From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8378 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: NNOs in different toposes "the same"? Date: Wed, 12 Nov 2014 08:44:20 +1030 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1415817236 5866 80.91.229.3 (12 Nov 2014 18:33:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 12 Nov 2014 18:33:56 +0000 (UTC) Cc: "categories@mta.ca list" To: Claudio Hermida Original-X-From: majordomo@mlist.mta.ca Wed Nov 12 19:33:50 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Xocjl-0002dE-GB for gsmc-categories@m.gmane.org; Wed, 12 Nov 2014 19:33:49 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58005) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Xocis-0007BD-6G; Wed, 12 Nov 2014 14:32:54 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Xocis-0005uo-5d for categories-list@mlist.mta.ca; Wed, 12 Nov 2014 14:32:54 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8378 Archived-At: Hi Claudio yes, I know this. One can prove it from Freyd's characterisation of N by finite colimits, as I hinted in my first message. What I'm after is the other situation, when the direct image functor preserves the NNO. I believe the geometric morphism being local is enough, but much more than needed, so I was after weaker/alternative conditions. David On 11 November 2014 23:31, Claudio Hermida wrote: > > On 2014-11-09, 10:42 PM, David Roberts wrote: >> >> Hi all, >> >> If have a geometric morphism f: E -> F, what's the/a sensible way to >> say that the natural number objects of E and F are 'the same'? If f is >> local, then f_* preserves colimits, and so both f^* and f_* respect >> natural numbers objects up to iso. But this is a little too strong, >> perhaps, since we only need f_* to respect finite limits to use the >> characterisation of |N by Freyd to show preservation. What other >> conditions could I impose, other than simply that f_* preserves the >> NNO? >> >> Secondly, what if E is the externalisation of an internal topos in F? >> For instance, F = Set and E the externalisation of a small topos, not >> necessarily an internal universe (in fact I don't want this to be the >> case!). Then if I can say what it means for the NNO in E to be 'the >> same as' that in F, I can say that the internal topos has the same NNO >> as the ambient category. >> >> Regards, >> >> David >> >> > Dear David, > > The short answer is: the inverse image functor f*:F -> E preserves NNO. > > The argument goes as follows: an NNO is an initial (1+)-algebra, that > is, initial for the endofunctor [X] -> [1+X]. Since f* preserves 1 and > +, it induces an isomorphism f*(1+) = (1+_)f*, which in turn induces a > functor > > (f*)-alg: (1 +)- alg -> (1+)-alg > > Fact (++): The right adjoint f_* induces a right adjoint to f*-alg > > Hence (f*)-alg preserves initial objects, aka, NNO. > > The Fact (++) can be easily calculated, but a more formal argument is > given as Thm A.5 in > > Hermida, Claudio, and Bart Jacobs. "Structural induction and coinduction > in a fibrational setting." Information and Computation 145.2 (1998): > 107-152. > > Regards, > > Claudio > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]