From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6706 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: Codensity and the ultrafilter monad Date: Mon, 13 Jun 2011 02:28:07 +0100 Message-ID: References: Reply-To: Tom Leinster NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1307938479 3857 80.91.229.12 (13 Jun 2011 04:14:39 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 13 Jun 2011 04:14:39 +0000 (UTC) To: "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Mon Jun 13 06:14:35 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 1QVyXq-0001yV-Rj for gsmc-categories@m.gmane.org; Mon, 13 Jun 2011 06:14:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:55567) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QVyVL-0000SM-0z; Mon, 13 Jun 2011 01:11:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QVyVK-0007sQ-4C for categories-list@mlist.mta.ca; Mon, 13 Jun 2011 01:11:58 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6706 Archived-At: A few days ago, I asked where in the literature I could find the fact tha= t=20 the codensity monad of the inclusion FinSet --> Set is the ultrafilter=20 monad. Michel Hebert replied: > 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 kn= ow > if this part of the exercise is solved or mentioned explicitly there. Thanks very much, Michel. The references to Lawvere's thesis (Section=20 III, Theorem 2) and Linton's 1966 La Jolla paper (Section 2) seem to be=20 general references for the structure-semantics adjunction, which is what=20 the earlier parts of the exercise are about. The word "ultrafilter" does= =20 not appear in either Lawvere or Linton. So I currently believe that Manes was the first to publish this fact. If= =20 someone knows better (perhaps Bill, Fred, Anders Kock or Myles Tierney), = I=20 hope they will let me know. (I suspect that John Isbell would have known it, at some level, when he=20 wrote his 1960 paper "Adequate subcategories", even though the language o= f=20 monads wasn't available then. But I haven't found it mentioned in his=20 words; Manes's exercise is the only written reference to this fact that I= =20 know of.) Incidentally, I've learned how many names the codensity monad has had=20 through history: it has also been called the model-induced monad/triple=20 (e.g. by Appelgate and Tierney), the coadequacy monad/triple (e.g. by=20 Lawvere), and the algebraic completion (e.g. by Manes). Thanks to all who replied. Best wishes, Tom > 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 = monad >> 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= . For >> example, the usual functor from Delta into Top induces the usual adjun= ction >> 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 i= t in >> 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 b= ut >> related characterization of the ultrafilter monad.) >> >> Thanks, >> Tom > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]