Entwined modules over linear categories and Galois extensions

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 $\mathcal D$.