Dear Peter, Dear David, Dear All, I was going to reply, but was too busy with non-mathematical things at the moment (a big move). The following are the papers relevant to David's question, beginning with the one that Peter mentions: M. C. Bunge, Classifying toposes and fundamental localic groupoids,in: Category Theory 1991, CMS Conference Procedings 13, American Mathematical Society (1992), 75-96. M. C. Bunge, Universal Covering Localic Toposes,Comptes Rendues Societe Royale du Canada 24 (1992), 245-250. M. C. Bunge and I.Moerdijk,On the construction of the Grothendieck fundamental group of a topos by paths,J. Pure and Applied Algebra 116 (1997), 99-113. Comments will have to wait, I'm afraid. Cordial regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************
From: P.T.Johnstone@dpmms.cam.ac.uk To: droberts@maths.adelaide.edu.au; categories@mta.ca Subject: categories: Re: fundamental localic groupoid? Date: Sun, 2 May 2010 10:49:59 +0100
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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