Re: Tensor of monads

[Sergey Goncharov <sergey@informatik.uni-bremen.de>]

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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