Re: Tensor of monads

[Gordon Plotkin <gdp@inf.ed.ac.uk>]

Dear Sergey, In the paper by Hyland,  Levy, Power and myself,
Combining algebraic effects with continuations, there is a proof that
the tensor of the continuations monad R^(R^-) (|R| >= 2) with itself,
or, indeed with any monad T with a constant (i.e. such that T(0) is
not empty), is the trivial monad.

It is not hard to see this directly, via large Lawvere theories. The
large Lawvere theory L of the continuations monad has:

   L(X,Y) = Set(R^X,R^Y)

and so the constants L(0,1) correspond to maps 1 --> R. Further, using
two distinct constants, any two operations  R^X --> R^Y can be coded
up into one operation R^(X +1) --> R^Y and then recovered via the two
constants. Given maps of large Lawvere theories L_T ---> M <----L such
that the images of any two operations in L_T and L commute, as L_T has
a constant all (the images of) constants in L are identified, as
usual, but then so are all images of any two operations R^X --> R^Y
(which will, for example, include all pairs of projections) and so M
is trivial.

A more general theorem is also proved in the paper which has as a
consequence that the tensor of any monad with rank with the
continuations monad exists.

On Thu, Aug 5, 2010 at 9:06 PM, Sergey Goncharov
<sergey@informatik.uni-bremen.de> wrote:
> Thank you Peter and André
> and all the participants of the discussion. It is indeed very helpful.
>
> Richard Garner wrote:
>>
>> 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
>
> I guess it applies both to the tensor and to the sum as well as to any other
> case where we need to form a span of monad morphism: S -> R <- T and which
> precisely can not be formed in this case.
>
> It looks like there are two counterexamples, both of which are based on the
> construction of tricky underlying categories. But what about existence of
> the tensor over Sets? I guess this is still open.  I tried to think about
> the tensor product of a continuation monad with itself as a possible
> counterexample, without any success though. Usually, continuation monad
> performs well when it comes to constructing counterexamples but it is
> difficult to see what the tensor product of it with itself  is supposed  to
> look like.
>
> Thanks again,
> Sergey.
>
>

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