Frobenius and separable functors for the category of entwined modules over cowreaths, I: General theory

TL;DR

This paper develops a general theoretical framework for entwined modules over cowreaths in monoidal categories, exploring conditions for Frobenius and separable properties of associated coalgebras and functors.

Contribution

It introduces the concept of entwined modules over cowreaths and characterizes when these structures are Frobenius or separable, linking algebraic properties to categorical conditions.

Findings

Conditions for coalgebras to be Frobenius or separable

Criteria for the forgetful functor to be Frobenius or separable

Equivalence of properties when the unit object is a $ extotimes$-generator

Abstract

Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). These properties are equivalent when the unit object of the category is a $\otimes$-generator.