On 29 April 2010 05:48, Ronnie Brown <ronnie.profbrown@btinternet.com>wrote:
I would like to make a further point about the topological fundamental groupoid of X. The information on this shows that iterating the fundamental groupoid does NOT lead to higher dimensional information on X.
If this means what I think it means: applying Pi_1 to the arrows and objects of the topologised fundamental groupoid, then I agree. In a suitable bicategory of topological groupoids (where internal weak equivalences a la Bunge-Pare or Everaert-Kieboom***-*van der Linden are formally inverted) the topologised fundamental groupoid is equivalent to a groupoid sans topology - in fact it is equivalent to itself where the topology is replaced by the discrete topology. The topologised fundamental groupoid in this way encodes only the 1-type of the space. David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]