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 ----------------------------------------------------------------------- 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