Topological Galois Theory

TL;DR

This paper develops a general topos-theoretic framework for Galois theories, unifying classical results and enabling new applications in areas like graph and finite group theory.

Contribution

It introduces an abstract topos-theoretic approach that generalizes and extends existing Galois theories to new mathematical contexts.

Findings

Unifies Galois theories under a topos-theoretic framework

Enables construction of Galois-type equivalences in graph theory

Provides a foundation for Galois theories in finite group contexts

Abstract

We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of continuous actions of a topological group. Our framework subsumes in particular Grothendieck's Galois theory and allows to build Galois-type equivalences in new contexts, such as for example graph theory and finite group theory.