From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5204 Path: news.gmane.org!not-for-mail From: Dusko Pavlovic Newsgroups: gmane.science.mathematics.categories Subject: Re: Conditions for adjoints -- another variant Date: Sun, 25 Oct 2009 01:03:36 +0000 (GMT) Message-ID: Reply-To: Dusko Pavlovic NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1256484576 7866 80.91.229.12 (25 Oct 2009 15:29:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 25 Oct 2009 15:29:36 +0000 (UTC) To: robin@ucalgary.ca, categories@mta.ca Original-X-From: categories@mta.ca Sun Oct 25 16:29:29 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 1N2528-0004I8-Cs for gsmc-categories@m.gmane.org; Sun, 25 Oct 2009 16:29:28 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N24Vg-0004IG-Nh for categories-list@mta.ca; Sun, 25 Oct 2009 11:55:56 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5204 Archived-At: cca 1991, i mentioned to mike barr that the functoriality of adjoints was derivable, and he knew it already. i think he said that this was used by isbell. (it's embarassing that i remember things from so long ago; and also that i don't remember them clearly enough to be confident. but does it really matter? when we optimize, we often find it more efficient not to store some data, but to regenerate when needed...) -- dusko On Sat, 24 Oct 2009, robin@ucalgary.ca wrote: > BTW. Here are some even cleaner conditions .... > > There is an adjoint between two categories iff > there are two object functions F and G (not required to be functors) and > For each X \in \X and Y \in \Y there are two functions: > > #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp > @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat > > (i)' @ and # are inverse @(#(f)) = f and #(@(g)) = g > (ii)' @(h k) = @(h) @(#(1) k) and dually #(xy) = #(x @(1)) #(y) > > Still hoping to find where these all are recorded!! > > -robin > > > > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]