From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7412 Path: news.gmane.org!not-for-mail From: Mark Weber Newsgroups: gmane.science.mathematics.categories Subject: Re: Yoneda Lemma when there is a monad Date: Wed, 22 Aug 2012 10:13:27 +1000 Message-ID: References: Reply-To: Mark Weber NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1345646367 11247 80.91.229.3 (22 Aug 2012 14:39:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 22 Aug 2012 14:39:27 +0000 (UTC) Cc: "categories@mta.ca" To: "info@christophertownsend.org" Original-X-From: majordomo@mlist.mta.ca Wed Aug 22 16:39:21 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 1T4C5N-0004jz-GR for gsmc-categories@m.gmane.org; Wed, 22 Aug 2012 16:39:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52386) by smtpy.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1T4C4m-00076l-I2; Wed, 22 Aug 2012 11:38:32 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1T4C4p-0007e7-HR for categories-list@mlist.mta.ca; Wed, 22 Aug 2012 11:38:35 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7412 Archived-At: Dear Christopher Your lemma is a combination of 2 results in the literature. The first is Pro= position(13) of 1. R. Street and B. Walters, Yoneda structures on 2-categories, Journal of Algebra 50:350-379, 1978 which says that for a functor J:B-->C between locally small categories, the f= unctor PB(C(J,1),1):PB-->PC corresponds to taking right kan extension along J -- here PB=3D[B^op,Set], C= (J,1):C-->PB sends (b,c) to the hom set C(Jb,c). Your lemma is the case J=3D= K (the comparison functor), and one uses the result that K is dense, due to = Fred Linton 2. F. Linton, Relative functorial semantics: adjointness results, LNM 99:384-418, 1969 to simplify the right kan extension to the F(A,a) of your lemma. Often there= are nice subcategories of C_T which are also dense in C^T, and this would g= ive other variants of your lemma. For more on this kind of situation see http://arxiv.org/abs/1101.3064 With best regards, Mark Weber On 21/08/2012, at 7:34 PM, info@christophertownsend.org wrote: > Hi >=20 > Does anyone have any references for the following generalisation of > the Yoneda lemma: >=20 > 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) >=20 > Nat[C^T(K_,(A,a)),F]=3DF(A,a) >=20 > 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=3DF(Ta)x. (And Nat[_] > means the set of natural transformations.) // >=20 > 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? >=20 > Thanks, Christopher >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]