From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6306 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Fri, 08 Oct 2010 18:53:31 -0300 Message-ID: References: ,<20101007010252.EA0FDCF26@mailscan2.ncs.mcgill.ca> ,<20101008004112.BE2B38572@mailscan2.ncs.mcgill.ca> 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 1286664274 14667 80.91.229.12 (9 Oct 2010 22:44:34 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 9 Oct 2010 22:44:34 +0000 (UTC) Cc: categories@mta.ca To: Marta Bunge Original-X-From: majordomo@mlist.mta.ca Sun Oct 10 00:44:33 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P4i9Y-0002Ii-BT for gsmc-categories@m.gmane.org; Sun, 10 Oct 2010 00:44:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44242) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P4i8o-0003G0-0L; Sat, 09 Oct 2010 19:43:46 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P4i8k-0008Vf-NX for categories-list@mlist.mta.ca; Sat, 09 Oct 2010 19:43:42 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6306 Archived-At: Marta Bunge wrote: > Dear Eduardo, Topological spaces or toposes, it is the same question. A > space is locally connected iff its topos of sheaaves is locally connected. Of course, 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, I had FORGOTTEN to say that in definition b) the V's are DISJOINT. ******** 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 ********** > In my view, the question of whether the notion of a covering space is a > structure or a property depends on the definition of covering space that > one adopts. If the definition is made for arbitrary spaces (as in Spanier, > whom you quote), 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 evenly > covered by p, then covering space is a structure, no matter what the nature > of the base space is. Well, for locally connected space B (or any locally connected topos as you pointed out), the forgetful functor into the topos of etale spaces over B is full and faithful, and for X over B, there is only one structure (up to isomorphism of structures). I wanted this to be considered under the analysis: *************** Michael Shulman wrote: > > property = forgetful functor is full and faithful > structure = forgetful functor is faithful > property-like structure = forgetful functor is pseudomonic *************** You see, with this criteria (property = forgetful functor is full and faithful) covering space is a property, something you do not think it is. I am not saying who is right, just putting in evidence that it is a matter not settled yet. May be full and faithfulness of the forgetful functor is not enough to call a covering space to be a property of a continuous map ? > It so happens that, in the case of a locally > connected space B, an alternative definition of a covering space can be > given (as in R. Brown, Topology and groupoids) that refers directly to > canonical neighborhoods of points of X (U open, connected, and each > connected component of the inverse image of U under p in X is mapped > homeomorphically onto U) and, with this definition, covering space is > indeed a property. So, in the locally connected case, the structure of > covering space can be equivalently replaced by a property - but I believe > that it is still a structure before those canonical choices are made. Can a > structure be equivalent to a property, yet not be a property?. Well, interesting question, but first we have to settle: What do we mean by structure ?, and, what do we mean by property ?. Finally, 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, in that case, there >>> is a canonical one. Since in this case all trivialization structures ARE isomorphic!. (if U and V are neighborhoods of b evenly covered, 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/ ]