Steven Vickers wrote:
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.
This is precisely where the construction falls down for a non-semilocally 1-connected space. If we restrict to looking at the automorphisms as a point (which supposedly form a topological group) the multiplication in the group pi_1 x pi_1 -> pi_1 (i.e. concatenating equivalence classes of loops) is not continuous in both variables, even though it is separately continuous (*). This can be traced back to the simple fact that the product of a pair of quotient maps in Top is not necessarily a quotient map. Note that this is the usual product in Top, not the compactly generated product. I presume there is a similar hitch with locales? (*) I learned of the non-continuity of the multiplication map from, among other places, the article J. Brazas, The topological fundamental group and hoop earring spaces, 2009, arXiv:0910.3685 which dashes optimistic earlier constructions. 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. One last question: has anyone studied the relation between (strong) shape of a space X and this localic groupoid G_X? David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]