From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6529 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 16:04:52 +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 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1297100579 1362 80.91.229.12 (7 Feb 2011 17:42:59 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 7 Feb 2011 17:42:59 +0000 (UTC) Cc: categories list To: Paul Levy Original-X-From: majordomo@mlist.mta.ca Mon Feb 07 18:42:54 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 1PmV6y-0005Y9-NT for gsmc-categories@m.gmane.org; Mon, 07 Feb 2011 18:42:52 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:55142) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PmV6L-00029e-CK; Mon, 07 Feb 2011 13:42:13 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PmV6E-0004QF-KG for categories-list@mlist.mta.ca; Mon, 07 Feb 2011 13:42:06 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6529 Archived-At: Actually, since the codomain of your functor P is actually Set, the argument I gave---which would be valid for any suitably complete codomain---can be rewritten to avoid the use of weighted limits entirely, by expressing those necessary in this case as hom-sets in a functor category. It then becomes the single calculation that End(P(-,F-)) =3D [C^op x C, Set](Hom_C, P.(1 x F)) =3D [C^op x D, Set](Hom_C.(1 x U), P) =3D [C^op x D, Set](Hom_D.(F^op x 1), P) =3D [C^op x D, Set](Hom_D, P.(U^op x 1)) =3D End(P(U-,-)) which requires no machinery beyond that of transpositions under adjunction. Richard On 7 February 2011 15:35, Richard Garner wrote: > 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} =3D {W, GX} =A0(**) > > 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) =3D WY > (by (4.28) of Kelly) and {Lan_X(W), G} =3D {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-)) =3D {H_C, P.(1 x F)} =3D {H_C.(1 x U), P} =3D {H_D.(F^op x 1= ), > P} =3D {H_D, P.(U^op x 1)} =3D End(P(U-,-)) > > by applying (**) twice to the adjointnesses 1 x F -| 1 x U =A0and =A0U^op > x 1 -| F^op x 1, and using the natural isomorphism H_C.(1 x U) =3D > H_D.(F^op x 1) obtained from the adjointness F -| U. > > Richard > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]