From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5910 Path: news.gmane.org!not-for-mail From: Peter May Newsgroups: gmane.science.mathematics.categories Subject: Re: covering spaces and groupoids Date: Tue, 01 Jun 2010 07:53:47 -0500 Message-ID: References: <4C02A580.2000606@math.upenn.edu> <4C02A698.9090706@math.uchicago.edu> <4C04CB41.9080705@btinternet.com> 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 1275485848 10823 80.91.229.12 (2 Jun 2010 13:37:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 2 Jun 2010 13:37:28 +0000 (UTC) Cc: jds@math.upenn.edu, categories@mta.ca To: Ronnie Brown Original-X-From: categories@mta.ca Wed Jun 02 15:37:26 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 1OJo8H-0000cb-SG for gsmc-categories@m.gmane.org; Wed, 02 Jun 2010 15:37:22 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OJnSQ-0005jg-MM for categories-list@mta.ca; Wed, 02 Jun 2010 09:54:06 -0300 In-Reply-To: <4C04CB41.9080705@btinternet.com> Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5910 Archived-At: I'm not against coverings of groupoids: 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. Peter On 6/1/10 3:56 AM, 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 > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]