From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6325 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Mon, 18 Oct 2010 17:04:28 -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 1287493554 16365 80.91.229.12 (19 Oct 2010 13:05:54 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 19 Oct 2010 13:05:54 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Tue Oct 19 15:05:52 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 1P8Bt2-00074s-Ba for gsmc-categories@m.gmane.org; Tue, 19 Oct 2010 15:05:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58873) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P8BrO-0007SM-9h; Tue, 19 Oct 2010 10:04:10 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P8BrK-0001TJ-5X for categories-list@mlist.mta.ca; Tue, 19 Oct 2010 10:04:06 -0300 In-Reply-To: <20101011201507.3A7732E52@mailscan3.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6325 Archived-At: Dear George=2C Thank you for reminding us of your old notion of Galois structure and cover= ing morphism in general categories. Although tangentially relevant to the d= iscussion initiated by Eduardo Dubuc=2C it relates to examples of propertie= s of continuous maps of spaces (or or morphisms of =A0toposes) studied in m= y book with Jonathon Funk=2C which may be relevant. I sent you this private= ly already=2C but on second thoughts I think it might be useful to make it = public. I begin by quoting a paragraph from your posting.=A0 > > >> Example 4. C =3D Fam(A) (or FiniteFam(A))=2C where A is an arbitrary cat= egory >> with terminal object and "multi-pullbacks" (which simply means that C ha= s >> pullbacks). This is a further generalization of the same thing=2C and >> everything can be repeated=2C but instead of "epimorphism" we should say >> "effective descent morphisms" (which is the same thing in the case of a >> topos). There are many non-topos-theoretic important special cases. For >> instance if C is the category of all (small) categories=2C then the cove= ring >> morphisms are as they should be=2C that is functors that are discrete >> fibrations and discrete opfibrations at the same time (this observation = is >> due to Steve Lack=2C although Steve never published it). If C is the cat= egory >> of all (small) groupoids=2C then this becomes even nicer since the discr= ete >> fibrations of groupoids are the same as discrete opfibrations=2C are Ron= nie >> Brown often tells us how nicely can they be used in homotopy theory... >=A0 > The notions of discrete fibration and discrete opfibration are lifted from = categories to geometric morphisms of toposes (in M. Bunge and J. Funk=2C Si= ngular Coverings of Toposes=2C LNM 1890=2C Springer=2C 2006=2C Chapter 9) r= elative to the symmetric KZ-monad called M therein for "measures" (M.Bunge = and A.Carboni=2C JPAA 105 (1995) 233-249). =A0They are=2C respectively=2C t= he local homeomorphisms and the complete spreads (singular coverings).=A0A = local homeomorphism over a locally connected space E with defining object X= is said to be an unramified covering if it is also a complete spread. Unra= mified coverings generalize covering morphisms =A0over a locally connected = space-- if X is a locally constant object of a locally connected space E=2C= then the corresponding local homeomorphism is a complete spread=2C hence a= n unramified covering. The class of unramified coverings is strictly larger= than the class of locally constant coverings=2C even over a locally connec= ted space (J. Funk and E.D. Tymchatyn=2C Unramified maps=2C J. Geometric To= pology 1(3) (2001) 249-280). Under hypotheses of the locally simply connect= ed kind=2C=A0unramified coverings are locally constant. The larger class of= unramified coverings =A0has some nice properties which the class of locall= y constant coverings fails to have -- for instance=2C they compose. Moreove= r=2C a van Kampen theorem holds not just for the class of locally constant = coverings but also for the larger class of unramified coveirngs (M.Bunge an= d S. Lack=2C Van Kampen theorems for toposes=2C Advances in Mathematics 179= /2 (2003) 291-317).=A0It is clear from your theory that both classes of mor= phisms are instances of what you call a Galois structure on the category of= (locally connected) topological spaces.=A0 > > Best wishes=2C Marta [For admin and other information see: http://www.mta.ca/~cat-dist/ ]