From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5203 Path: news.gmane.org!not-for-mail From: Newsgroups: gmane.science.mathematics.categories Subject: Re: Conditions for adjoints Date: Sun, 25 Oct 2009 10:48:39 +0100 Message-ID: Reply-To: 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 1256484392 7473 80.91.229.12 (25 Oct 2009 15:26:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 25 Oct 2009 15:26:32 +0000 (UTC) To: 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 1N24z2-0003J5-1d 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 1N24Wa-0004Lt-N8 for categories-list@mta.ca; Sun, 25 Oct 2009 11:56:52 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5203 Archived-At: robin@ucalgary.ca a =C3=A9crit : > (Apologies to those who received the earlier type mashed version ...) >=20 > Jeremy Dawson and I were discusing whether one can express the conditio= ns > for an adjoint without requiring functors ... this is what we came up > with: >=20 >=20 > 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: >=20 > #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp > @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat >=20 > 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. >=20 > 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? >=20 > -robin Hello, Such conditions are discussed in detail in: Kosta Do=C5=A1en, Cut Elimination in Categories, Trends in Logic 6, Kluwe= r, 1999. Those you mention already appear on p. 258 of: Kosta Do=C5=A1en, Deductive Completeness, Bull. Symbolic Logic Volume 2, = Number 3 (1996), 243-283. (http://www.math.ucla.edu/~asl/bsl/0203/0203-001.ps). Regards, Laurent M=C3=A9hats [For admin and other information see: http://www.mta.ca/~cat-dist/ ]