Probability monad () |
Category |
Elements/points of |
Extra structure of |
-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 |
-regular probability measures |
topology induced by cartesian closedness |
Full characterization unknown |
K-P '24 |