From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4341 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: A question on adjoints Date: Wed, 19 Mar 2008 13:43:23 -0500 (EST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019882 12830 80.91.229.2 (29 Apr 2009 15:44:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:44:42 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Wed Mar 19 19:42:25 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Mar 2008 19:42:25 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jc6pa-0002ac-9e for categories-list@mta.ca; Wed, 19 Mar 2008 19:32:22 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 105 Original-Lines: 54 Xref: news.gmane.org gmane.science.mathematics.categories:4341 Archived-At: Actually, F isn't even a functor. The unique arrow 0 --> Ub has to give a canonical arrow F0 = 1 --> b, which there isn't. You could choose one, of course, but it could not be functorial. Actually, I realized the answer to my question cannot be yes. Here's why. Let A be some complete category to be specified later. Let d be a fixed object of A. Let B be set\op and Fa = Hom(a,d). The right adjoint is given by b |---> d^b. It is not entirely trivial to show this, but if my answer were "yes", then you could show that the class of objects that were equalizers of powers of d would be complete. It is obviously closed under products but, over 40 years ago, Isbell gave an example in which it was not closed under equalizers. This much is true: if there is an equalizer of the form a --> UFa ===> Ub, then a ---> UFa ===> UFUFa is an equalizer. Michael On Tue, 18 Mar 2008, Vaughan Pratt wrote: > 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 >> >> > >