Well Peter, I imagine that the reason is that the question was asking for "locale of path", and not for the "local of automorphisms of the fiber", which is completely developed and solved. I copy and paste the question: "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." As you know, there are many papers on the fundamental localic groupoid, but all of then deal (with variations) with the locale of automorphism. The works I know (of course they may be also other authors I ignore) were made (in a cronological order by first contribution, but later mixed in time) by: Grothendieck-Tierney (Tierney first observed that the actions of a grothendiek progroup were the same thing that the actions of the localic group inverse image in the category of locales) Moerdijk, Kennison, Bunge and Dubuc. In the case of topological spaces there is work done by Hernandez Paricio. There is also the often cited memoir of Joyal-Tierney. However, there is one paper i know which in some sense deals with paths and which may have some relevance to the question asked: Bunge M., Moerdijk I.,On the construction of the Grothendieck fundamental group of a topos by paths, J. Pure Appl. Alg. 116 (1997). P.T.Johnstone@dpmms.cam.ac.uk wrote:
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
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