From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2437 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Logic preserved in double negation subtopos? Date: Tue, 2 Sep 2003 20:52:12 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018662 4162 80.91.229.2 (29 Apr 2009 15:24:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:22 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 2 20:45:17 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Sep 2003 20:45:17 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19uKnF-00009P-00 for categories-list@mta.ca; Tue, 02 Sep 2003 20:42:37 -0300 X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *19uHCH-0006G6-53*ymcE9joHdyg* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 56 Xref: news.gmane.org gmane.science.mathematics.categories:2437 Archived-At: The inclusion of double-negation sheaves is an example of what I called a sub-open map in my paper "Open maps of toposes" (Manuscripta Math. 31 (1980), 217-247). Sub-open maps have the property that their inverse image functors commute with implication -- indeed, one could take that as a definition, although it wasn't how I defined them in the paper. Peter Johnstone ---------- On Tue, 2 Sep 2003, Jonas Eliasson wrote: > While writing a joint paper with Steve Awodey, we came to think about the > following question: > > Given a Grothendieck topos Sh(C), what logic is preserved by the > associated sheaf functor from Sh(C) to the double negation subtopos of > Sh(C)? > > We know that a: Sh(C) --> DNSh(C) preserves geometric logic. Since it is > double negation it also preserves 0 (falsehood), negation and implication. > >From this you can draw the conclusion that a preserves the validity of > formulas built up from double negation stable predicates without universal > quantifiers. > > Presumably this has been studied in the literature, can something stronger > be said about what validities are preserved, could anyone provide a > reference for a general result of this kind? > > Grateful for any help, > Jonas Eliasson > > > > > ------------------------------------------ > | Jonas Eliasson | > | Department of Mathematics | > | Uppsala University | > | Sweden | > | E-mail: jonase@math.uu.se | > | Homepage: http://www.math.uu.se/~jonase/ | > ------------------------------------------ > > > > > > > >