# The equivalence between the categories of Giry-algebras and convex spaces

## TL;DR

This paper establishes an equivalence between the categories of Giry-algebras and convex spaces by exploring dualities, monads, and measurable structures rooted in the properties of the unit interval.

## Contribution

It provides a new proof of the categorical equivalence between Giry-algebras and convex spaces using duality and monad structures.

## Key findings

- Giry-algebras are equivalent to convex spaces.
- The unit interval plays a central role in the duality.
- Convex spaces can be associated with measurable spaces via Boolean subobjects.

## Abstract

A duality between the category of convex spaces and measurable spaces arises from the existence of the unit interval, which is an object in both these categories. The full subcategory of the category of convex spaces, consisting of just the single object, the unit interval, is both a dense and codense subcategory in the category of convex spaces. Combined with the the symmetric monoidal closed category structure of the category, one obtains the double dualization monad into the unit interval, which sends a point to the evaluation map at that point. The restriction of the codomain of the unit of this monad to the weakly averaging affine functionals is an isomorphism. Moreover, every convex space has an associated measurable space, whose {\sigma}-algebra is generated by the Boolean subobjects of that convex space. The resulting {\sigma}- algebra of that measurable space makes it a separated measurable space. These properties are used to give a proof that the category of Giry algebras is equivalent to the category of convex spaces.