From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4947 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Categorical problems Date: Sun, 7 Jun 2009 11:13:58 +1000 Message-ID: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: multipart/alternative; boundary=Apple-Mail-1--257933298 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1244415399 14982 80.91.229.12 (7 Jun 2009 22:56:39 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 7 Jun 2009 22:56:39 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Mon Jun 08 00:56:36 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 1MDRI4-0007vg-8i for gsmc-categories@m.gmane.org; Mon, 08 Jun 2009 00:56:36 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MDQbD-0004ps-IO for categories-list@mta.ca; Sun, 07 Jun 2009 19:12:19 -0300 Original-Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4947 Archived-At: Walter Tholen's recent message reminded me of a conjecture. Perhaps we have been too shy about stating our conjectures because, even among mathematicians, they may have seemed too technical. I seem to remember Peter Freyd saying once that the problem in category theory of proving sets were small (to find adjoints to functors for example) was analogous to finding numerical bounds in mathematical analysis. Surely by now, there are as many people who understand what a sheaf is as understand what the Riemann Hypothesis asserts (for example, local to global versus analytic continuation). So here is a problem I came up with in the 1970s. As with Fermat's Last Theorem, I don't particularly remember having any application for it. However, similar solved problems were used by Rosebrugh-Wood to characterize the category of sets in terms of adjoint strings involving the Yoneda embedding. By locally small I mean having homs in a chosen category Set of small sets. Problem. Suppose A is a locally small site whose category E of Set- valued sheaves is also locally small. Is E a topos? == Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]