Re: Tensor of monads

[Michael Barr <barr@math.mcgill.ca>]

I am afraid you are right.  Here is what Manes says on the subject in
LNM80:

It is an open question whether or not
$\t\otimes\widetilde{\t}$ always exists. A constructive proof can be given
if both $\t$ and $\widetilde{\t}$ have a rank (in the sense of
\cite[2.2.6]{man}) by generalizing Freyd's proof in \cite{fre}.

What I had in mind involved two isomorphic, but distinct triples.  The one
coming from the underlying functor of complete sup semilattices (whose mu
would be union) and the other coming from complete inf semilattices (whose
mu is intersection).  The problem comes when you make those operations
commute with each other.  It would seem to me that would force complete
distributivity.  But completely distributive boolean algebras do give a
tripleable category.  Perhaps the moderator could throw some light on this
from his work on CCD lattices.

Could this still be unknown?  I guess it could.  It is not a topic that
has aroused a great deal of interest.

Sorry, Michael

On Sat, 31 Jul 2010, Richard Garner wrote:

> Isn't P * P isomorphic to P, by the Eckmann-Hilton argument?
>
> On 29 July 2010 20:29, Michael Barr <barr@math.mcgill.ca> wrote:
>
>> There are examples in Ernie Manes's 1967 thesis.  Perhaps the simplest
>> (although it piggybacks on the non-existence of free complete boolean
>> algebras that had been know for only a few years at the time) is that the
>> tensor product of the complete sup semilattice triple with itself doesn't
>> exist.  The triple takes a set X to 2^X and can be interpreted also as the
>> complete inf semilattice triple.  On the other hand, I think Manes showed
>> that the tensor product of the beta triple with itself exists, but is one
>> of the two inconsistent triples, the one that fixes the empty set and
>> takes all non-empty sets to one point.  (The other inconsistent triple
>> takes all sets to one point.)
>>
>>
>> On Wed, 28 Jul 2010, Sergey Goncharov wrote:
>>
>>  Dear categorists,
>>>
>>> in "Combining algebraic eects 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,
>>>
>>>


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