From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6585 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: A conditon on maps between sheaves Date: Mon, 14 Mar 2011 14:27:20 +0000 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1300191151 25113 80.91.229.12 (15 Mar 2011 12:12:31 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 15 Mar 2011 12:12:31 +0000 (UTC) Cc: zoran skoda , categories list To: Andrej Bauer Original-X-From: majordomo@mlist.mta.ca Tue Mar 15 13:12:26 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PzT6w-0000rW-ER for gsmc-categories@m.gmane.org; Tue, 15 Mar 2011 13:12:26 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42188) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PzT6s-0001ZI-2e; Tue, 15 Mar 2011 09:12:22 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PzT6i-0000oV-C9 for categories-list@mlist.mta.ca; Tue, 15 Mar 2011 09:12:12 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6585 Archived-At: Dear Andrej, For example: take B to be the circle and E' its Moebius double cover, which has no global sections. Then for every x in B you can take U = B and your condition holds vacuously for any f whatsoever. If E = E'+E' then the codiagonal f has your property but is not mono. Regards, Steve. zoran skoda wrote: > Dear Andrej, > > I do not see that the condition as you stated it implies that the map is > mono. For example, one can take an example such that for every x the U with > above property is the whole base B, hence we have a mono on global sections > over B, but this solely is very weak and does not imply we have mono > locally, hence on stalks. Maybe you wanted that, in fact, for every nei W > around x there is open U around x which is within U ? > > Zoran > > On Sat, Mar 12, 2011 at 2:33 AM, Andrej Bauer wrote: > >> Dear categorists, >> >> I have come across a condition on maps between sheaves which I am >> unable to recognize as with my feeble knowledge of sheaf theory. I >> would appreciate any hints as to what this condition is about. >> >> Succinctly but imprecisely my condition can be expressed as: the >> inverse image of a sufficiently small section is again a section. >> >> More precisely, let p : E -> B be p' : E' -> B be two etale maps over >> a base space B and let f : E -> E' be a continuous map such that p = f >> p'. The mystery condition on f is as follows: for every x in B there >> is a neighborhood U of x, such that for every section s : U -> E' of >> p' there exists a unique section t : U -> E of p for which t(U) = >> f^(-1)(s(U)). >> >> It follows from this condition that f is mono as a morphism in Sh(B) >> because such an f is injective on each fiber. But I think the >> condition says more than that. Am I looking at a standard notion? >> >> With kind regards, >> >> Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]