From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4955 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: Categorical problems Date: Mon, 08 Jun 2009 15:30:05 -0300 Message-ID: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: lo.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 1244547625 1658 80.91.229.12 (9 Jun 2009 11:40:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 9 Jun 2009 11:40:25 +0000 (UTC) To: Ross Street , Original-X-From: categories@mta.ca Tue Jun 09 13:40:18 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 1MDzgf-00009y-Ee for gsmc-categories@m.gmane.org; Tue, 09 Jun 2009 13:40:17 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MDyvk-0000hV-Gn for categories-list@mta.ca; Tue, 09 Jun 2009 07:51:48 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4955 Archived-At: 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. Another, the Etale "topos" in the sense of Joyal's axiomatic theory of etal maps (which is even a subcategory of a topos). Another (solved), to show the existence of colimits in the category of topoi, the only hard part is to get the generators. Concerning the other thread (not Ross question) > > My question is, What would be candidates for the Fundamental Theorem > > of Category Theory? > > > > Yoneda Lemma comes to my mind. What do you think? Of course, Yoneda Lemma, at the birth of category theory, is the fundamental result that makes of category theory something more than a convenient language. Related to this, the definition of category should include small hom sets, and categories with large hom sets should be called "illegitimate" (in the manner of the definition of topoi, which include generators, the others being illegitimate or "faux" in Grothendieck's terminology). (*) 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 ?). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]