David Roberts writes about topologising the fundamental groupoid that `this should be well known'. Here are some references on that: 1. (R.BROWN with G. DANESH-NARUIE), ``The fundamental groupoid as a topological groupoid'', {\em Proc. Edinburgh Math. Soc.} 19 (1975) 237-244. 2. various editions of my book published now as `Topology and Groupoids', see 10.5.8, p.385 (the result is more general since it deals with topologising (\pi X)/N ). The method relies on the result that if X is reasonably nice, then a covering morphism of groupoids G \to \pi_1 X determines a `lifted' topology on Y = Ob(G) for which \pi_1(Y) \cong G. Presumably these methods cannot be adapted or modified for locales??? Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

David Roberts raised some questions that gave rise to several answers which, being interesting in themselves, "miss the mark" (as Joyal said) concerning the original questions. I copy and paste from Davis Roberts postings: "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." "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." The fundamental group, progroup, localic group, groupoid, localic groupoid, etc etc are all required (fundamentally !!) to represent first degree cohomology. This of course characterize them. Their classifying toposes should be the categories of "covering projections" (= locally constant sheaves in the locally-connected case). Classically Pi_1(X) was a set of classes of paths. Grothendieck started a departure (of course, out of facts already known at the time) from this, and constructed Pi_1(X) via fiber functors, where a topology appears for Pi_1. This Pi_1 can also be done combinatorialy by associating groupoids to family covers, and with several variations yields pro-things or localic things or even pro-localic things. Now, the equivalence with the Poincare Groupoid of paths holds (for topological spaces) strictly for the case of "locally connected, semilocally 1-connected space X" and no more. The proof does not work in any other case. Notice that this case correspond to the existence of a universal covering, that is, the fiber functor is representable. Bunge-Moerdijk studied paths in the general topos case, (the topos of sheaves on the unit interval playing the role of the unit interval), but their approach yields the good Pi_1 only in the representable case. So, it seems that under the present ideas concerning what a path is or should be, paths can not be used to construct the correct Pi_1 outside the representable case, where only discrete groupoids are pertinent, and no topology (or locales) is necessary. Some new ideas on a correct generalization of the notion of path ? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]