From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8379 Path: news.gmane.org!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: NNOs in different toposes "the same"? Date: Wed, 12 Nov 2014 11:37:13 +0000 (GMT) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1415817354 7981 80.91.229.3 (12 Nov 2014 18:35:54 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 12 Nov 2014 18:35:54 +0000 (UTC) Cc: "categories@mta.ca list" To: David Roberts Original-X-From: majordomo@mlist.mta.ca Wed Nov 12 19:35:48 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 1Xoclg-0003oL-80 for gsmc-categories@m.gmane.org; Wed, 12 Nov 2014 19:35:48 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58012) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Xocl0-0007JJ-NX; Wed, 12 Nov 2014 14:35:06 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Xocl0-00060A-Qg for categories-list@mlist.mta.ca; Wed, 12 Nov 2014 14:35:06 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8379 Archived-At: As Claudio has said, inverse image functors always preserve the NNO, but David's question was: when does the direct image f_* preserve the NNO? A sufficient condition for this which is weaker than being local is that f should be connected, i.e. that f^* should be full and faithful, since then the unit map N -> f_*f^*N is an isomorphism. But this is certainly not necessary: for example, the inclusion from sheaves to presheaves on any locally connected internal site in a topos preserves N (see C3.3.10 in the Elephant). I don't know any necessary and sufficient condition (other than the condition that f_* preserves N!); if you restrict to countably cocomplete toposes (where N is always a countable copower of 1), then f_* preserves N iff it preserves all countable coproducts, iff f^* is full and faithful on morphisms with codomain N. But it's not clear to me what significance this latter condition has. Peter Johnstone On Mon, 10 Nov 2014, 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]