From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5202 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: Conditions for adjoints -- another variant Date: Sun, 25 Oct 2009 10:05:48 +0000 Message-ID: Reply-To: Paul Levy NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1256484387 7461 80.91.229.12 (25 Oct 2009 15:26:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 25 Oct 2009 15:26:27 +0000 (UTC) To: robin@ucalgary.ca, categories@mta.ca Original-X-From: categories@mta.ca Sun Oct 25 16:26:20 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 1N24z2-0003J6-1d for gsmc-categories@m.gmane.org; Sun, 25 Oct 2009 16:26:16 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N24XR-0004P4-M6 for categories-list@mta.ca; Sun, 25 Oct 2009 11:57:45 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5202 Archived-At: Hi Robin, In my thesis http://www.cs.bham.ac.uk/~pbl/papers/thesisqmwphd.pdf Def. 109 - 110, pages 220-222 I listed six (equivalent) definitions of adjunction, one of which (Def. 110(4)) resembles yours, and two of which (Def. 110(4)-(5)) don't mention any functors.=20 (Some of these definitions - though not the one that resembles yours - us= e the notion of "representing object", which itself can be defined in eithe= r element style or naturality style.) The list also appears in my "Call-by-push-value" book, Def. 9.33 (page 23= 5) and Def. 11.17 (pages 278-280). Also see the discussion in Sect. 1.2 of my TAC paper "Adjunction models f= or call-by-push-value with stacks" http://www.tac.mta.ca/tac/volumes/14/5/14-05abs.html regards, Paul On Sat, 24 Oct 2009 17:11:26 -0600 (MDT), robin@ucalgary.ca wrote: > BTW. Here are some even cleaner conditions .... >=20 > There is an adjoint between two categories iff > there are two object functions F and G (not required to be functors) an= d > For each X \in \X and Y \in \Y there are two functions: >=20 > #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp > @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat >=20 > (i)' @ and # are inverse @(#(f)) =3D f and #(@(g)) =3D g > (ii)' @(h k) =3D @(h) @(#(1) k) and dually #(xy) =3D #(x @(1)) #(y) >=20 > Still hoping to find where these all are recorded!! >=20 > -robin >=20 >=20 >=20 >=20 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]