From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7409 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: Yoneda Lemma when there is a monad Date: Tue, 21 Aug 2012 19:08:54 +0100 Message-ID: References: Reply-To: Tom Leinster NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset="US-ASCII" X-Trace: ger.gmane.org 1345646228 10106 80.91.229.3 (22 Aug 2012 14:37:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 22 Aug 2012 14:37:08 +0000 (UTC) Cc: "categories@mta.ca" , Tom Leinster To: "info@christophertownsend.org" Original-X-From: majordomo@mlist.mta.ca Wed Aug 22 16:37:05 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 1T4C3M-0002Uw-4k for gsmc-categories@m.gmane.org; Wed, 22 Aug 2012 16:37:04 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52369) by smtpy.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1T4C2b-0006kH-7s; Wed, 22 Aug 2012 11:36:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1T4C2e-0007Zz-62 for categories-list@mlist.mta.ca; Wed, 22 Aug 2012 11:36:20 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7409 Archived-At: Dear Chris, > C^T embeds in [(C_T)^op,Set]; i.e. any category of algebras embeds in > the presheaf category of the Kleisli category. This part is morally clear, I think, via the notion of density. A functor F: A --> B is called dense if the induced functor Hom(F, -): B --> [A^op, Set] is full and faithful. An equivalent condition (stated loosely) is that every object of B is a colimit of objects of the form F(a) (with a in A) in a canonical way. (See e.g. Categories for the Working Mathematician.) Now take a monad T on a category C. Every T-algebra is canonically a coequalizer of free T-algebras. So every object of C^T is canonically a colimit of objects of the subcategory C_T. As long as "canonically" means what I assume it does (and I haven't checked the details), this tells us that the inclusion C_T --> C^T is dense. Hence the induced functor C^T ---> [(C_T)^op, Set] is full and faithful - which I guess is what you mean by an embedding. All the best, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]