Characterizing Giry-algebras as coseparable super convex spaces
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.
Cite
@article{arxiv.1907.03209,
title = {Characterizing Giry-algebras as coseparable super convex spaces},
author = {Kirk Sturtz},
journal= {arXiv preprint arXiv:1907.03209},
year = {2022}
}
Comments
There is a fundamental error in the article based on the property that the category of Giry algebras itself has no coseparator - hence any representation of it should reflect that. The problem can be circumvented as discussed in a more recent article on algebras using the support of a probability measure