From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8377 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: Re: NNOs in different toposes "the same"? Date: Tue, 11 Nov 2014 10:01:21 -0300 Message-ID: References: Reply-To: Claudio Hermida NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1415737222 31816 80.91.229.3 (11 Nov 2014 20:20:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 11 Nov 2014 20:20:22 +0000 (UTC) To: David Roberts , "categories@mta.ca list" Original-X-From: majordomo@mlist.mta.ca Tue Nov 11 21:20:16 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 1XoHvD-0007Xe-Ml for gsmc-categories@m.gmane.org; Tue, 11 Nov 2014 21:20:15 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:56693) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XoHuk-0004Ml-JA; Tue, 11 Nov 2014 16:19:46 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XoHuk-0002tp-Tt for categories-list@mlist.mta.ca; Tue, 11 Nov 2014 16:19:46 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8377 Archived-At: 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/ ]