Topological Galois theory

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 extends Grothendieck's theory of Galois categories and allows to build Galois-type equivalences in new contexts, such as for example graph theory and finite group theory.