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Generalized topological covering systems on quantum events’ structures. (English) Zbl 1100.81002

The author gives a sheaf-theoretic view on the foundational problems of quantum mechanics. Various algebraic and categorical concepts are interpreted in this context. Part of the interpretation may already be read off the given definitions: The author calls quantum events algebra any orthomodular orthoposet $L$. Mappings $\varphi :K\to L$ between such algebras $K$ and $L$, which respect the unit 1, the orthocomplementation $☆$, the partial order, and have the property that joining of orthogonal elements is submonotone, are called quantum algebraic homomorphisms. Thus, objects and morphisms of a category $ℒ$ are given which is called quantum events structure by the author. Boolean algebras $B$ with their usual morphisms constitute a category $ℬ$ which in this context is called a Boolean events structure.

Assuming that $ℬ$ is a generating subcategory of $ℒ$ it is shown that a quantum events algebra $L$ may be represented by a sheaf for a suitable chosen Grothendieck topology $𝐉$ on $ℬ$. The main result is that a quantum events algebra $L$ is isomorphic to a colimit in the category of elements of the sheaf functor of Boolean frames of $L$.

MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic 81P15 Quantum measurement theory 06C15 Complemented lattices, etc. 18F20 Categorical methods in sheaf theory 18F10 Grothendieck topologies 03G30 Categorical logic, topoi