From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6334 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Sun, 24 Oct 2010 17:15:02 -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 1288001198 13481 80.91.229.12 (25 Oct 2010 10:06:38 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 25 Oct 2010 10:06:38 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Mon Oct 25 12:06:36 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 1PAJwj-0001y4-Sc for gsmc-categories@m.gmane.org; Mon, 25 Oct 2010 12:06:30 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49685) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PAJvc-00067A-DX; Mon, 25 Oct 2010 07:05:20 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PAJvZ-0001Gt-Vl for categories-list@mlist.mta.ca; Mon, 25 Oct 2010 07:05:18 -0300 In-Reply-To: <20101021002656.76CCDD13F@mailscan3.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6334 Archived-At: Dear George=2C> I haste to correct a possible misconception arising from my previous postin= g=2C and to propose an idea in connection with it. You wrote:=A0 > > I am not sure I fully understood what you say about unramified coverings > versus locally constant coverings. Are you even saying that you found a > Galois structure on TOP=2C or on any subcategory of TOP=2C whose covering= s are > exactly the unramified coverings (and the situation is non-trivial in the > sense that unramified coverings are not the same as locally constant > coverings? That would be wonderful! > Let C =3D LoCo/E=2C defined as in your book Galois Theories (with F. Borceu= x). An object p =A0of C with domain F=A0is said to be a covering morphism i= f there exists a morphism e of effective descent in Top with codomain E suc= h that (F=2Cp) is split by e. =A0A complete spread p of C with domain F =A0= - that is=2C an unramified morphism=2C need not be a covering morphism in C= in your sense=2C as we know. Whether there is a Galois structure on C whos= e coverings are precisely the unramified coverings without it forcing them = to be identified with the locally constant coverings does not seem likely. = At least we know that the class of unramified coverings in C is stable unde= r pullbacks and has other nice properties=2C so C is a natural choice of un= iverse. The fact that we have called "coverings" the unramified morphisms m= ay then be misleading if coverings are to be tied up with Galois theory.=A0 > Nevertheless=2C it is the case in topology that the notion of a covering in= the traditional sense has been enlarged to include branchings but not fold= s. A branched covering of a locally connected space E (R.H. Fox 1957) =A0is= the spread completion of a locally constant covering on a pure open subspa= ce U of E=2C thought off as "the complement of a knot". At least in this ca= se it is meaningful to consider the "branched fundamental groupoid" (or "kn= ot groupoid") of E with non-singular part U. It might be of interest to con= sider a notion of =A0"generalized Galois theory" to encompass this notion o= f "generalized covering morphism". Let me know what you think. Relevant dis= cussions in topos theory can be found in (Bunge-Niefield 2000)=2C (Funk 200= 0)=2C =A0(Bunge-Lack 2003)=2C as well as in=A0(Bunge-Funk 2006=2C 2007).=A0 > With best regards=2C > Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Burnside Hall=2C Office 1005 805 Sherbrooke St. West Montreal=2C QC=2C Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]