From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6826 Path: news.gmane.org!not-for-mail From: Steven Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: exponentials of perfect maps and local homeomorphisms Date: Tue, 16 Aug 2011 08:54:46 +0100 Message-ID: References: Reply-To: Steven Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1313537779 16596 80.91.229.12 (16 Aug 2011 23:36:19 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 16 Aug 2011 23:36:19 +0000 (UTC) Cc: To: claudio pisani Original-X-From: majordomo@mlist.mta.ca Wed Aug 17 01:36:15 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QtTB9-0005HB-F5 for gsmc-categories@m.gmane.org; Wed, 17 Aug 2011 01:36:15 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53584) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QtT9J-0001Ri-Uo; Tue, 16 Aug 2011 20:34:21 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QtT9J-0001yB-94 for categories-list@mlist.mta.ca; Tue, 16 Aug 2011 20:34:21 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6826 Archived-At: Dear Claudio, For locales, the questions about bundles can be reduced to results about spaces (=3D locales) internal in the topos of sheaves over the base space= . By "bundle" I just mean "map" (always assumed continuous), but it is useful to have the alternative word because we take an alternative perspective in which the bundle is thought of as a space (the fibre) continuously parametrized by a base point. Then there are correspondences between properties of the bundle as map and its properties "fibrewise", the latter corresponding to properties of the internal space. local homeomorphism <-> fibrewise discrete perfect <-> fibrewise compact regular (or completely regular? They become inequivalent in a topos.) Now the localic version of your questions can be reduced to the topos-validity of questions about exponentiation of locales. If X compact regular and Y discrete, (1) Is Y^X discrete? (2) Is X^Y compact regular? I'm reasonably sure (2) is known, but I can't think of the references offhand. The hard part is Tychonoff, which is topos-valid localically. As for (1), I'm inclined to believe it but can't think of reasons or references at the moment. The "localic bundle theorem" that relates bundles over a topos to locales in it is fundamental and goes back - I believe - to Fourman, Scott, Joyal= , Tierney. It is a result that depends on using point-free rather than point-set topology. All the same, I guess there's a chance of exploiting it to prove results about classical point-set bundles. "Topos-valid locale" sounds scary - Do we have to use lattices instead of spaces? Do we have to know all about toposes? A lot of my work has been about the beneficial effect of using the "geometric" fragment of topos-valid logic, using which one can reason with the points of a point-free space. All the best, Steve Vickers. On Mon, 15 Aug 2011 20:03:22 +0100 (BST), claudio pisani wrote: > Dear Steve, by "perfect" I mean proper and separated (proper =3D closed with compact fibers > =3D stably closed). I prefer the topological setting because I am not v= ery acquainted with locales, > but any result which seems to confirm the conjecture is good to me. > So you are saying that (apart from the separation condition) the second question has an > affirmative answer in Loc. Any idea about the first one? > > Regards, Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]