From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5911 Path: news.gmane.org!not-for-mail From: Peter May Newsgroups: gmane.science.mathematics.categories Subject: Re: covering spaces and groupoids Date: Wed, 02 Jun 2010 08:41:58 -0500 Message-ID: References: <4C02A580.2000606@math.upenn.edu> <4C02A698.9090706@math.uchicago.edu> <4C04CB41.9080705@btinternet.com> <4C0502DB.5030603@math.uchicago.edu> Reply-To: Peter May NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1275525107 26709 80.91.229.12 (3 Jun 2010 00:31:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 3 Jun 2010 00:31:47 +0000 (UTC) Cc: jds@math.upenn.edu, categories@mta.ca To: Ronnie Brown Original-X-From: categories@mta.ca Thu Jun 03 02:31:46 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OJyLZ-0008WS-Oo for gsmc-categories@m.gmane.org; Thu, 03 Jun 2010 02:31:45 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OJxpl-0002RF-2D for categories-list@mta.ca; Wed, 02 Jun 2010 20:58:53 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5911 Archived-At: 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]