Re: Tensor of monads

[Paul Levy <pbl@cs.bham.ac.uk>]

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











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