David Roberts wrote:
Going back to more general thoughts, given an arbitrary space X, the topos Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney), which I thought (!) could be interpreted as the fundamental groupoid in nice situations. Here I suppose is my real question: what relation is there between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a test case could be the Warsaw circle W: Pi_1(W) is equivalent to the discrete groupoid on the set {a,b}, and surely G_W is more than this.
I would like to make a few observations. If my recollection is right, during the 1980's the dream of some topos theorists (including myself) was to use atomic toposes as generalised K(pi,1)-spaces. The dream has not materialised and it maybe foolish. Another avenue is to extend Grothendieck Galois theory, since the fundamental groupoid of a space classifies the covers of that space. Eduardo Dubuc is presently developing a general theory of Galois toposes. The fundamental Galois topos of a topos E could be defined as the reflection E--->E(1) of E into the full subcategory of Galois toposes over the base topos S. I am not supposing that S is the category of sets. I am writing E(1) because I want to suggest that it is a stage of a Postnikov tower of S-toposes: E(-1)<---E(0)<---E(1)<----E(2)<---- The topos E(-1) is of course the image of the geometric morphisme E-->S. The construction of E(-1) depends on the factorisation system (epimorphisms of toposes, sub-toposes) The topos E(0) is the totally disconnected localic reflection of E over the base topos S. Its construction depends on the factorisation system (connected morphisms of toposes, totally disconnected localic morphisms) The topos E(1) is the Galois topos reflection of E over the base topos S. Its construction depends on the factorisation system (1-connected morphisms of toposes, Galois morphisms) The topos E(2) does not exists in general for the simple reason that it is a 2-topos, not an ordinary topos. A 2-topos is enriched over groupoids. The whole theory depends on three factorisation systems (epimorphisms of toposes, sub-toposes) (connected morphisms of toposes, totally disconnected localic morphisms) (1-connected morphisms of toposes, Galois morphisms) The first factorisation system is well known and was constructed a long time ago. Excuse my ignorance, but I do not know if the second factorisation system has been fully constructed. Can someone tell me? The third factorisation system was partially constructed by Dubuc: he can factor the morphism E-->S when S is the category of sets. My observations concerning your problems could be off the mark. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]