The Fundamental Progroupoid of a General Topos

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004. introduction. It is well known that if E is a locally connected topos then the category of covering projections (that is, locally constant objects) is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid π(E), and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The subject that concern us here was developed (in the context of grothendieck topos over Set) for a pointed locally connected topos by Grothendieck-Verdier in a series of commented exercises in Expose IV of the SGA4 [1], and by Artin-Mazur [2]. Later Moerdijk [15] treats the subject over a general base topos S, and replace progroups by prodiscrete localic groups. Bunge [3] does the unpointed case and works with prodiscrete localic groupoids. See also Bunge-Moerdijk [4], and the Appendix in Dubuc [6] for a resume of this theory. The salient feature of the theory is that covering projections are considered as a full subcategory of the topos, and this fact is essential in the proofs of the validity of the statements. Covering projections can not be considered as a full subcategory when the topos is not locally connected. The principal source of inspiration for our work was the paper of Hernandez-Paricio [8], where he treats successfully the case of non locally connected topological spaces. There it is possible to see that there is a descent datum underneath the notion of covering projection, and that this datum has to be taken into account in the definition of covering projection. We can see an implicit situation of classical topological descent as described in the introduction to “Categories Fibrees et Descente”, [7], Expose VI. Once the descent datum is made explicit, the category of covering projections of a topological space trivialized by a (fix) covering is, by its very definition, the classifying topos of a discrete groupoid, and this groupoid can be explicitly constructed as the free category over the nerve of the covering. The assignemment of this groupoid is functorial on the filtered poset of covering sieves and determines the fundamental progroupoid of the space. We explain all this in section 1. Given any topos, there is no problem to construct the topos of locally constant objects trivialized by a (fix) cover. The problem is that when the topos is not locally connected, the resulting topos is not atomic because it fails to be both