A simplified categorical approach to several Galois theories

TL;DR

This paper introduces a simplified categorical framework for understanding various Galois theories by defining Galois structures and epimorphisms within a general categorical setting.

Contribution

It presents a unified, simplified approach to Galois theories using categorical concepts like group objects and principal homogeneous spaces.

Findings

Defines Galois structures in a general categorical context

Introduces Galois epimorphisms as principal homogeneous spaces

Provides a unified framework for multiple Galois theories

Abstract

We discuss the concept of Galois structure and Galois epimorphism in a general setting. Namely, a Galois structure for an epimorphism $\pi\colon M\to B$ in some category ${\mathcal C}$ is the action of a group object that gives to $M$ the structure of principal homogeneous space in the relative category ${\mathcal C}_B$.