From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7410 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Yoneda Lemma when there is a monad Date: Wed, 22 Aug 2012 07:26:41 +1000 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1345646256 10358 80.91.229.3 (22 Aug 2012 14:37:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 22 Aug 2012 14:37:36 +0000 (UTC) Cc: categories@mta.ca To: info@christophertownsend.org Original-X-From: majordomo@mlist.mta.ca Wed Aug 22 16:37:33 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1T4C3o-0002y9-GM for gsmc-categories@m.gmane.org; Wed, 22 Aug 2012 16:37:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52375) by smtpy.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1T4C38-0006oQ-Oz; Wed, 22 Aug 2012 11:36:50 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1T4C3B-0007am-NQ for categories-list@mlist.mta.ca; Wed, 22 Aug 2012 11:36:53 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7410 Archived-At: Dear Christopher On 21/08/2012, at 7:34 PM, info@christophertownsend.org wrote: > The usual Yoneda lemma is recovered by taking the trivial monad. The > Lemma gives a generalised Yoneda embedding: C^T embeds in > [(C_T)^op,Set]; i.e. any category of algebras embeds in the presheaf > category of the Kleisli category. I wasn't aware of this quite trivial > result and was hoping for some guidance as to where it is covered > already in the literature? Indeed, the category C^T of Eilenberg-Moore algebras is the pullback of the Yoneda embedding y : C --> [C^op,Set] and the functor [K^op,1] : [(C_T)^op,Set] --> [C^op,Set] which restricts along K. The pullback appears with proof on page 166 of=20 [The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149--168]=20= and is proved in the setting of Yoneda structures in [(with R.F.C. Walters) Yoneda structures on 2-categories, J. Algebra 50 = (1978) 350--379]. I attribute the result to=20 [FEJ Linton, Relative functorial semantics: adjointness results, Lecture = Notes in Math 99 (1969) 166--177] but cannot remember whether the pullback is explicitly there. Best wishes, Ross= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]