Entwined modules over linear categories and Galois extensions

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 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 E ⊆ D of categories. Under suitable conditions, we show that entwined modules over a C-Galois extension may be described as modules over the subcategory E of C-coinvariants of D.

Entwined Modules Over Representations of Categories

“…Proof For each y ∈ X , it follows by [4,Lemma 2.4] that H (x,r ) y is object C R y (ψ y ). We consider β : y −→ z in X and suppose we have α : x −→ y, i.e., x ≤ y.…”

“…When the coalgebra C is fixed, we have the subcategory E nt C . Given an entwining structure (R, C, ψ), we have a category M C R (ψ) of modules over it (see our earlier work in [4]). These entwined modules over (R, C, ψ) may be seen as modules over a certain categorical quotient space of R, which need not exist in an explicit sense, but is studied only through its category of modules.…”

“…In the next three sections, we study separability and Frobenius conditions for functors relating Mod C − R to the category Mod − R of modules over the underlying representation R : X −→ E nt C −→ L in. For this, we have to adapt the techniques from [9] as well as our earlier work in [4]. For more on Frobenius and separability conditions for Doi-Hopf modules and modules over entwining structures of algebras, we refer the reader to [6,[15][16][17].…”

“…Similarly, given an entwined C-representation R : X −→ E nt C with C finitely generated as a K -vector space, we can replace the induced representation(C, R) ψ : X −→ L in by (C * ⊗ R) ψ . Then, Mod − (C, R) ψ may be replaced by Mod − (C * ⊗ R) ψ .Proposition 10 4. Let X be a small category and R : X −→ E nt C an entwined C-representation.…”

Cyclic cohomology of entwining structures

“…Together, an entwining structure (A, C, ψ) behaves like a bialgebra or more generally, a comodule algebra over a bialgebra, as pointed out by Brzeziński [6]. There is also a well developed theory of entwined modules, with applications to diverse objects such as Doi-Hopf modules, Yetter-Drinfeld modules and coalgebra Galois extensions (see, for instance, [1], [2], [5], [7], [8], [9], [11], [12]).…”