I'm not against coverings of groupoids: I reworked that theory from scratch when writing ``A concise course in algebraic topology''. Chapter 3 (pp21-32) does covering spaces, covering groupoids, the orbit category and the various equivalences of categories among them. I like it, but that chapter is maybe the main reason that my book is less popular than others: non-categorical types find it too difficult for young minds to absorb the first time around. Peter On 6/1/10 3:56 AM, Ronnie Brown wrote:

Peter May wrote: --------------------------- Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).) --------------------------

I would like to put in a case here for the groupoid approach ( see `Elements of modern topology' (1968), and subsequent editions; got the idea from Gabriel/Zisman, so not entirely idiosyncratic). If TopCov(X) is the category of covering spaces of X, and X admits a universal cover, then the fundamental groupoid functor \pi induces an equivalence of categories

\pi: TopCov(X) \to GpdCov(\pi X)

to the category of groupoid covering morphisms of \pi X. This seems to me to be the most intuitive version - a covering map is modelled by a covering morphism. I prefer the proof in this version, since it does not involve choices of base point, and allows the non connected case. It also allows one to discuss the case X is a topological group and to look at topological group covering maps. (Brown/Mucuk, Math ProcCamb Phil Soc 1994, following up ideas of R.L. Taylor).

The notion of covering morphism of groupoids goes back to P.A. Smith (Annals, 1951), called a regular morphism, and nowadays a discrete fibration, I think.

Is there an analogous version for Galois theory? Ronnie Brown

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