From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6337 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Mon, 25 Oct 2010 10:26:57 -0400 Message-ID: References: Reply-To: Marta Bunge NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1288136267 29930 80.91.229.12 (26 Oct 2010 23:37:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 26 Oct 2010 23:37:47 +0000 (UTC) To: George Janelidze , Original-X-From: majordomo@mlist.mta.ca Wed Oct 27 01:37:45 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PAt5M-0001gw-Ft for gsmc-categories@m.gmane.org; Wed, 27 Oct 2010 01:37:44 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43515) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PAt4W-0008Oz-P9; Tue, 26 Oct 2010 20:36:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PAt4Q-0000uV-43 for categories-list@mlist.mta.ca; Tue, 26 Oct 2010 20:36:46 -0300 In-Reply-To: <20101025111544.E773E63DD@mailscan3.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6337 Archived-At: Dear George=2C > Thanks for your interesting response.=A0Let me just comment here on your "a= ddition 1" below. It is my contention that "unramified coverings" is not an= appropriate expression to describe those "coverings" not associated with a= Galois theory in your sense=2C as first=2C there is a specific meaning att= ached to it=2C and secondly=2C a "branched Galois theory" already exists in= formally in the subject of knot groupoids. One may possibly generalize your= Galois theories to include these phenomena. I devote one paragraph to each= contention.=A0 > 1. When R.H.Fox ("Covering spaces with singularities"=2C R.H. Fox et al=2C = editors=2C Algebraic Geometry and Topology: A Symposium in honor of S. Lefs= chetz=2C Princeton University Press=2C 1957=2C 243-257) introduced spreads = and their completions=2C he had in mind what the title of his paper says=2C= that is=2C "coverings with singularities" so=2C not the traditional locall= y constant coverings. There could be "ramifications"=2C or branchings over = points of the base.=A0But no folds. Specifically=2C he was thinking of bran= ched coverings =A0(branching over a knot in the base) as the spread complet= ions of locally constant coverings=2C in which branching points were added = to the domain space. This is what led him to define a notion of spread=2C a= nd then perform a completion process leading to another spread singled out = among all such corresponding to a given cosheaf on the base space. The bran= ched coverings=2C and more generally the complete spreads of which they are= the motivating example=2C are "ramified". Now=2C add the condition that th= e complete spread (e.g. a branched covering) be a local homeomorphism. This= does not force it to be locally constant=2C as we know=2C but it cannot th= en have ramifications. Hence the expression "unramified coverings".=A0 > 2. As I said in my last posting=2C the =A0"branched coverings"=2C which are= very important in topology=2C yet do not correspond to any Galois theory i= n your sense=2C should correspond to a "generalized Galois theory" or to a = "branched Galois theory". To support my contention=2C note that=2C in (M. B= unge and S. Lack=2C van Kampen theorems for toposes=2C Advances in Mathemat= ics 179/2 (2003) 291-317)=2C we obtain=2C as an application of the van Kamp= en theorems we prove therein=2C =A0a connection with the use of the automor= phism group of a (universal) branched covering in the calculation of knot g= roups=2C as advocated by Fox. In particular=2C and the point I am making he= re=2C the expression "unramified coverings" does not describe them accurate= ly=2C as there may be ramifications. For a topos E=2C there is a biequivale= nce of the 2-categories of branched coverings of E branching over an object= Y (the latter thought of as the complement of a knot K) =A0on the one hand= =2C and that of all locally constant coverings of the slice topos E/Y on th= e other. The latter may in turn be viewed as the fundamental groupoid of E/= Y=2C or as the knot groupoid G(K) of K. There is a "Galois theory" there no= t associated with coverings in your sense=2C that is=2C with locally consta= nt coverings.=A0 > All of this requires further investigation=2C for which I will have no time= possibly until December=2C due to my trip to Buenos Aires.=A0 > With best regards=2C > Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics=20 McGill UniversityBurnside Hall=2C Office 1005 805 Sherbrooke St. West Montreal=2C QC=2C Canada H3A 2K6 Office: (514) 398-3810/3800 =A0 Home: (514) 935-3618 marta.bunge@mcgill.ca=20 http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]