From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7406 Path: news.gmane.org!not-for-mail From: info@christophertownsend.org Newsgroups: gmane.science.mathematics.categories Subject: Yoneda Lemma when there is a monad Date: Tue, 21 Aug 2012 10:34:30 +0100 Message-ID: Reply-To: info@christophertownsend.org NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; DelSp="Yes"; format="flowed" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1345551491 14029 80.91.229.3 (21 Aug 2012 12:18:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 21 Aug 2012 12:18:11 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Aug 21 14:18:09 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.134]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1T3nPJ-0004GD-7e for gsmc-categories@m.gmane.org; Tue, 21 Aug 2012 14:18:05 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35337) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1T3nOO-0006WY-6E; Tue, 21 Aug 2012 09:17:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1T3nON-0002W0-9T for categories-list@mlist.mta.ca; Tue, 21 Aug 2012 09:17:07 -0300 Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7406 Archived-At: 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/ ]