Re: covering spaces and groupoids

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

Dear Peter,

You wrote:
--------------------------------
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.
--------------------------------

It seems to me that you give a complicated route via the universal cover
to the inverse equivalence from GpdCov(\pi X) to TopCov(X), which
assumes connectivity and so requires a  choice of base points.

My account starts with any covering morphism q: G \to \pi_1 X of
groupoids and gives precise local conditions on X for there to be a
`lifted topology' on Ob(G) which makes it a covering space of X with
fundamental groupoid canonically isomorphic to G. No connectivity is
assumed, which makes it useful for discussing coverings of fundamental
groupoids of non connected topological groups. It has other uses, such
as topologising \pi_1 X.

I now find something quite unintuitive, even bizarre, in any emphasis on
`fundamental groups and change of base point': it is like giving railway
schedules in terms of return journeys and change of start points.

The later editions of my book also give a full account of orbit spaces
and orbit groupoids under the action of a group, giving conditions for
the fundamental groupoid of the orbit space to be naturally isomorphic
to the orbit groupoid of the fundamental groupoid.

Ronnie









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