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/ ]