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Probability monads on categories of topological spaces are classical objects of study in the categorical approach to probability theory, with important applications in the semantics of probabilistic programming languages. We construct a probability monad on the category of compactly generated weakly Hausdorff (CGWH) spaces, a (if not the) standard choice of convenient category of topological spaces. Because a general version of the Riesz representation theorem adapted to this setting plays a fundamental role in our construction, we name it the Riesz probability monad. We show that the Riesz probability monad is a simultaneous extension of the classical Radon and Giry monads that is topologically enriched. Topological enrichment corresponds to a strengthened continuous mapping theorem (in the sense of probability theory). In addition, restricting the Riesz probability monad to the Cartesian closed subcategory of weakly Hausdorff quotients of countably based (QCB) spaces results in a probability monad which is strongly affine, ensuring that the notions of independence and determinism interact as we would expect.
@article{TAC_2024_41_a54,
author = {Peter Kristel and Benedikt Peterseim},
title = {A topologically enriched probability monad on the {Cartesian} closed category of {CGWH} spaces},
journal = {Theory and applications of categories},
pages = {1983--2012},
publisher = {mathdoc},
volume = {41},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a54/}
}
TY - JOUR AU - Peter Kristel AU - Benedikt Peterseim TI - A topologically enriched probability monad on the Cartesian closed category of CGWH spaces JO - Theory and applications of categories PY - 2024 SP - 1983 EP - 2012 VL - 41 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TAC_2024_41_a54/ LA - en ID - TAC_2024_41_a54 ER -
%0 Journal Article %A Peter Kristel %A Benedikt Peterseim %T A topologically enriched probability monad on the Cartesian closed category of CGWH spaces %J Theory and applications of categories %D 2024 %P 1983-2012 %V 41 %I mathdoc %U http://geodesic.mathdoc.fr/item/TAC_2024_41_a54/ %G en %F TAC_2024_41_a54
Peter Kristel; Benedikt Peterseim. A topologically enriched probability monad on the Cartesian closed category of CGWH spaces. Theory and applications of categories, Tome 41 (2024), pp. 1983-2012. http://geodesic.mathdoc.fr/item/TAC_2024_41_a54/