From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6102 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 04:03:02 +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 X-Trace: dough.gmane.org 1283608444 17300 80.91.229.12 (4 Sep 2010 13:54:04 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Sep 2010 13:54:04 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Sep 04 15:54:02 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OrtBx-0003nY-JW for gsmc-categories@m.gmane.org; Sat, 04 Sep 2010 15:54:01 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37849) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OrtAt-0002hc-RH; Sat, 04 Sep 2010 10:52:55 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OrtAr-0008CX-5B for categories-list@mlist.mta.ca; Sat, 04 Sep 2010 10:52:53 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6102 Archived-At: Thank you all for your replies. If I understood you well, the adjunction is as follows: [C,2](P o Delta, Q) =~ [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: P(x,x) |- Q(x) ============== P(x,y) |- Q(x) /\ x=y But in Pitts it was reversed: Q(x) |- P(x,x) ============== Q(x) /\ x=y |- P(x,y) Where is my mistake? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]