I'm surprised that no-one has yet replied to David's original question by citing the work of Marta Bunge; she has a paper called (I think) "Classifying toposes and fundamental localic groupoids" dating from the early 1990s. (Being away from home at present, I don't have the exact reference to hand; I was hoping Marta would provide it.) Peter Johnstone On Apr 28 2010, David Roberts wrote:

Hi all,

it is (or should be) well known that the fundamental groupoid of a locally connected, semilocally 1-connected space X can be given a topology such that it is a topological groupoid (composition continuous etc). Without these assumptions on X there are counterexamples to the continuity of composition (c.f. misguided attempts to build a topological fundamental group). I was wondering, though, if there is a localic fundamental groupoid of an arbitrary space. One could presumably pass the the topos Sh(X) and then consider the fundamental groupoid of that, but I was wondering if there was a way to pass directly from the description of the arrows of Pi_1(X) as a set of classes of paths to a locale of classes of paths, and thence to a localic groupoid. My only 'evidence' that this might be the case is that the product in Loc is different to the product in Top, and so this may provide a work around the non-continuity of composition.

Thanks,

David Roberts

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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 -------- Message d'origine-------- De: categories@mta.ca de la part de David Roberts Date: jeu. 29/04/2010 10:09 À: categories@mta.ca Objet : categories: Re: fundamental localic groupoid? Steven Vickers wrote:

It's not obvious to me that the path composition can be transferred to that coequalizer. I think we would like the coequalizer to be stable under pullback, but that is not always true for locales. Perhaps it is in this case.

This is precisely where the construction falls down for a non-semilocally 1-connected space. If we restrict to looking at the automorphisms as a point (which supposedly form a topological group) the multiplication in the group pi_1 x pi_1 -> pi_1 (i.e. concatenating equivalence classes of loops) is not continuous in both variables, even though it is separately continuous (*). This can be traced back to the simple fact that the product of a pair of quotient maps in Top is not necessarily a quotient map. Note that this is the usual product in Top, not the compactly generated product. I presume there is a similar hitch with locales? (*) I learned of the non-continuity of the multiplication map from, among other places, the article J. Brazas, The topological fundamental group and hoop earring spaces, 2009, arXiv:0910.3685 which dashes optimistic earlier constructions. 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. One last question: has anyone studied the relation between (strong) shape of a space X and this localic groupoid G_X? David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]