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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 ϕ:KL 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 are given which is called quantum events structure by the author. Boolean algebras B with their usual morphisms constitute a category which in this context is called a Boolean events structure.

Assuming that is a generating subcategory of it is shown that a quantum events algebra L may be represented by a sheaf for a suitable chosen Grothendieck topology 𝐉 on . 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:
81P10Logical foundations of quantum mechanics; quantum logic
81P15Quantum measurement theory
06C15Complemented lattices, etc.
18F20Categorical methods in sheaf theory
18F10Grothendieck topologies
03G30Categorical logic, topoi