Question (fwd)

Question (fwd)

[Thomas Streicher]

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

Re: Question (fwd)

[wlawvere]

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
>
>
>
>