  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.

Category Theory;Probability