From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6006 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor of monads Date: Thu, 29 Jul 2010 14:24:43 +0100 (BST) Message-ID: References: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1280449617 5033 80.91.229.12 (30 Jul 2010 00:26:57 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 30 Jul 2010 00:26:57 +0000 (UTC) Cc: categories@mta.ca To: Sergey Goncharov Original-X-From: majordomo@mlist.mta.ca Fri Jul 30 02:26:54 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OedR7-0001T3-Km for gsmc-categories@m.gmane.org; Fri, 30 Jul 2010 02:26:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33349) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OedP4-0004Wb-66; Thu, 29 Jul 2010 21:24:46 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OedP0-0003Ac-23 for categories-list@mlist.mta.ca; Thu, 29 Jul 2010 21:24:42 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6006 Archived-At: Sorry, the previous posting was nonsense -- a bisemilattice is the same thing as a semilattice, by the Eckmann-Hilton argument. However, if you leave out the zero, and consider the "set of nonempty subsets" monad, this time on the category of sets of cardinality 2^n - 1 for some n, you do get a counterexample. Peter Johnstone On Thu, 29 Jul 2010, Prof. Peter Johnstone wrote: > Here's a slightly artificial counterexample: let C be the category > of finite sets whose cardinality is a power of 2, and all functions > between them. The covariant power-set functor restricts to a > functor C --> C, and has a monad structure whose algebras are > semilattices. If the tensor product of this monad with itself > existed, its algebras would be bisemilattices, i.e. sets with two > semilattice structures which "commute with each other" in the > obvious sense. Free bisemilattices exist, but they don't > necessarily have cardinality a power of 2: by my calculation, the > free bisemilattice on two generators has seven elements. So the > free-bisemilattice functor doesn't exist as an endofunctor of C. > > Peter Johnstone > ----------------------- > On Wed, 28 Jul 2010, Sergey Goncharov wrote: > >> Dear categorists, >> >> in "Combining algebraic e?ects with continuations", by Hyland et al. the >> authors say carefully: "In general, the tensor product of two arbitrary >> monads seems not to exist.." without providing a counterexample though, >> presumably because they did not have any. Was there any progress reported >> on this issue since then? Or maybe someone can even make up a >> counterexample right on the nail? >> >> Thanks, >> >> -- >> Sergey Goncharov, Junior Researcher >> >> DFKI Bremen Phone: +49-421-218-64276 >> Safe and Secure Cognitive Systems Fax: +49-421-218-9864276 >> Cartesium, Enrique-Schmidt-Str. 5 Email: Sergey.Goncharov@dfki.de >> D-28359 Bremen Site: www.dfki.de/sks/staff/sergey >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]