From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6335 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Mon, 25 Oct 2010 13:15:34 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1288056268 8018 80.91.229.12 (26 Oct 2010 01:24:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 26 Oct 2010 01:24:28 +0000 (UTC) To: "Marta Bunge" , Original-X-From: majordomo@mlist.mta.ca Tue Oct 26 03:24:25 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 1PAYH2-0005wL-P8 for gsmc-categories@m.gmane.org; Tue, 26 Oct 2010 03:24:25 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60360) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PAYGC-0005dF-Tg; Mon, 25 Oct 2010 22:23:32 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PAYGA-0003ov-Hn for categories-list@mlist.mta.ca; Mon, 25 Oct 2010 22:23:30 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6335 Archived-At: Dear Marta, I fully agree with every word you say, and I can only add: 1. After these many years I trust myself that covering morphisms should only be defined via Galois theory. But exactly for this reason any adjective should indicate that Galois theory is not applicable (or we do not know yet, how to apply it). So for me "unramified coverings" is one of possible good names for something that is not presented as coverings with respect to some Galois theory. 2. More importantly than terminology, I think to find what you call "generalized Galois theory" would be very interesting, and, as I already said many times, I should study your work seriously. And again, in my opinion the aim would be to find a general-categorical definition that gives good examples in all (or in the most of) those categories I mentioned before (that is, not just in Top and TOP, but also, say, in CR^o = the opposite category of commutative rings). Such investigations will - I believe - soon or late lead to a beautiful unification of certain big parts of algebraic topology and algebraic geometry. With best regards, George ----- Original Message ----- From: "Marta Bunge" To: Sent: Sunday, October 24, 2010 11:15 PM Subject: categories: Re: property_vs_structure Dear George, I haste to correct a possible misconception arising from my previous posting, and to propose an idea in connection with it. You wrote: > > 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, or on any subcategory of TOP, whose coverings 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 = LoCo/E, defined as in your book Galois Theories (with F. Borceux). An object p of C with domain F is said to be a covering morphism if there exists a morphism e of effective descent in Top with codomain E such that (F,p) is split by e. A complete spread p of C with domain F - that is, an unramified morphism, need not be a covering morphism in C in your sense, as we know. Whether there is a Galois structure on C whose 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 under pullbacks and has other nice properties, so C is a natural choice of universe. The fact that we have called "coverings" the unramified morphisms may then be misleading if coverings are to be tied up with Galois theory. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]