Re: covering spaces and groupoids

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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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