From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7414 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Yoneda Lemma when there is a monad Date: Wed, 22 Aug 2012 07:29:12 -0700 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1345646432 11886 80.91.229.3 (22 Aug 2012 14:40:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 22 Aug 2012 14:40:32 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Wed Aug 22 16:40:28 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 1T4C6d-000694-DE for gsmc-categories@m.gmane.org; Wed, 22 Aug 2012 16:40:27 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52399) by smtpy.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1T4C5v-0007RN-4z; Wed, 22 Aug 2012 11:39:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1T4C5y-0007hZ-BF for categories-list@mlist.mta.ca; Wed, 22 Aug 2012 11:39:46 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7414 Archived-At: If this has an answer I'd like to know. In my paper Pratt, V.R. ``Communes via Yoneda, from an Elementary Perspective'', Fundamenta Informaticae 103, 203-218, DOI 10.3233/FI-2010-325, IOS Press, 2010 which among other things generalizes it in section 1.5 to the Chu setting, I treated it as part of the uncitable Yoneda folklore. Why uncitable? Well, one of the many Norbert Weiner stories is that he was challenged in class on some point. He stared at the board for a bit, wandered outside, came back twenty minutes later and said "It's obvious." The Yoneda lemma is like that. Vaughan Pratt On 8/21/2012 2:34 AM, info@christophertownsend.org wrote: > Hi > > Does anyone have any references for the following generalisation of > the Yoneda lemma: > > Lemma: If (T,i,m) is a monad on a locally small category C then for > any functor F:(C_T)^op->Set, contravariant from the Kleisli category > to Set and for any T algebra (A,a) > > Nat[C^T(K_,(A,a)),F]=F(A,a) > > where K is the usual comparison functor from the Kleisli category, > C_T, to the category of algebras of T, C^T. F(A,a) means the subset > of F(A) consisting of elements x such that F(m_A)x=F(Ta)x. (And Nat[_] > means the set of natural transformations.) // > > 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? > > Thanks, Christopher > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]