Re: Tensor of monads

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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
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>

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