From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7408 Path: news.gmane.org!not-for-mail From: Zhen Lin Low Newsgroups: gmane.science.mathematics.categories Subject: Re: Yoneda Lemma when there is a monad Date: Tue, 21 Aug 2012 20:25:16 +0800 Message-ID: References: Reply-To: Zhen Lin Low NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1345646218 9972 80.91.229.3 (22 Aug 2012 14:36:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 22 Aug 2012 14:36:58 +0000 (UTC) Cc: categories@mta.ca To: info@christophertownsend.org Original-X-From: majordomo@mlist.mta.ca Wed Aug 22 16:36:55 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 1T4C3C-0002G0-A6 for gsmc-categories@m.gmane.org; Wed, 22 Aug 2012 16:36:54 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52363) by smtpy.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1T4C1s-0006iF-DX; Wed, 22 Aug 2012 11:35:32 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1T4C1v-0007Yw-Mj for categories-list@mlist.mta.ca; Wed, 22 Aug 2012 11:35:35 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7408 Archived-At: Hi, This can be regarded as a special case of the generalised nerve theorem of Mark Weber [2007, Familial 2-functors and parametric right adjoints], which states that for a suitable subcategory A ("arities") of C, the Eilenberg-Moore category for T embeds into the presheaf category on the full subcategory of the Kleisli category spanned by A. Modulo Weber's requirement that A be small, C is always suitable. Best wishes, -- Zhen Lin On 21 August 2012 17:34, 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/ ]