From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6705 Path: news.gmane.org!not-for-mail From: Michel Hebert Newsgroups: gmane.science.mathematics.categories Subject: Re: Codensity and the ultrafilter monad Date: Fri, 10 Jun 2011 10:19:25 +0200 Message-ID: References: Reply-To: Michel Hebert NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1307695778 25572 80.91.229.12 (10 Jun 2011 08:49:38 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 10 Jun 2011 08:49:38 +0000 (UTC) Cc: Michel To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Jun 10 10:49:32 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QUxPI-0005BU-6s for gsmc-categories@m.gmane.org; Fri, 10 Jun 2011 10:49:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:32820) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QUxNH-0000DX-Ra; Fri, 10 Jun 2011 05:47:27 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QUxNH-0000MO-7y for categories-list@mlist.mta.ca; Fri, 10 Jun 2011 05:47:27 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6705 Archived-At: Hi Tom, This appears as exercise 3.2.12(e) in Manes' book (Algebraic Theories). The references there are Lawvere's thesis and Linton's 1966, but I don't know if this part of the exercise is solved or mentioned explicitly there. Best regards, Michel On Fri, Jun 10, 2011 at 12:25 AM, Tom Leinster wrote: > Dear all, > > Any functor from a small category A to a complete category E induces a > contravariant adjunction between E and Set^A. This in turn induces a mon= ad > on E, the "codensity monad" of the functor. > > (The construction of the adjunction is better known in its dual form, > starting with a functor from a small category to a COcomplete category. F= or > example, the usual functor from Delta into Top induces the usual adjuncti= on > between topological spaces and simplicial sets.) > > The codensity monad of the inclusion FinSet --> Set is the ultrafilter > monad. This seems a rather basic fact, but I've been unable to find it i= n > the literature. I'd be grateful if someone could tell me a reference. > > (I'm aware of the 1987 paper by Reinhard B=F6rger giving a different but > related characterization of the ultrafilter monad.) > > Thanks, > Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]