From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6310 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE: property_vs_structure Date: Sat, 9 Oct 2010 18:26:10 -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 1286664418 15168 80.91.229.12 (9 Oct 2010 22:46:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 9 Oct 2010 22:46:58 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Sun Oct 10 00:46:57 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 1P4iBt-00039n-10 for gsmc-categories@m.gmane.org; Sun, 10 Oct 2010 00:46:57 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37695) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P4iBB-0006TP-Ca; Sat, 09 Oct 2010 19:46:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P4iB9-000092-EF for categories-list@mlist.mta.ca; Sat, 09 Oct 2010 19:46:11 -0300 In-Reply-To: <20101009210755.68229A98F@mailscan2.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6310 Archived-At: Dear Eduardo=2C>In a =A0previous message of mine and =A0in response to your= questions=A0> >>=A0What do we mean by structure ?=2C and=2C what do we=A0mean by property= ? > =A0I wrote> >> On a given data=2C a structure is additional data on it such=A0that=2C i= f it exists=2C it is not necessarily unique up to isomorphism.=A0>>A=A0prop= erty is additional data such that=2C if it exists=2C it is unique up to=A0i= somorphism (in the model theoretic sense).> to which you replied>=A0=A0 >> Well=2C here it is necessary first to establish what do we mean by=A0 >> "isomorphism". To do this we need a way to compare the structures=2C tha= t is=2C we=A0 >> have to define morphism of structures (see below). Without this=2C the a= bove is=A0 >> meaningless.> Notice that I specified "in the model theoretic sense". In that context=2C = the notions of a structure and of a morphism of structures are defined and = it is that sense that I meant them. They are perfectly meaningful.=A0 > You added>=A0 >> People discussing structure vs property were giving examples where all t= his=A0 >> was clear and straightforward (invertibility in a monoid=2C neutral elem= ent in a=A0 >> semigroup=2C etc). My original purpose when I wrote my first mail was to= =A0 >> consider a less trivial example testing the following "definitions"=2C t= hat I=A0 >> see you subscribe above at least in what it concerns "structure" and "pr= operty": >=A0 > ********* > Michael Shulman wrote: >> (**) >> property =3D forgetful functor is full and faithful >> structure =3D forgetful functor is faithful >> property-like structure =3D forgetful functor is pseudomonic > ********** > I do not "subscribe" to these notions. I simply use the well-known notions = from first-order logic and model theory as I said earlier. I am aware of th= e difference between the locally connected =A0and the general case concerni= ng covering projections and it is not the mathematics that I was disputing.= But I still have trouble following your analysis of this situation as was = your purpose. The difference is that=2C whereas you consider the notion of = a covering projection to be a property of a continuous map p from X to B wh= ich=2C in the general (non locally connected case) is a "hidden structure"= =2C I view the notion of a covering projection to be a structure in the fir= st place and=2C in the specific locally connected case=2C one that may be e= quivalently reduced to a property. Every property is a structure=2C but not= every structure is =A0reducible to a property.=A0 > All the best=2C =A0Marta = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]