Question (fwd)

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

My colleague Klaus Keimel has the following question and would be glad
if someone could answer it.
If you want to answer him directly his e-mail address is

   keimel@mathematik.tu-darmstadt.de

I have come across a different, but similar question concerning the
distribution monad on Froelicher spaces (as studied by Froelicher, Kriegl
and Michor). Can one characterize elementarily the algebras for the monad
T(X) = Lin(R^X,R) on \SS, the cartesian closed category of Froelicher
spaces and "smooth" maps between them? I guess smooth and linear is not
sufficient...

Thomas Streicher


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  The monad of probability measures over compact Hausdorf spaces

If we assign to every compact Hausdorff space X the set PX of all
probability measures on X endowed with the vague (= weak*topology if
we consider PX embedded in the dual of C(X)), then we have a monad.
The unit e assign the Dirac measure to every point x in X, the
multiplication m assigns the barycentre of every probability measure
on PX.

My question is: What are the algebras and the algebra homomorphisms
of this monad?

It should be straightforward, that compact convex sets in locally
convex vector spaces are algebras. Probability measures on such spaces
have a barycentre. Continuous affine maps should be homomorphisms.

Are these all algebras and homomorphisms?

One thinks that this should be known. Who knows about this? I would be
interested in hints and in relevant references.

Klaus Keimel