The factorization of the Giry monad

Abstract

We construct a factorization of the Giry monad through the category of convex spaces, and show that, provided that no measurable cardinals exist, probability measures can be viewed as natural transformations. Using the adjunction of this factorization, we then show the category of Giry algebras is equivalent to the category of convex measurable spaces where the $\sigma$-algebra structure associated with a convex space satisfies an elementary property.

arXiv commentLemma 9.5 page 13 is incorrect. The Giry monad cannot be factored through the category of convex spaces except for finite spaces. Using standard measurable spaces one can obtain an adjoint factorizaton, and is described in the article Giry algebras for standard measurable spaces (2022)