## Categorical Foundations of Quantum Logics and Their Truth Values Structures (2004)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Zafiris04categoricalfoundations,

author = {Elias Zafiris},

title = {Categorical Foundations of Quantum Logics and Their Truth Values Structures},

year = {2004}

}

### OpenURL

### Abstract

We introduce a foundational sheaf theoretical scheme for the comprehension of quantum event structures, in terms of localization systems consisting of Boolean coordinatization coverings induced by measurement. The 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 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 event structures. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of quantum systems.

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Citation Context ...omC(X, Y ) × HomC(Y, Z) → HomC(X, Z) For morphisms g ∈ HomC(X, Y ), h ∈ HomC(Y, Z) the image of the pair (g, h) is called the composition; it is denoted h◦g. The composition operation is associative. =-=[4]-=-. For any f ∈ HomC(X, Y ) we have idY ◦ f = f ◦ idX = f. For an arbitrary category C the opposite category C op is defined in the following way: the objects are the same, but HomC op(X, Y ) = HomC(Y, ... |

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Citation Context ... this paper we will avoid to mention Grothendieck topologies explicitly in order to avoid unnecessary technical complications in the exposition of the arguments, but the interested reader may consult =-=[10]-=- for details. The category of sheaves is a topos, and consequently, comes naturally equipped with an object of generalized truth values, called subobject classifier. This object of truth values, being... |