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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可数加性效应理论中确定性模型和概率模型之间的二分法

效应理论是一种相对较新的分类逻辑方法，可以看作是广义概率理论 (GPT) 的抽象形式。虽然 GPT 的标量始终是实际单位区间 $[0,1]$，但在效果器中，它们可以形成任何效果幺半群。因此，更多的病理效应幺半群会产生非常奇特的效应。在本文中，我们介绍了 $\sigma$-effectuses，其中定义了某些可数的态射和。我们特别研究了 $\sigma$-effectuses，其中非规范化的状态可以被规范化。我们证明了具有归一化的非平凡 $\sigma$-effectus 具有作为标量的双元素效应幺半群 $\{0,1\}$ 或实际单位区间 $[0,1]$。当状态和/或谓词分离态射时，我们发现在 $\{0, 1\}$ 情况下，类别必须嵌入到集合和偏函数的类别中（因此也是布尔代数的类别），表明它实现了确定性模型，而在 $[0,1]$ 情况下，我们发现它嵌入归入 Banach 阶单元空间和 Banach 前基范数空间（满足附加属性）的类别，恢复 GPT 中存在的结构。因此，从抽象的分类和操作考虑，我们发现物理理论的确定性和凸概率模型之间存在二分法。