From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5902 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: covering spaces and groupoids Date: Tue, 01 Jun 2010 16:44:43 -0300 Message-ID: References: <4C02A580.2000606@math.upenn.edu> <4C02A698.9090706@math.uchicago.edu> Reply-To: "Eduardo J. Dubuc" 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 1275485504 9507 80.91.229.12 (2 Jun 2010 13:31:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 2 Jun 2010 13:31:44 +0000 (UTC) Cc: Peter May , jds@math.upenn.edu, To: Ronnie Brown Original-X-From: categories@mta.ca Wed Jun 02 15:31:40 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 1OJo2l-000631-Dk for gsmc-categories@m.gmane.org; Wed, 02 Jun 2010 15:31:39 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OJnXO-0005yO-5f for categories-list@mta.ca; Wed, 02 Jun 2010 09:59:14 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5902 Archived-At: > Is there an analogous version for Galois theory? In SGA 1 A.G. shows an equivalence C ~= GpoidActions(\pi(C)) where C is a category with certain axioms (that he callls Galoisiene) and \pi(C) is the fundamental groupoid of C (its objects are the set I of fiber functors of C, and its vertice groups are profinite, discrete only in case of existence of universal covering), and GpoidActions(\pi(C)) is the category of families indexed by I with an action of \pi(C). This includes as examples both the case of covering spaces and classical Galois Theory (Artin theory with the algebraic closure) I imagine that the category GpoidActions(\pi(C)) should be equivalent to GpdCov(\pi X) in a general abstract setting. Subsequently, this theory was extended and generalized in a well determined direction (progroupoids, localic groupoids, localic progroupoids) in several steps by A.G. himself, Moerdiejk, Bunge, Dubuc and Joyal-Tierney. Other authors extended the basic theory (presence of universal covering and discrete groups) in different directions. e.d. 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/ ]