States of Convex Sets.

State spaces in probabilistic and quantum computation are convex sets, that is, Eilenberg-Moore algebras of the distribution monad. This article studies some computationally relevant properties of convex sets. We introduce the term effectus for a category with suitable coproducts (so that predicates, as arrows of the shape X -> 1 + 1, form effect modules, and states, arrows of the shape 1 -> X, form convex sets). One main result is that the category of cancellative convex sets is such an effectus. A second result says that the state functor is a "map of effecti". We also define 'normalisation of states' and show how this property is closed related to conditional probability. This is elaborated in an example of probabilistic Bayesian inference.