I'm eclectic, and prefer closer contact with the real world of existing applications. There the overwhelming majority of the literature uses universal covers as usually constructed. I didn't go into it, but the dependence of that on the basepoint is also ephemeral: you get a universal covering space functor from the fundamental groupoid to such coverings easily enough. That is also used in applications (quite recently by Kate Ponto in work on fixed point theory). In any case, I don't place the emphasis you do on this matter, which I regard as minor from the point of view of algebraic topology. Peter On 6/2/10 2:03 AM, Ronnie Brown wrote:
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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