Is there an analogous version for Galois theory?
In SGA 1 A.G. shows an equivalence C ~= GpoidActions(\pi(C)) where C is a category with certain axioms (that he callls Galoisiene) and \pi(C) is the fundamental groupoid of C (its objects are the set I of fiber functors of C, and its vertice groups are profinite, discrete only in case of existence of universal covering), and GpoidActions(\pi(C)) is the category of families indexed by I with an action of \pi(C). This includes as examples both the case of covering spaces and classical Galois Theory (Artin theory with the algebraic closure) I imagine that the category GpoidActions(\pi(C)) should be equivalent to GpdCov(\pi X) in a general abstract setting. Subsequently, this theory was extended and generalized in a well determined direction (progroupoids, localic groupoids, localic progroupoids) in several steps by A.G. himself, Moerdiejk, Bunge, Dubuc and Joyal-Tierney. Other authors extended the basic theory (presence of universal covering and discrete groups) in different directions. e.d. 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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