On the Representation Theory of Galois and Atomic Topoi

The notion of a (pointed) Galois pretopos (“catégorie galoisienne”) was considered originally by Grothendieck in [12] in connection with the fundamental group of an scheme. In that paper Galois theory is conceived as the axiomatic characterization of the classifying pretopos of a profinite group G. The fundamental theorem takes the form of a representation theorem for Galois pretopos (see [10] for the explicit interpretation of this work in terms of filtered unions of categories the link to filtered inverse limits of topoi and its relation to classical Galois’s galois theory). An important motivation was pragmatical. The fundamental theorem is tailored to be applied to the category of etal coverings of a connected locally noetherian scheme pointed with a geometric point over an algebraically closed field. We quote: “Cette équivalence permet donc de interpréter les opérations courantes sur des revêtements en terms des opérations analogues dans BG, i.e. en terms des opérations évidentes sur des ensembles finis où G opére”. Later, in collaboration with Verdier ([1] Ex IV), he considers the general notion of pointed Galois Topos in a series of commented exercises (specially Ex IV, 2.7.5). There, specific guidelines are given to develop the theory of classifying topoi of progroups. It is stated therein that Galois topoi correspond exactly, as categories, to the full subcategories generated by locally constant objects in connected locally connected topoi (this amounts to the construction of Galois closures), and that they classify progroups. In [19], Moerdiejk developed this program in a rather sketchy way under the light of the concept of localic group. He proves the fundamental theorem in the form of a characterization of pointed Galois topoi as the classifying topoi of prodiscrete localic groups. We develop the theory of locally constant objects as defined in [1] Ex. IX in an apendix-section 5. We take from [6] the idea of presenting the topos of objects split by a cover as a push-out topos. We show how the existence of galois closures follows automatically by the fact that this topos has essential points. Connected groupoids are considered already in [12] because of the lack of a canonical point. The groupoid whose objects are all the points and with arrows the natural transformations, imposes itself as the natural mathematical object to be considered (although all the information is already in any one of its vertex groups). The theory is developed with groups for the sake of simplicity, but the appropriate formulation of the groupoid version is not straightforward (see [12] V 5). On general grounds the association of a localic groupoid to the set of points of a topos is evident by means of an enrichment over localic spaces of the categories of set-valued functors. Localic spaces are formal duals of locales, and it is not evident how this enrichment can be made in a way that furnish a manageable theory for the some-times unavoidable work in the category of locales. Generalizing the construction in [9] of the localic group of automorphisms of a set-valued functor we develop this enrichment in section 2.1, and in section 2.2 we construct the