Dichotomy between Deterministic and Probabilistic Models in Countably Additive Effectus Theory

Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any effect monoid. Hence, there are quite exotic effectuses resulting from more pathological effect monoids.In this paper we introduce a-effectuses, where certain countable sums of morphisms are defined. We study in particular a-effectuses where unnormalized states can be normalized. We show that a non-trivial a-effectus with normalization has as scalars either the two-element effect monoid {0, 1} or the real unit interval [0, 1]. When states and/or predicates separate the morphisms we find that in the {0, 1} case the category must embed into the category of sets and partial functions (and hence the category of Boolean algebras), showing that it implements a deterministic model, while in the [0, 1] case we find it embeds into the category of Banach order-unit spaces and of Banach pre-base-norm spaces (satisfying additional properties), recovering the structure present in GPTs.Hence, from abstract categorical and operational considerations we find a dichotomy between deterministic and convex probabilistic models of physical theories.