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Stream: community: our work

Topic: Benedikt Peterseim

Benedikt Peterseim (Apr 16 2024 at 09:18):

Hello everyone! I'm happy to announce that I've finished my very first paper ever, joint with Peter Kristel.

You can find a preprint here.

Its title is long, but informative: "A Topologically Enriched Probability Monad on the Cartesian Closed Category of CGWH Spaces".

As the title suggests, the main result is the construction of a (commutative enriched) probability monad on the category of compactly generated weakly Hausdorff (CGWH) spaces.

If you don't know what a probability monad is, check out this nLab page, or Julian Asilis's Bachelor's thesis for an elementary introduction to the topic, or this video tutorial by @Paolo Perrone. You can think of a probability monad on a catgeory as providing a coherent setting for doing probability theory in that category. Under Moggi's interpretation of monads as "effects", they have important applications for modelling randomness in (prossibly "probabilistic") programming languages (or type theories). If you're still confused about what probability monads are supposed to be or what they're good for, I'd also be very happy to answer any general questions of this type here.

Now, you may have seen on the nLab page that quite a number of probability monads have been defined before. So you may ask: why yet another proability monad? There are at least three reasons why:

CGWH spaces are a (if not the) standard cartesian closed substitute for topological spaces. So it's natural to ask whether the known probability monads on categories of topological spaces have an analogue for CGWH spaces.
The aforementioned applications to programming languages require the base category to be cartesian closed. For a comparison to previous approaches tackling this problem and some merits that ours might have, see the paper.
The construction is interesting. Naively, one might simply try to equip the set of Borel (Baire/Radon/...) measures on a CGWH space with a suitable CGWH topology. However, it's very much unclear how to do this in a way that makes all the required maps continuous. The solution is a functional-analytic, double-dualisation based approach, using a very pretty and categorically natural version of the Riesz representation theorem, which you may find interesting in itself.

If you found any of this interesting, I encourage you to give the paper a read -- it's not too long. Of course, any feedback or suggestions are highly appreciated.

Reid Barton (Apr 24 2024 at 15:14):

Interesting stuff!

It seems like the same constructions would work in the category of condensed sets (which contains CGWH as a full subcategory), and produce (probability) measure monads that restrict to the ones you constructed on CGWH.

I wonder how it is related to the measures that are described in and around Example 3.3 of
https://www.math.uni-bonn.de/people/scholze/Analytic.pdf
(The discussion is very brief because this class of measures turns out not to be the correct thing for what they want to do in the rest of the lecture notes.)
My understanding is that they are supposed to be the same in the case of compact Hausdorff spaces.

Is there a second class of measures that can be obtained from your construction but taking all functions instead of just the bounded ones--possibly the compactly supported measures?

Benedikt Peterseim (Apr 24 2024 at 15:35):

This is true at least if you take the closed subspace generated by finitely supported measures in $C(X)^\wedge$ (notation of the paper). This is part of Theorem 5.0.5. in my Master's thesis. (Beware that the thesis is less polished than the paper and uses slightly different terminology.) I don't know how this subspace compares to the full dual $C(X)^\wedge$ , in general, but for compact Hausdorff spaces, they are the same. I'd conjecture that this is also the case for any weakly Hausdorff QCB space.

So here is a conjecture: Let $X$ be a Hausdorff QCB space. Then $C(X)^\wedge$ can be identified with the space of compactly supported Radon measures on $X$ .

Concerning your question about the relation to condensed mathematics, you're absolutely right, one could extend this construction to condensed sets. However, this would "factor" through the one on CGWH spaces, since if $X$ is a condensed set, then $\mathbb{R}^X$ (internal hom of condensed sets) "is" a CGWH space (i.e. in the essential image of the inclusion of CGWH spaces into condensed sets), see the appendix of the aforementioned thesis. So, in some sense it's no loss of generality to restrict to CGWH spaces, in this case.

Finally, Example 3.3 in the notes you linked is concerned with measures on compact Hausdorff spaces, and since $C_b(X)=C(X)$ in this case, the space of measures mentioned there is exactly the one from our paper (also topologically)!

Benedikt Peterseim (Apr 24 2024 at 15:36):

Thanks for your interest, by the way. :)