TOPOS-THEORETIC CLASSIFICATION OF QUANTUM EVENTS STRUCTURES IN TERMS OF BOOLEAN REFERENCE FRAMES

Abstract

We construct a sheaf-theoretic representation of quantum events structures, in terms of Boolean localization systems. These covering systems are constructed as ideals of structure-preserving morphisms of quantum event algebras from varying Boolean domains, identified with physical contexts of measurement. The modeling sheaf-theoretic scheme is based on the existence of a categorical adjunction between presheaves of Boolean event algebras and quantum event algebras. On the basis of this adjoint correspondence, we also prove the existence of an object of truth values in the category of quantum logics, characterized as subobject classifier. This classifying object plays the equivalent role that the two-valued Boolean truth values object plays in classical events structures. We construct the object of quantum truth values explicitly, and furthermore, demonstrate its functioning for the valuation of propositions in a typical quantum measurement situation.