Hi Eduardo, Marta, Andre, Peter and everyone else. Thank you all for your replies. I knew Marta's work would be applicable, but wasn't sure how much. This was the sort of clarification I was looking for

Bunge-Moerdijk studied paths in the general topos case, (the topos of sheaves on the unit interval playing the role of the unit interval), but their approach yields the good Pi_1 only in the representable case.

It was my lack of knowledge of things localic that was hindering me. But I am now intrigued by Andre's mention of the 2-topos E(2). Given the number of approaches to 2-topoi (Street, Weber, Shulman) I know this is a little bit more involved than a "groupoid-enriched topos" (although I imagine Andre was just emphasising that it is a (2,1)-category, not a general bicategory). The reason I am interested is that in my thesis I extend the (geometric*) fundamental bigroupoid Pi_2 from spaces to topological groupoids, and construct a topological groupoid over a 'nice enough' space which is, according Pi_2 2-connected. The construction is functorial, and will also work for manifolds to give a (Frechet) Lie groupoid (this joint work with Andrew Stacey). There is a subtlety in that 'nice enough' means locally simply connected and locally relatively 2-connected -- 2-well connected in the terminology of my thesis -- but for this to be locally trivial, like the universal covering space, 'nice enough' means having a basis of relatively contractible open sets. The construction mimics the construction of the universal covering space, in that it is the source fibre of the fundamental bigroupoid with an appropriate topology. (*geometric, in the sense that it isn't some sort of composition of functors through (bi)simplicial sets, or using geometric realisation, or an adjoint to N:2Gpd -> sSet; but using paths and surfaces and so on) Now I'm trying to imagine what relation the tower of topoi that Andre sketched has to the usual Whitehead tower in the case of a Grothendieck topos Sh(X) for a space X. It is probably the case that the 'Joyal tower' corresponds to the classical notion up to the maximum dimension n that X is locally n-connected (or more generally: n-well connected = 'locally (n-1)-connected and locally relatively n-connected') - and then one needs to use techniques more along the lines of Marta and coworkers. Actually (and this is thinking out loud) E(1), being a Galois topos, is sheaves on some groupoid, so I imagine E(2) is probably stacks (of groupoids) on a bigroupoid, and hence of course it is a (2,1)-category.. hmm. In the nice case that there is an initial object in E(2), I suppose I could conjecture that this initial stack is presented by my 2-connected cover, but I have no evidence whatsoever for it beyond a vague analogy. Food for thought. Thanks again, David PS: Despite being called Robert (or Rob) more times than I can remember, I have picked up a swag of new names in the past few days! Very peculiar :-) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]