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Giry algebras are barycenters maps, which are coequalizers of contractiblecoequalizer pairs (like any algebras), and their existence, in general,requires the measurable space be coseparated by the discrete two point space,and the hypothesis that no measurable cardinals exist. Under that hypothesis,every measurable space which is coseparated has an algebra, and the category ofGiry algebras provides a convenient setting for probability theory because itis a symmetric monoidal closed category with all limits and colimits, as wellas having a seperator and coseperator. This is in stark contrast to the Kleisicategory of the Giry monad, which is often used to model conditionalprobability, which has a seperator but not much else.

The existence and utility of Giry algebras in probability theory