From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6528 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: theorem about ends Date: Mon, 7 Feb 2011 15:35:26 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1297100542 1111 80.91.229.12 (7 Feb 2011 17:42:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 7 Feb 2011 17:42:22 +0000 (UTC) Cc: categories list To: Paul Levy Original-X-From: majordomo@mlist.mta.ca Mon Feb 07 18:42:16 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 1PmV6F-000541-A2 for gsmc-categories@m.gmane.org; Mon, 07 Feb 2011 18:42:07 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:32858) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PmV5Z-000223-IO; Mon, 07 Feb 2011 13:41:25 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PmV5Q-0004P3-7R for categories-list@mlist.mta.ca; Mon, 07 Feb 2011 13:41:16 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6528 Archived-At: Dear Paul, I do not know anywhere that it appears explicitly, but it can be pieced together quite quickly from results about weighted limits in Kelly's book. First, given any adjunction X -| Y : B --> A, any W : A --> Set, and any G : B --> C, we have {WY, G} = {W, GX} (**) in the sense that the one exists if the other does, and the canonical comparison is then an isomorphism. This follows since Lan_X(W) = WY (by (4.28) of Kelly) and {Lan_X(W), G} = {W, GX} (by (4.58) ibid). Since the end of a functor T: K^op x K --> E is by definition ((3.59) ibid) the limit of H weighted by the hom-functor H_K: K^op x K --> Set, we have, in the situation you describe, that End(P(-,F-)) = {H_C, P.(1 x F)} = {H_C.(1 x U), P} = {H_D.(F^op x 1), P} = {H_D, P.(U^op x 1)} = End(P(U-,-)) by applying (**) twice to the adjointnesses 1 x F -| 1 x U and U^op x 1 -| F^op x 1, and using the natural isomorphism H_C.(1 x U) = H_D.(F^op x 1) obtained from the adjointness F -| U. Richard On 7 February 2011 11:25, Paul Levy wrote: > Dear all, > > 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). > > > Paul > > > -- > Paul Blain Levy > School of Computer Science, University of Birmingham > +44 (0)121 414 4792 > http://www.cs.bham.ac.uk/~pbl > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]