From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4933 Path: news.gmane.org!not-for-mail From: tholen@mathstat.yorku.ca Newsgroups: gmane.science.mathematics.categories Subject: Re: Famous unsolved problems in ordinary category theory Date: Thu, 4 Jun 2009 21:53:10 -0400 Message-ID: Reply-To: tholen@mathstat.yorku.ca 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 1244224426 13609 80.91.229.12 (5 Jun 2009 17:53:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 5 Jun 2009 17:53:46 +0000 (UTC) To: Michael Shulman , categories@mta.ca, Original-X-From: categories@mta.ca Fri Jun 05 19:53:41 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 1MCdbn-0002WO-Cy for gsmc-categories@m.gmane.org; Fri, 05 Jun 2009 19:53:39 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MCcqB-0007Xq-T1 for categories-list@mta.ca; Fri, 05 Jun 2009 14:04:27 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4933 Archived-At: Finding the "right" questions and notions is certainly a prominent theme in category theory, perhaps more prominently than in other fields. Still, just like in other fields, solving open problems was always part of the agenda. For example, half a century ago people asked whether every "standard construction" (=monad) is induced by an adjunction, and it took a few years to have two interesting answers. And there is ceratinly a string of examples leading all the way to today. I don't know whether there are any >famous< unsolved problems in ordinary category theory, but there are certainly non-trivial questions. Here is one that we formulated in an article with Reinhard B"orger (Can. J. Math 42 (1990) 213-229) two decades ago: A category A is total (Street-Walters) if its Yoneda embedding A ---> Set^{A^{op}} has a left adjoint. Then 1. A has small colimits, and 2. any functor A-->B that preserves all existing colimits of A has a right adjoint. Do properties 1 and 2 imply totality for A? I must admit that, after formulating the question we never considered it again, so there may well be a known or quick answer. So don't hold back please, especially since I plan to incorporate several questions of this type in my CT09 talk. Walter. Quoting Michael Shulman : > Probably people are going to jump on me for saying this, but it seems to > me that category theory is different from much of mathematics in that > often the difficulty is in the definitions rather than the theorems, and > in the questions rather than the answers. Thus, there are probably many > unsolved problems in category theory, but we don't know what they are > yet, because figuring out what they are is the main aspect of them > that is unsolved. (-: > > Mike > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]