From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6308 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: FW: property_vs_structure Date: Sat, 9 Oct 2010 10:12:13 -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 1286664347 14927 80.91.229.12 (9 Oct 2010 22:45:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 9 Oct 2010 22:45:47 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Sun Oct 10 00:45:45 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 1P4iAh-0002iC-I1 for gsmc-categories@m.gmane.org; Sun, 10 Oct 2010 00:45:43 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37683) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P4iA3-0006KN-HD; Sat, 09 Oct 2010 19:45:03 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P4i9z-00006C-VF for categories-list@mlist.mta.ca; Sat, 09 Oct 2010 19:45:00 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6308 Archived-At: Dear Eduardo=2C > You ask > >>What do we mean by structure ?=2C and=2C what do we mean by property ? > On a given data=2C a structure is additional data on it that=2C if it exist= s=2C it is not necessarily unique up to isomorphism. A property is additio= nal data such that=2C if it exists=2C it is unique up to isomorphism (in th= e model theoretic sense).=A0 > >You also write >>Finally=2C I still do not understand what do you mean (in your first mail= ) by:>> >>>>> Even in the locally connected case there are several non isomorphic >>>>> trivialization structures. The difference is that=2C in that case=2C = there >>>>> is a canonical one. >> >>> Since in this case all trivialization structures ARE isomorphic!. >> >>> (if U and V are neighborhoods of b evenly covered=2C then the structure= s are >>> isomorphic in a connected W contained in the intersection) >>> > Precisely=2C it comes down to what definition of covering space one adopts.= If the general definition is adopted then=2C even in the locally connected= case=2C it is a structure=2C as any two trivialization structures given by= U=2C V=2C need not be isomorphic except on a connected W contained in the = intersection. If=2C on the other hand=2C the specifi definition is adopted= =2C where canonical neighborhoods (U open=2C connected=2C and each connecte= d component of the inverse image of U under p in X is mapped homeomorphical= ly onto U) are given as part of the structure=2C then the structure is a pr= operty.=A0 >=A0 I think that I have nothing else to say on his matter. > Best=2CMarta > ---------------------------------------- >> Date: Fri=2C 8 Oct 2010 18:53:31 -0300 >> From: edubuc@dm.uba.ar >> To: marta.bunge@mcgill.ca >> CC: categories@mta.ca >> Subject: Re: property_vs_structure >> >> >> >> Marta Bunge wrote: >>> Dear Eduardo=2C Topological spaces or toposes=2C it is the same questio= n. A >>> space is locally connected iff its topos of sheaaves is locally connect= ed. >> >> Of course=2C it is only that I wanted to focus in topological spaces to = fix the >> ideas and so that the following two definitions can be compared. >> >> ALSO=2C I had FORGOTTEN to say that in definition b) the V's are DISJOIN= T. >> >> ******** >> Let f: X --> B a continuous function of topological spaces: >> [assume surjective to simplify=2C and if b \in B=2C write X_b for the fi= ber >> X_b =3D f-1(b)]. >> >> Then=2C we have the two familiar definitions a) and b): >> >> f is "fefesse" if given b \in B=2C then >> >> a) for each x \in X_b=2C there is U=2C b \in U=2C such that >> >> b) there is U=2C b \in U=2C such that for each x \in X_b=2C >> >> there is V=2C x \in V=2C and f|V : V --> U homeo. >> >> (the non commuting quantifiers again !) >> >> a) fefesse =3D local homeomorphism >> >> b) fefesse =3D covering map >> ********** >> >>> In my view=2C the question of whether the notion of a covering space is= a >>> structure or a property depends on the definition of covering space tha= t >>> one adopts. If the definition is made for arbitrary spaces (as in Spani= er=2C >>> whom you quote)=2C where a continuous map p from X to B is said to be a >>> covering projection if each point of X has an open neighborhood U evenl= y >>> covered by p=2C then covering space is a structure=2C no matter what th= e nature >>> of the base space is. >> >> Well=2C for locally connected space B (or any locally connected topos as= you >> pointed out)=2C the forgetful functor into the topos of etale spaces ove= r B is >> full and faithful=2C and for X over B=2C there is only one structure (up= to >> isomorphism of structures). >> >> I wanted this to be considered under the analysis: >> >> *************** >> 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 >> *************** >> >> You see=2C with this criteria (property =3D forgetful functor is full an= d >> faithful) covering space is a property=2C something you do not think it = is. I am >> not saying who is right=2C just putting in evidence that it is a matter = not >> settled yet. May be full and faithfulness of the forgetful functor is no= t >> enough to call a covering space to be a property of a continuous map ? >> >>> It so happens that=2C in the case of a locally >>> connected space B=2C an alternative definition of a covering space can = be >>> given (as in R. Brown=2C Topology and groupoids) that refers directly t= o >>> canonical neighborhoods 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. So=2C in the locally connected case=2C the structure= of >>> covering space can be equivalently replaced by a property - but I belie= ve >>> that it is still a structure before those canonical choices are made. C= an a >>> structure be equivalent to a property=2C yet not be a property?. >> >> Well=2C interesting question=2C but first we have to settle: >> >> What do we mean by structure ?=2C and=2C what do we mean by property ?. >> >> Finally=2C I still do not understand what do you mean (in your first mai= l) by: >> >>>>> Even in the locally connected case there are several non isomorphic >>>>> trivialization structures. The difference is that=2C in that case=2C = there >>>>> is a canonical one. >> >> Since in this case all trivialization structures ARE isomorphic!. >> >> (if U and V are neighborhoods of b evenly covered=2C then the structures= are >> isomorphic in a connected W contained in the intersection) >> >> best e.d. > = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]