From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9284 Path: news.gmane.org!.POSTED!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: Do there exist nontrivial locally bounded geometric morphisms and/or locally (pre)sheaf toposes? Date: Wed, 2 Aug 2017 17:20:00 +0100 (BST) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: blaine.gmane.org 1501772168 12290 195.159.176.226 (3 Aug 2017 14:56:08 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 3 Aug 2017 14:56:08 +0000 (UTC) Cc: categories list To: Mamuka Jibladze Original-X-From: majordomo@mlist.mta.ca Thu Aug 03 16:56:03 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1ddHXV-0002Pb-OU for gsmc-categories@m.gmane.org; Thu, 03 Aug 2017 16:55:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40761) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ddHWf-0001lT-Re; Thu, 03 Aug 2017 11:55:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ddHWV-0001vI-UC for categories-list@mlist.mta.ca; Thu, 03 Aug 2017 11:54:51 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9284 Archived-At: The answer to Mamuka's question (1) is no. Observe first that YY/f^*(X) is bounded over XX/X iff it's bounded over XX, since XX/X --> XX is bounded. And one has: Proposition: If Y has global support in YY and G is a bound for YY/Y over XX, then \Sigma_Y(G) is a bound for YY over XX. Proof: By assumption, any object B of YY/Y is a subquotient of some object G \times Y^*f^*(I), with I an object of XX. But the Frobenius reciprocity condition \Sigma_Y(G\times Y^*f^*(I)) \cong \Sigma_Y(G) \times f^*(I) holds, so \Sigma_Y(B) is a subquotient of \Sigma_Y(G)\times f^*(I). Finally, since Y has global support, any object A of XX is a quotient of \Sigma_Y(Y^*(A)) \cong A \times Y. I have no thoughts at present about question (2). Peter Johnstone On Mon, 31 Jul 2017, Mamuka Jibladze wrote: > Recently I posted this question > > https://mathoverflow.net/q/277582/41291 > > to mathoverflow and now it occurred to me that most likely I can get a > quick answer here. > > Are there geometric morphisms f: YY -> XX which are > > (1) locally but not globally bounded, or > (2) locally but not globally presheaf, or > (3) as in (2) and bounded? > > In more detail, I mean this: there must be an object X in XX with > global support (X->1 epic) such that the pullback f/X: YY/f^*(X) -> XX/X > is > > (1) bounded, while f is not bounded, or > (2) equivalent over XX/X to the topos (XX/X)^{CC^op} of internal > presheaves on some internal category CC of XX/X, while YY is not > equivalent to any such over XX, or > (3) same as (2) and in addition f bounded. > > Can any of these happen? > > Hoping, > Mamuka > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]