On Galois categories and perfectly reduced schemes

Abstract

It turns out that one can read off facts about schemes up to universal homeomorphism from their Galois categories. Here we propose a first modest slate of entries in a dictionary between the geometric features of a perfectly reduced scheme (or morphism of such) and the categorical properties of its Galois category (or functor of such). The main thing that makes this possible is Schr\"oer's total separable closure.