From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6104 Path: news.gmane.org!not-for-mail From: David Leduc Newsgroups: gmane.science.mathematics.categories Subject: Re: Equality as an adjunction Date: Sat, 4 Sep 2010 07:21:41 +0000 Message-ID: References: Reply-To: David Leduc 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: dough.gmane.org 1283610437 24120 80.91.229.12 (4 Sep 2010 14:27:17 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Sep 2010 14:27:17 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Sep 04 16:27:14 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Orti5-0001Wd-Rs for gsmc-categories@m.gmane.org; Sat, 04 Sep 2010 16:27:14 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:57554) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Orth2-0005Ri-Bv; Sat, 04 Sep 2010 11:26:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ortgz-0000Ah-VU for categories-list@mlist.mta.ca; Sat, 04 Sep 2010 11:26:06 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6104 Archived-At: Sorry, I guess my mistake is that I've got the adjunction in the wrong direction: it is a left Kan extension: [CxC,2](Lan_Delta P, Q) =3D~ [C,2](P, Q o Delta) And equality is given by Lan_Delta T : CxC -> 2. Does it always exist? And what was the right Kan extension in my previous mail when it exists? I wrote: > Thank you all for your replies. If I understood you well, the > adjunction is as follows: > > =A0[C,2](P o Delta, Q) =3D~ [CxC,2](P, Ran_Delta Q) > > where Ran_Delta Q is the right Kan extension if Q along Delta. > (I write [C,D] for the functor category, and 2 for the category with > only two objects True and False and identities). If T is the constant > functor that sends every object to True, then Ran_Delta T : CxC -> 2 > is the equality predicate, right? > > Now I write this adjunction in logical form: > > =A0 =A0P(x,x) |- Q(x) > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > P(x,y) |- Q(x) /\ x=3Dy > > But in Pitts it was reversed: > > =A0 Q(x) |- P(x,x) > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > Q(x) /\ x=3Dy |- P(x,y) > > Where is my mistake? > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]