Concerning Thomas Streicher's question about the internal linear functional (=distribution of compact support) monad on Froelicher's smooth category : see volume 1 of the journal Functional Analysis for two papers by Waelbroeck which show that the algebras are essentially determined by complete bornological spaces. Concerning Klaus Keimel's question about "abstract" compact convex sets, I do not recall a precise reference but is seems that Linton or Semadeni or both proved that they all do embed in locally convex linear spaces. This of course is in contrast with the noncompact finitary part of the probability theory, where there are those special algebras which are often discarded as spurious, but which in fact by a very natural adjoint to an algebraic functor record the face structure of any convex set as a semilattice. That raises the question: is there no equally natural way to record face structure for COMPACT convex sets ? Quoting 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
----------------------------------------------------------------------- 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