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 generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again generalising 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 formalised, 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.