From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8077 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Poset adjoints Date: Thu, 17 Apr 2014 01:26:28 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1397831480 10373 80.91.229.3 (18 Apr 2014 14:31:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 18 Apr 2014 14:31:20 +0000 (UTC) Cc: "categories" To: "Vasili I. Galchin" Original-X-From: majordomo@mlist.mta.ca Fri Apr 18 16:31:15 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Wb9ov-00048I-4l for gsmc-categories@m.gmane.org; Fri, 18 Apr 2014 16:31:13 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35121) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Wb9o1-0007JD-21; Fri, 18 Apr 2014 11:30:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Wb9nz-0000eg-CH for categories-list@mlist.mta.ca; Fri, 18 Apr 2014 11:30:15 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8077 Archived-At: Properly posed, Vasili's question should be not > Given that poset adjoints are considered miniature versions of > adjoints, what are are the unit and co-unit natural transformations > say between poset A and B? but: = : Given order-preserving functions r: A --> B and l: B --> A between : posets A and B, with r right adjoint to l, what are the unit and = : co-unit natural transformations 1_B =3D=3D> rl and lr =3D=3D> 1_A ? And the answer, as always, is that 1_B: b --> rlb (for b in B) is = the order relation b < rlb that adjointness correlates to lb < lb, = while 1_A: lra --> a (for a in A) is the order relation lra < a = that adjointness correlates to ra < ra. (Above I'm writing simply < for the (reflexive, transitive) order relatio= n on either of the two posets.) HTH. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]