David Roberts writes: ---------------------- 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. ------------------------- Isn't this a bit like a sledgehammer to crack a nutshell? The fact is that if X has a universal cover then the topologised fundamental groupoid is such that (s,t): \pi_1(X) \to X \times X is a covering map, so one easily determines that the fundamental groups of \pi_1(X) are determined by those of x. So some new ideas are needed to get at the 2-type of X. I'd like to use this occasion to advertise some of the aims of Grothendieck's writings in the 1980s in the following excerpt, and which may even be relevant to David's question in this thread (??!): In a letter dated 02/05/1983 Alexander Grothendieck wrote: "Don't be surprised by my supposed efficiency in digging out the right kind of notions--I have just been following, rather let myself be pulled ahead, by that very strong thread (roughly: understand non commutative cohomology of topoi!) which I kept trying to sell for about ten or twenty years now, without anyone ready to ``buy'' it, namely to do the work. So finally I got mad and decided to work out at least an outline by myself." This is relevant to `Pursuing Stacks', and to `Derivateurs', for both of which see work by Georges Maltsiniotis. I had found that groupoids were in some sense more powerful than nonabelian cohomology, so in higher dimensions looked for, and eventually found, higher groupoids and colimit results, giving more information than could seemingly be obtained from exact sequences. Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]