On 1 Aug 2010, at 01:31, Richard Garner wrote:
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.
Here is a suggestion; I don't know how it relates to yours. Let S and T be monads on a category C. An "S,T-algebra" is a C-object X together with an S-algebra structure theta and T-algebra structure phi. An S,T-algebra morphism is a C- morphism that is homomorphic in both components. Let D be the category of S,T-algebras and homomorphisms, and U : D --> C the forgetful functor. Then U creates U-split coequalizers. If it has a left adjoint, we call the monad the "sum" of S and T. I think the sum of S and T, if it exists, has to be a coproduct in the category of monads, but haven't checked the details. Next suppose that C is cartesian, and S and T are strong. Now D will be a locally C-indexed (by this I mean [C^op,Set]-enriched) category. A morphism from (X,theta,phi) to (X',theta',phi') over Z is a C- morphism from Z x X to X' that's homomorphic in its second argument, with respect to both structures. If U has a (locally C-indexed) left adjoint, we get the "sum" of strong monads. Again, I think it's a coproduct in the category of strong monads. Next suppose C is cartesian closed and S and T are strong. For an S,T-algebra (X,theta,phi), the following are equivalent: (1) for all C-objects Y and Z, the two C-morphisms from SY x TZ x X^(YxZ) to X are equal (2) for every C-object Y, the two C-morphisms from SY x T(X^Y) to X are equal (2') for every C-object Z, the two C-morphisms from TZ x S(X^Z) to X are equal. When these hold, we say that (X,theta,phi) "commutes". (I'd like to express this without quantification over objects, but I can't see how.) We thus obtain a full (locally C-indexed) subcategory D' of D consisting of the commuting S,T-algebras and homomorphisms, and U' : D' --> C the restriction of U. Then U' creates U'-split coequalizers. If it has a left adjoint, we call the induced monad the "tensor" of S and T. Now a cocone of strong monads S -----> M <----- T is said to "commute" when for all C-objects X and Y the two C- morphisms from SX x TY to M(X x Y) are equal. I think a tensor of S and T will always give an initial commuting cocone, but haven't checked the details. Paul -- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]