Characterizing Giry-algebras as coseparable super convex spaces

TL;DR

This paper characterizes Giry-algebras using super convex spaces, establishing a full and faithful Yoneda embedding for coseparable spaces, and constructs a barycenter map to factorize the Giry monad, leading to an equivalence of categories.

Contribution

It introduces a new categorical framework for Giry-algebras via coseparable super convex spaces and constructs a barycenter map to factorize the Giry monad.

Findings

Yoneda embedding is full and faithful for coseparable super convex spaces.

A barycenter map is constructed to factorize the Giry monad.

Establishes an equivalence of categories for Giry-algebras.

Abstract

We investigate the Eilenberg-Moore algebras for the Giry monad defined on the category of measurable spaces using super convex spaces. The category of super convex spaces has a subcategory consisting of the one point extension of the real line, and the truncated Yoneda embedding arising from the full subcategory with that one object is full, although it is not faithful. By restricting to those super convex spaces which are coseparable by the one point extension of the real line, the truncated Yoneda embedding is full and faithful. This permits the construction of a barycenter map used to factorize the Giry monad, and obtain an equivalence of categories.