From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6305 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE: property_vs_structure Date: Fri, 8 Oct 2010 15:19:17 -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 1286664243 14579 80.91.229.12 (9 Oct 2010 22:44:03 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 9 Oct 2010 22:44:03 +0000 (UTC) To: Eduardo Dubuc , Original-X-From: majordomo@mlist.mta.ca Sun Oct 10 00:44:02 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P4i8z-000283-VH for gsmc-categories@m.gmane.org; Sun, 10 Oct 2010 00:43:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37665) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P4i8H-0006AC-NQ; Sat, 09 Oct 2010 19:43:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P4i8F-0008Ud-Vb for categories-list@mlist.mta.ca; Sat, 09 Oct 2010 19:43:12 -0300 Importance: Normal In-Reply-To: <20101008004112.BE2B38572@mailscan2.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6305 Archived-At: Dear Eduardo=2C Topological spaces or toposes=2C it is the same question. A space is locall= y connected iff its topos of sheaaves is locally connected.=A0 In my view=2C the question of whether the notion of a covering space is a s= tructure or a property depends on the definition of covering space that one= adopts.=A0 If the definition is made for arbitrary spaces (as in Spanier=2C whom you q= uote)=2C where a continuous map p from X to B is said to be a covering proj= ection if each point of X has an open neighborhood U evenly covered by p=2C= then covering space is a structure=2C no matter what the nature of the bas= e space is. =A0It so happens that=2C in the case of a locally connected spa= ce B=2C an alternative definition of a covering space can be given (as in R= . Brown=2C Topology and groupoids) that refers directly to canonical neighb= orhoods of points of X (U open=2C connected=2C and each connected component= of the inverse image of U under p in X is mapped homeomorphically onto U) = and=2C with this definition=2C covering space is indeed a property. =A0 So=2C in the locally connected case=2C the structure of covering space can = be equivalently replaced by a property - but I believe that it is still a s= tructure before those canonical choices are made.=A0Can a structure be equi= valent to a property=2C =A0yet not be a property? =A0 This is all I meant. I was not disputing a well known fact about covering s= paces of locally connected spaces (or of toposes=2C for that matter). Best regards=2CMarta > Date: Thu=2C 7 Oct 2010 21:40:54 -0300 > From: edubuc@dm.uba.ar > To: marta.bunge@mcgill.ca > CC: categories@mta.ca > Subject: Re: property_vs_structure >=20 > I am talking about topological spaces >=20 > Marta Bunge wrote: >>=20 >>=20 >> Even in the locally connected case there are several non isomorphic=20 >> trivialization structures. The difference is that=2C in that case=2C the= re=20 >> is a canonical one. >>=20 >>> Date: Wed=2C 6 Oct 2010 09:34:51 -0300 >>> From: edubuc@dm.uba.ar >>> To: edubuc@dm.uba.ar >>> CC: categories@mta.ca >>> Subject: categories: errata >>> >>> >>>> >>>> The difference is only manifest when the space B is not locally >>>> connected. In >>>> this case we may have homeomorphisms from X to X over B which do not >>>> preserve >>>> this structure (Spanier=2C Algebraic Topology). >>>> >>>> >>> >>> is not quite it should be=2C >>> >>> there is a clear notion of isomorphism of trivialization structure=2C= =20 >> and a same >>> space X over B may have non isomorphic structures. Alternatively=2C a= =20 >> continuous >>> function over B does not necessarily preserve the trivialization=20 >> structures. >>> >>> however=2C if B is locally connected=2C trivialization structures are= =20 >> like a pure >>> property of X. >>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]