From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6703 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Codensity and the ultrafilter monad Date: Thu, 9 Jun 2011 23:25:55 +0100 Message-ID: Reply-To: Tom Leinster NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset="ISO-8859-15" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1307663410 3608 80.91.229.12 (9 Jun 2011 23:50:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 9 Jun 2011 23:50:10 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Fri Jun 10 01:50:06 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 1QUozC-00072w-FE for gsmc-categories@m.gmane.org; Fri, 10 Jun 2011 01:50:02 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53945) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QUoxF-0005Lk-Ij; Thu, 09 Jun 2011 20:48:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QUoxE-0006y0-P1 for categories-list@mlist.mta.ca; Thu, 09 Jun 2011 20:48:00 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6703 Archived-At: Dear all, Any functor from a small category A to a complete category E induces a=20 contravariant adjunction between E and Set^A. This in turn induces a=20 monad on E, the "codensity monad" of the functor. (The construction of the adjunction is better known in its dual form,=20 starting with a functor from a small category to a COcomplete category.=20 For example, the usual functor from Delta into Top induces the usual=20 adjunction between topological spaces and simplicial sets.) The codensity monad of the inclusion FinSet --> Set is the ultrafilter=20 monad. This seems a rather basic fact, but I've been unable to find it i= n=20 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=20 related characterization of the ultrafilter monad.) Thanks, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]