I would like to take up on one small point in Jean Pradines' excellent email on terminology. He also writes On 02/10/2011 15:48, jpradines wrote:
Gabriel-Zisman, which introduces an adjoint for the "nerve functor" from groupoids to simplicial sets. This approach yields immediately a van Kampen theorem by preservation of colimits, according to Ronnie's presentation.
I agree with the words `a van Kampen theorem', with emphasis on `a'. What seems to be little appreciated is the notion of fundamental groupoid on a set A of base points; for a simplicial set K this notion is not seen as the set of base points is usually thought of as K_0. The whole point of the more general construction is to choose A according to the geometry at hand (see a Grothendieck quote on my web page for `Topology and Groupoids'); for the topological circle one conveniently chooses A to consist of 2 points. The general situation with covers requires A to meet every path component of every 1,2,3 fold intersection of the sets of the cover, so this format of the theorem does not follow immediately from adjoint functor arguments. The higher dimensional theorems of van Kampen type have NOT been shown to follow from simple adjoint functor arguments, or indeed any simple arguments, but have led to extensive (but restricted) non abelian colimit calculations in higher homotopy. The discussion of codiscrete or banal (or .....) groupoids raises the question of who was first to see the banal groupoid on {0,1} as a kind of unit interval groupoid? This use was made in my 1968 book `Elements of modern topology', now reissued as T&G. I should mention Dedecker, Paul; Valderrama, Jerko Graphes et cographes sur une catégorie abstraite. Application à l'homotopie. C. R. Acad. Sci. Paris Sér. A-B 262 1966 A377–A380. From the review: "The notion of cograph leads to a definition of homotopy in a category". Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]