Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems

TL;DR

This paper investigates the structure of entwined comodules and contramodules over coalgebras with multiple objects, establishing Frobenius, separability, and Maschke theorems for related functors, advancing categorical algebra theory.

Contribution

It introduces the concepts of entwined comodules and contramodules over coalgebras with several objects and proves fundamental theorems relating to their functor categories.

Findings

Proved Frobenius theorems for entwined module categories

Established separability conditions for functors between these categories

Derived Maschke type theorems in the context of coalgebras with multiple objects

Abstract

We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of ``entwined comodules'' and ``entwined contramodules'' over a triple $(\mathscr C,A,\psi)$, where $A$ is an algebra, $\mathscr C$ is a coalgebra with several objects and $\psi$ is a collection of maps that ``entwines'' $\mathscr C$ with $A$. Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.