On the Galois Theory of Grothendieck

In this paper we deal with Grothendieck's interpretation of Artin's interpretation of Galois's Galois Theory (and its natural relation with the fundamental group and the theory of coverings) as he developed it in Expose V, section 4, “Conditions axiomatiques d'une theorie de Galois” in the SGA1 1960/61. This is a beautiful piece of mathematics very rich in categorical concepts, and goes much beyond the original Galois's scope (just as Galois went much further than the non resubility of the quintic equation). We show explicitly how Grothendieck's abstraction corresponds to Galois work. We introduce some axioms and prove a theorem of characterization of the category (topos) of actions of a discrete group. This theorem corresponds exactly to Galois fundamental result. The theorem of Grothendieck characterizes the category (topos) of continuous actions of a profinite topological group. We develop a proof of this result as a "passage into the limit” (in an inverse limit of topoi) of our theorem of characterization of the topos of actions of a discrete group. We deal with the inverse limit of topoi just working with an ordinary filtered colimit (or union) of the small categories which are their (respective) sites of definition. We do not consider generalizations of Grothendieck's work, except by commenting briefly in the last section how to deal with the prodiscrete (not profinite) case. We also mention the work of Joyal-Tierney, which falls naturally in our discussion. There is no need of advanced knowledge of category theory to read this paper, exept for the comments in the last section.