From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3158 Path: news.gmane.org!not-for-mail From: "Tom Leinster" Newsgroups: gmane.science.mathematics.categories Subject: Monads on finite categories Date: Sat, 25 Mar 2006 21:32:47 -0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-15 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241019126 7486 80.91.229.2 (29 Apr 2009 15:32:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:06 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNS0H-0004x6-VP for categories-list@mta.ca; Sun, 26 Mar 2006 05:57:45 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 104 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:3158 Archived-At: Dear All, 1. Are there significant or interesting examples of monads on finite categories? I want to look beyond monads on posets, a.k.a. closure operators. (Since a finite category with binary sums or products is necessarily a poset, some of the usual examples of monads reduce to this case.) I can only think of one class of examples (described below), and I don't know if it's particularly significant. 2. Any monad on a finite category is idempotent. Is this widely known? Thanks. Tom * * * The class of examples: let A be a finite Cauchy-complete category. Let M be the 2-element monoid consisting of the identity and an idempotent, so that [M, A] is the category of idempotents in A. Then the diagonal functor A ---> [M, A] has adjoints on both sides. The induced monad on A is trivial, but that on [M, A] is not. (It sends an idempotent e to 1_a, where a is the object through which e splits.)