From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4338 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: A question on adjoints Date: Tue, 18 Mar 2008 22:43:35 -0700 Message-ID: NNTP-Posting-Host: main.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 1241019880 12814 80.91.229.2 (29 Apr 2009 15:44:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:44:40 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Wed Mar 19 15:18:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 15:18:54 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc2mT-00015Y-P2 for categories-list@mta.ca; Wed, 19 Mar 2008 15:12:53 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 102 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:4338 Archived-At: Isn't the following a counterexample? Let A = Set and let B = A\{0} (the category of nonempty sets). Let F send the empty set in A to the singleton set in B, and otherwise let F and U be the evident identity functors between A and B. Similarly let \eta and \epsilon be the identity natural transformations, except for \eta_0 which can only be the unique function from 0 to 1. Naturality of \eta and \epsilon depends on both being the identity, except for \eta_0 but that's from the initial object so all its diagrams commute. Then 0 equalizes the two arrows from U1 to U2 but \eta_0 does not equalize UF\eta a and \eta UFa since the latter two are both 1_1 in A whence they are equalized by 1. Vaughan Michael Barr wrote: > I guess I am getting old and dumb. This question should have been a snap > for me years ago. It is old fashioned, only a 1-categorical question and > not about internal vs. external. > > Suppose F: A --> B is left adjoint to U: B --> A. Suppose a is an object > of A and b, b' objects of B such that there is an equalizer > a ---> Ub ===> Ub'. (The two arrows Ub to UB' are not assumed to be U > of arrows from B.) Does it follow that a ---> UFa ===> UFUFa is an > equalizer? The arrows are \eta a, UF\eta a and \eta UFa of course. > > Michael > >