  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 sigma-effectuses, where certain countable sums of
morphisms are defined. We study in particular sigma-effectuses where
unnormalized states can be normalized. We show that a non-trivial
sigma-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.
