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
Dear David, I don't know the full answer, but here are some thoughts. Let I be the (localic) compact real interval [0,1]. I is locally compact and so for any locale X the locale exponential X^I exists and is the locale of paths in X. The endpoints of I give two maps d0 and d1: X^I -> X and we have the groupoid operations in the obvious way. (For composition this needs that I is a pushout of two copies of itself, glued end to end. This is not obvious, but I believe I once proved it to myself and perhaps it is known anyway. Exponentiation then transforms that pushout into a pullback, showing that X^I is in fact homeomorphic to the locale of composable pairs of paths.) The groupoid laws cannot hold until we have factored out homotopy, which must therefore be the next step. If gamma and delta are two paths agreeing at the endpoints, then a homotopy from gamma to delta, relative to the endpoints, can be expressed as a map D -> X where D is the closed Euclidean 2-ball (a closed disc). This uses two maps I -> D, taking I to the upper and lower half boundaries of D, and giving two maps du and dl: X^D -> X^I. Hence the fundamental groupoid Pi_1(X) should be the coequalizer of du and dl. 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. Once the groupoid operations have been established on the coequalizer, I conjecture it's not going to be too hard to prove the groupoid laws, using the usual constructions of homotopy theory. Regards, Steve Vickers. On Wed, 28 Apr 2010 11:54:29 +0930, David Roberts <droberts@maths.adelaide.edu.au> wrote: provide
a work around the non-continuity of composition.
Thanks,
David Roberts
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