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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