From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6619 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Constitutive Structures Date: Sat, 16 Apr 2011 10:53:33 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1302956435 30296 80.91.229.12 (16 Apr 2011 12:20:35 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 16 Apr 2011 12:20:35 +0000 (UTC) Cc: "Ellis D. Cooper" , categories To: "Prof. Peter Johnstone" Original-X-From: majordomo@mlist.mta.ca Sat Apr 16 14:20:31 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QB4UI-0005P8-DC for gsmc-categories@m.gmane.org; Sat, 16 Apr 2011 14:20:30 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60779) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QB4UE-0005Hp-0V; Sat, 16 Apr 2011 09:20:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QB4UB-0006hT-5g for categories-list@mlist.mta.ca; Sat, 16 Apr 2011 09:20:23 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6619 Archived-At: In fact, I think the condition of being an E-map in this new sense says: f : E --> F over Set[O] is such a map just when, on taking the hyperconnected-localic factorisations E --> E' --> Set[O] and F --> F' --> Set[O], the induced geometric morphism f' : E' --> F' is hyperconnected. In particular, if F --> Set[O] is localic, then f is an E-map if and only if it is hyperconnected. So this gets us no further than before. Oh well! Richard On 16 April 2011 10:31, Richard Garner wrote: > Thanks Peter. It did occur to me last night that this probably was the > hyperconnected-localic factorisation and it is nice to have this > feeling confirmed! The problem is that the factorisation system I > described allows one to adjoin n-ary relations to _arbitrary_ objects > of Set[O], rather than merely to the generic object. In particular, as > you point out, of the first group of maps I listed it is only > necessary to consider the case n=1, and in fact on looking at at your > proof, orthogonality to this immediately implies orthogonality to the > last of the maps I listed. > > Here's an attempt to overcome this; I suspect it will end up suffering > the same fate as the previous one but you never know! Rather than > describing a factorisation system on GTop, I am going to describe one > on GTop / Set[O]. The generating right maps will simply be the maps > from the classifying topos of an object equipped with an n-ary > relation into Set[O], though now these maps are viewed as maps over > Set[O]. If this generates a factorisation system (E, M), then its > M-maps with codomain E --> Set[O] will correspond to those things > constructible by repeatedly adjoining n-ary relations or equations > between n-ary relations to the specified object of E. Every such map > will be localic, but I think that the E-maps are no longer the > hyperconnected morphisms; the inverse image part of such a map need > only be full on subobjects of the specified object of its domain. > > Now on factorising the unique map from R: Set --> Set[O] into the > terminal object of GTop / Set[O], it is possible that we obtain > something non-trivial which captures the structures (in geometric > logic) supported by the reals. I am however a bit hesitant about this > as my feeling is that if p: E --> F is an E-map of toposes over > Set[O], and F --> Set[O] is localic, then p probably is actually > hyperconnected (i.e., fullness on subobjects of the (image of) the > generic object implies fullness on all subobjects) so that we are back > in the situation we were in before... > > Richard > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]