From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6005 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 10:18:28 +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 1280449574 4957 80.91.229.12 (30 Jul 2010 00:26:14 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 30 Jul 2010 00:26:14 +0000 (UTC) Cc: categories@mta.ca To: Sergey Goncharov Original-X-From: majordomo@mlist.mta.ca Fri Jul 30 02:26:13 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OedQR-0001FT-Oy for gsmc-categories@m.gmane.org; Fri, 30 Jul 2010 02:26:11 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46694) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OedLB-000466-SQ; Thu, 29 Jul 2010 21:20:45 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OedL8-00031g-GP for categories-list@mlist.mta.ca; Thu, 29 Jul 2010 21:20:42 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6005 Archived-At: 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/ ]