nLab table of probability monads

Probability monad ( P P ) Category Elements/points of P X P X Extra structure of P X P X P P -Algebras References
distribution monad (a.k.a. finitary Giry monad, convex combination monad) Set convex combinations or finitely-supported probability measures (just a set) convex spaces Fritz '09, Jacobs '18
Giry monad Meas probability measures initial σ-algebra of evaluation maps Full characterization unknown. See also here. Lawvere '62, Giry '80
Giry monad Pol Borel probability measures initial topology of integration maps Full characterization unknown. See also here. Giry '80
Probability monad on QBS? quasi-Borel spaces Equivalence classes of random variables quasi-Borel structure inherited by the cartesian closed structure Full characterization unknown. HKSY '17
Radon monad Comp Radon probability measures (or continuous valuations) compact convex subsets of locally convex topological vector spaces Swirszcz '74, Keimel '08
ordered Radon monad CompOrd Radon probability measures (or continuous valuations) weak topology w.r.t. continuous functions, stochastic order compact convex subsets of ordered? locally convex topological vector spaces Swirszcz '74, Keimel '08
measure monad on Top Top τ-additive measures A-topology, stochastic order Full characterization unknown. See also here. F-P-R '19
probabilistic powerdomain? dcpo, continuous domains? continuous valuations stochastic order abstract probabilistic domains? (continuous case) J-P '89
extended probabilistic powerdomain Top, stably compact spaces continuous valuations initial topology of evaluation maps, stochastic order Full characterization unknown. Dedicated section here Heckmann '96, A-J-K '04, GL-J '19, F-P-R '19
valuation monad on locales? Loc continuous valuations initial topology of evaluation maps Vickers '11
Kantorovich monad complete metric spaces Radon probability measures of finite first moment Kantorovich-Wasserstein metric closed convex subsets of Banach spaces van Breugel '05, F-P '19
ordered Kantorovich monad complete L-ordered? metric spaces Radon probability measures of finite first moment Kantorovich-Wasserstein metric, stochastic order closed convex subsets of ordered Banach spaces? F-P '20
Baire monad Weakly Hausdorff quotients of countably-based topological spaces Baire measures topology induced by cartesian closedness Full characterization unknown K-P '24
Riesz monad compactly generated weakly Hausdorff topological spaces k k -regular probability measures topology induced by cartesian closedness Full characterization unknown K-P '24

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Created on January 9, 2025 at 04:09:30. See the history of this page for a list of all contributions to it.