Giry algebras for standard measurable spaces

TL;DR

This paper explores Giry algebras within standard measurable spaces by introducing super convex spaces, utilizing Isbell duality, and characterizing the Giry monad's algebraic structure.

Contribution

It develops a novel framework using super convex spaces and duality techniques to analyze Giry monad algebras on standard measurable spaces.

Findings

Characterization of Giry algebras via super convex spaces

Application of Isbell duality in the analysis

Identification of the algebraic structure of Giry monad

Abstract

The notion of "super convex spaces" generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums. Using this notion permits a very elegant approach for analysis of the Giry monad on standard measurable spaces and identifying the $\mathcal{G}$-algebras for that monad. We use Isbell duality and restrict the adjunction $\mathbf{Spec} \dashv \mathcal{O}$ to a proper subcategory of super convex spaces and separated standard measurable spaces.