From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6293 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: property_vs_structure Date: Tue, 05 Oct 2010 12:42:44 -0300 Message-ID: References: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1286367596 4322 80.91.229.12 (6 Oct 2010 12:19:56 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 6 Oct 2010 12:19:56 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Oct 06 14:19:55 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 1P3SyP-0006WZ-4m for gsmc-categories@m.gmane.org; Wed, 06 Oct 2010 14:19:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40538) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P3Sxd-0004og-0s; Wed, 06 Oct 2010 09:19:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P3SxX-0000HM-Ft for categories-list@mlist.mta.ca; Wed, 06 Oct 2010 09:18:59 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6293 Archived-At: Michael Shulman wrote: > > property = forgetful functor is full and faithful > structure = forgetful functor is faithful > property-like structure = forgetful functor is pseudomonic > On the thread "property" "structure" "property-like structure" and may be some other etceteras. I put on the table the following example to be analyzed: Let f: X --> B a continuous function of topological spaces: [assume surjective to simplify, and if b \in B, write X_b for the fiber X_b = f^-1(b)]. Then, we have the two familiar definitions a) and b): f is "fefesse" if given b \in B, then a) for each x \in X_b, there is U, b \in U, such that b) there is U, b \in U, such that for each x \in X_b, there is V, x \in V, and f|V : V --> U homeo. (the non commuting quantifiers again !) a) fefesse = local homeomorphism b) fefesse = covering map Well, both are "properties" of a continuous function, but they are not of the same kind. in b) is hidden a structure, namely a trivialization structure associated to an open cover of B. If B is locally connected, then "covering map" behaves like a perfectly pure property. 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, Algebraic Topology). have fun ! e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]