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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