Localic Galois Theory

Abstract

In Proposition I of "Memoire sur les conditions de resolubilite des equations par radicaux", Galois established that any intermediate extension of the splitting field of a polynomial with rational coefficients is the fixed field of its galois group. We first state and prove the (dual) categorical interpretation of of this statement, which is a theorem about atomic sites with a representable point. In the general case, the point determines a proobject and it becomes (tautologically) prorepresentable. We state and prove the, mutatus mutatis, prorepresentable version of Galois theorem. In this case the classical group of automorphisms has to be replaced by the localic group of automorphisms. These developments form the content of a theory that we call "Localic Galois Theory". An straightforward corollary of this theory is the theorem: "A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point". This theorem was first proved in print in Joyal A, Tierney M. "An extension of the Galois Theory of Grothendieck", Mem. AMS 151, Theorem 1, Section 3, Chapter VIII. Our proof is completely independent of descent theory and of any other result in that paper.