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After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state of the von Neumann algebra of physical quantities of a quantum system determines a certain measure on the spectral presheaf of the system.
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G. Birkhoff and J. von Neumann · 1936
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A.M. Gleason · 1957
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Sheaves in Geometry and Logic: A First Introduction to Topos Theory
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Noncommutative Geometry
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C.J. Isham and J. Butterfield · 1998
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Categories for the Working Mathematician
S. MacLane · 1998
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A topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category
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Some possible roles for topos theory in quantum theory and quantum gravity
C.J. Isham and J. Butterfield · 2000
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Space-time and the philosophical challenge of quantum gravity
C.J. Isham and J. Butterfield · 2001
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A topos perspective on the Kochen-Specker theorem: IV. Interval valuations
C.J. Isham and J. Butterfield · 2002
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A sheaf theoretic approach to measure theory
M. Jackson · 2006
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T. Coquand, B. Spitters · 2008
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Topos theory and ‘neo-realist’ quantum theory
A. Döring · 2008
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A topos foundation for theories of physics: I. Formal languages for physics,
A. Döring and C.J. Isham · 2008
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A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory
A. Döring and C.J. Isham · 2008
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A topos foundation for theories of physics: IV. Categories of systems
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M. Takesaki · 2002
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Kochen-Specker theorem for von Neumann algebras
A. Döring · 2005
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Observables I: Stone spectra
H.F. de Groote · 2005
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A globalisation of the Gelfand duality theorem
B. Banaschewski and C.J. Mulvey · 2006
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A topos foundation for theories of physics: III. Quantum theory and the representation of physical quantities with arrows δ ˘ ( A ^ ) : → ℝ ⪰ ¯ \breve{\delta}(\hat{A}):\rightarrow\underline{\mathbb{R}^{\succeq}}
A. Döring and C.J. Isham
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‘What is a Thing?’: Topos Theory in the Foundations of Physics
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