Re: Tensor of monads

[Richard Garner <r.h.g.garner@gmail.com>]

I must admit to feeling slightly confused by both Peter's and André's
examples. In both cases, the monads considered arise on a category other
than the category of sets; and it is not clear to me what is meant by
forming the tensor product of two such monads. In the case of finitary
monads on Set, the meaning is clear, since a finitary monad corresponds to a
Lawvere theory, a Lawvere theory is a special kind of small category with
finite products, and we know how to form the tensor product of two
categories with finite products (essentially because the doctrine of finite
products is a pseudo-commutative one). This extends without problem to
monads with rank on Set; and even to monads without rank on Set, so long as
one recognises that the correlate notion of Lawvere theory will be a large
one, so that the tensor product may not always exist.  In the case of base
categories other than Set, one would have to use a generalisation of the
notion of Lawvere theory: such has been given by Nishizawa-Power (see also
Lack-Power) but they require that the base category be locally presentable,
which is not the case in the examples of André and Peter.

My own take on what Peter's example is doing is the following. Given any
finitary monad L on Sets, there is a corresponding Lawvere theory T, and so
for any category C, we can consider the category Mod(T,C) of T-models in C:
it's the category of finite-product preserving functors T -> C. There is a
forgetful functor Mod(T,C)-->C given by evaluation at 1, and this will be
monadic so long as it has a left adjoint; in which case we induce a monad L'
on C. In particular, letting L be the finitary monad on Set whose algebras
are sup-semilattices-without-a-unit, letting L * L be its tensor with
itself, and letting C be the category of finite sets of odd cardinality,
then Peter's example seems to show that:

   -- The induced monad L' on C exists but the induced monad (L * L)' does
not.

On the other hand, André's example raises a question which I find quite
interesting. André describes two reflective subcategories of the ordered
class of ordinal numbers, and then says that, their intersection being
empty, the tensor of the corresponding idempotent monads cannot exist. I
would be inclined to say that this shows that the coproduct of these monads
does not exist, but this leads on to my question, which is:

   -- Given idempotent monads S, T on a category C for which we can speak of
the tensor of S and T, is it always the case that S * T is isomorphic to S +
T?

Here is a test case. Power's 2000 paper "Enriched Lawvere theories" shows
that, for a symmetric monoidal closed V which is lfp as a closed category,
finitary V-monads are equivalent to enriched Lawvere theories, which are
certain small V-categories with finite cotensors. Using the tensor of such
V-categories, we can define a tensor of such monads. Consider in particular
when V is [D^op, Set] for some small D, and let S and T be two idempotent
strong (=V-enriched) monads on V. Now it should be possible to calculate S +
T and S * T in terms of the corresponding Lawvere theories and to see if
they coincide. I haven't tried this yet but it should be an interesting
exercise. (The obvious thing to try first is to take S and T corresponding
to sheaf subtoposes of [D^op, Set]. Then these monads, being
product-preserving, are commutative, and so it's natural to think that S * T
should be the monad corresponding to the intersection of these subtoposes. S
+ T, on the other hand, I'm inclined to think may be something bigger, and
not necessarily cartesian.)

Richard


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