arXiv - CS - Logic in Computer Science Pub Date : 2020-03-23 , DOI: arxiv-2003.10245
Kenta Cho; Bas Westerbaan; John van de Wetering

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.

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