Convex sets appear as state spaces of computation in an effectus, and form an effectus.

State spaces in probabilistic and quantum computation are convex sets, that is, Eilenbergâ€“Moore algebras of the distribution monad. This article studies some computationally relevant properties of convex sets.

We introduce the term effectus for a base category with suitable coproducts (so that predicates, as arrows of the shape X -> 1 + 1, form effect modules, and states, as 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". As a result conditional states can be defined abstractly. We show how they capture probabilistic Bayesian inference in this setting via an example.