From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5200 Path: news.gmane.org!not-for-mail From: robin@ucalgary.ca Newsgroups: gmane.science.mathematics.categories Subject: Conditions for adjoints Date: Sat, 24 Oct 2009 13:51:38 -0600 (MDT) Message-ID: Reply-To: robin@ucalgary.ca NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1256432165 27374 80.91.229.12 (25 Oct 2009 00:56:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 25 Oct 2009 00:56:05 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sun Oct 25 02:55:58 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 1N1rOn-0007qR-DE for gsmc-categories@m.gmane.org; Sun, 25 Oct 2009 02:55:57 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N1qqQ-0002Fh-VC for categories-list@mta.ca; Sat, 24 Oct 2009 21:20:27 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5200 Archived-At: (Apologies to those who received the earlier type mashed version ...) Jeremy Dawson and I were discusing whether one can express the conditions for an adjoint without requiring functors ... this is what we came up with: There is an adjoint between two categories if and only if there are object functions F and G (not functors) and for each X in \X and Y in \Y there are functions: #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat between the homsets such that (1) @(#(1)) =3D 1 and dually #(@(1)) =3D 1 (inverse on identities) (2) @(1) @(#(1) #(f)) =3D f and dually #(@(g) @(1)) #(1) =3D g (3) @(#(f @(1)) h k) =3D f @(h) @(#(1) k) and dually #(x y @(#(1) z)) =3D #(x @(1)) #(y) z. I find it hard to believe that such conditions have not been recorded.=20 Does anyone have a reference or similar conditions which do not require functors? -robin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]