\subseteq \mathcal D$ of categories. Under suitable conditions, we show that entwined modules over a $C$-Galois extension may be described as modules over the subcategory $\mathcal E$ of $C$-coinvariants of $\mathcal D$.