From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>
To: Sergey Goncharov <sergey@informatik.uni-bremen.de>
Cc: categories@mta.ca
Subject: Re: Tensor of monads
Date: Thu, 29 Jul 2010 14:24:43 +0100 (BST) [thread overview]
Message-ID: <E1OedP0-0003Ac-23@mlist.mta.ca> (raw)
In-Reply-To: <alpine.LRH.2.00.1007291006210.5174@siskin.dpmms.cam.ac.uk>
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/ ]
next prev parent reply other threads:[~2010-07-29 13:24 UTC|newest]
Thread overview: 24+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-07-28 14:02 Sergey Goncharov
2010-07-29 8:21 ` N.Bowler
2010-07-29 9:18 ` Prof. Peter Johnstone
2010-07-29 10:29 ` Michael Barr
2010-07-31 8:45 ` Richard Garner
[not found] ` <AANLkTinxyVQ1fXu7DLWu4CUF3AP2KPX6PLQFDB+zG4Ef@mail.gmail.com>
2010-07-31 12:48 ` Michael Barr
[not found] ` <alpine.LRH.2.00.1007291006210.5174@siskin.dpmms.cam.ac.uk>
2010-07-29 13:24 ` Prof. Peter Johnstone [this message]
[not found] ` <alpine.LRH.2.00.1007291422370.5174@siskin.dpmms.cam.ac.uk>
2010-07-30 1:02 ` Sergey Goncharov
2010-07-31 20:34 ` Eckmann-Hilton (Was: Tensor of monads) Toby Bartels
[not found] ` <4C5224A4.4000105@informatik.uni-bremen.de>
2010-07-30 10:37 ` Tensor of monads Prof. Peter Johnstone
2010-07-30 22:41 ` Tom Leinster
2010-08-01 19:49 ` Ronnie Brown
2010-08-02 9:47 ` Ronnie Brown
2010-08-01 0:31 ` Richard Garner
2010-08-02 19:55 ` Paul Levy
2010-08-03 6:39 ` Richard Garner
[not found] ` <AANLkTimd202AX=3hUqU9ABkKUy9Z4Loh1RXTiDgVZ3Ku@mail.gmail.com>
2010-08-03 11:03 ` Paul Levy
2010-08-09 20:26 ` Paul Levy
2010-08-05 20:06 ` Sergey Goncharov
2010-08-08 19:24 ` Gordon Plotkin
2010-07-30 3:44 ` Joyal, André
[not found] ` <AANLkTin5+paq8sP-eVjdf8rZOyA-z=t6QzAhCqVUsyQi@mail.gmail.com>
2010-08-01 9:13 ` Richard Garner
2010-08-02 14:17 ` Prof. Peter Johnstone
[not found] ` <alpine.LRH.2.00.1008021514290.18118@siskin.dpmms.cam.ac.uk>
2010-08-02 21:12 ` Richard Garner
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