From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6530 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: theorem about ends Date: Mon, 7 Feb 2011 22:26:50 +1100 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1297102149 10651 80.91.229.12 (7 Feb 2011 18:09:09 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 7 Feb 2011 18:09:09 +0000 (UTC) Cc: categories list To: Paul Levy Original-X-From: majordomo@mlist.mta.ca Mon Feb 07 19:09:04 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PmVWK-0003qv-Id for gsmc-categories@m.gmane.org; Mon, 07 Feb 2011 19:09:04 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46047) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PmVVc-00039m-HF; Mon, 07 Feb 2011 14:08:20 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PmVVR-0004YK-Rt for categories-list@mlist.mta.ca; Mon, 07 Feb 2011 14:08:10 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6530 Archived-At: Dear Paul On 07/02/2011, at 11:25 AM, Paul Levy wrote: > Does the following result (which I learnt from Rasmus Mogelberg) > appear in the literature somewhere? > > Given categories C and D, a functor P : C^op x D --> Set and an > adjunction F -| U : D --> C > > the end over c in C of P(c,Fc) is (isomorphic to) the end over d in D > of P(Ud,d). This is the sort of thing that would be used in the course of things without explicit enunciation as a Lemma or Proposition. It involves two applications of the end version of Yoneda (take V = Set for the ordinary case): end_c P(c,Fc) =~ end_{c,d} V(D(Fc,d),P(c,d)) =~ end_{c,d} V(C(c,Ud),P(c,d)) =~ end_d P(Ud,d). It is quite pretty, I agree. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]