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States of Convex Sets

22 Oct 2014 [ preprint · slides ]

doi 10.1007/978-3-662-46678-0_6
with Bart Jacobs and Bas Westerbaan

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, Eilenberg–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.