Localic Galois Theory

In this article we prove the following: 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. We explain and give the necessary definitions to understand this statement. The hard direction in this equivalence was first proved in print in [4], Theorem 1, Section 3, Chapter VIII, and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales. We develop our version and our proof of this theorem, which is completely independent of descent theory and of any other result in [4]. Here the theorem follows as an straightforward consequence of a direct generalization of the fundamental theorem of Galois. 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. These developments correspond exactly to Classical Galois Theory. 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.