From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4961 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Categorical problems Date: Tue, 9 Jun 2009 14:44:02 +0100 (BST) Message-ID: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; format=flowed; charset=US-ASCII X-Trace: ger.gmane.org 1244682200 8773 80.91.229.12 (11 Jun 2009 01:03:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 11 Jun 2009 01:03:20 +0000 (UTC) To: "Eduardo J. Dubuc" , categories@mta.ca Original-X-From: categories@mta.ca Thu Jun 11 03:03:17 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MEYhI-0000mi-QM for gsmc-categories@m.gmane.org; Thu, 11 Jun 2009 03:03:16 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MEY6C-0000Jb-E6 for categories-list@mta.ca; Wed, 10 Jun 2009 21:24:56 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4961 Archived-At: On Mon, 8 Jun 2009, Eduardo J. Dubuc wrote: > Ross Street wrote: > >> Problem. Suppose A is a locally small site whose category E of Set- >> valued sheaves is also locally small. Is E a topos? (see (*) below) > > This is one of (probably) many problems of Girau topoi [satisfy all > conditions > in Girau's Theorem exept (may be) the set of generators] which are not known > to be a topos. > > (*) It seems Not: Take a Girau (really faux but locally small) topos E, with > the canonical topology. Then the topos of sheaves should be E again, which is > not a topos (am I missing something ?). > I presume that Ross was using the word "topos" to mean "elementary topos". But in any case, Eduardo was missing something: the proof that, if E is an \infty-pretopos (my preferred name for what he calls a "Girau(d) topos"), then every canonical sheaf on E is representable, requires the existence of a generating set (see C2.2.7 in the Elephant). For a counterexample in the absence of generators, let G be the "large" group of all functions from the ordinals to {0,1} having finite support, the group operation being pointwise addition mod 2 (or, if you prefer, the group of finite subsets of the ordinals under symmetric difference), and let E be the (elementary) topos of G-sets. For each ordinal \alpha, let A_\alpha be the set {0,1} with G acting via its \alpha-th factor; then any G-set admits morphisms into only a set of the A_\alpha, from which it follows that the coproduct of all the A_\alpha exists as a (set-valued) canonical sheaf on E, though it clearly isn't a set. Moreover, this coproduct admits a proper class of maps to itself, so the category of sheaves on E isn't locally small; hence it doesn't violate Ross's conjecture. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]