Probabilities, distribution monads, and convex categories

Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalizes key properties of the unit interval [0,1]. Such effect monoids can be used to define a probability distribution monad, again generalizing the situation for [0,1]-probabilities. It will be shown that there are translations back and forth, in the form of an adjunction, between effect monoids and “convex” monads. This convexity property is formalized, both for monads and for categories. In the end, this leads to “triangles of adjunctions” (in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories.