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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    van den Berg, Benno and Heunen, Chris 2012. No-go theorems for functorial localic spectra of noncommutative rings. Electronic Proceedings in Theoretical Computer Science, Vol. 95, Issue. , p. 21.

    Zafiris, Elias and Karakostas, Vassilios 2013. A Categorial Semantic Representation of Quantum Event Structures. Foundations of Physics, Vol. 43, Issue. 9, p. 1090.

    Spitters, Bas Vickers, Steven and Wolters, Sander 2014. Gelfand spectra in Grothendieck toposes using geometric mathematics. Electronic Proceedings in Theoretical Computer Science, Vol. 158, Issue. , p. 77.

    Schreiber, Urs and Shulman, Michael 2014. Quantum Gauge Field Theory in Cohesive Homotopy Type Theory. Electronic Proceedings in Theoretical Computer Science, Vol. 158, Issue. , p. 109.

    Nakayama, Kunji 2014. Topos quantum theory on quantization-induced sheaves. Journal of Mathematical Physics, Vol. 55, Issue. 10, p. 102103.

    Heunen, Chris 2014. Characterizations of Categories of Commutative C*-Subalgebras. Communications in Mathematical Physics, Vol. 331, Issue. 1, p. 215.

    Döring, Andreas 2015. Spectral presheaves as quantum state spaces. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 373, Issue. 2047, p. 20140247.

    Karakostas, Vassilios and Zafiris, Elias 2017. Contextual semantics in quantum mechanics from a categorical point of view. Synthese, Vol. 194, Issue. 3, p. 847.

    Kornell, Andre 2017. Quantum collections. International Journal of Mathematics, Vol. 28, Issue. 12, p. 1750085.

    de Silva, Nadish and Barbosa, Rui Soares 2019. Contextuality and Noncommutative Geometry in Quantum Mechanics. Communications in Mathematical Physics,

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  • Print publication year: 2011
  • Online publication date: June 2011

6 - Bohrification

from I - Beyond the Hilbert Space Formalism: Category Theory

Summary

Introduction

More than a decade ago, Chris Isham proposed a topos-theoretic approach to quantum mechanics, initially in the context of the Consistent Histories approach, and subsequently (in collaboration with Jeremy Butterfield) in relationship with the Kochen–Specker Theorem [21–23] (see also [20] with John Hamilton). More recently, jointly with Andreas Döring, Isham expanded the topos approach so as to provide a new mathematical foundation for all of physics [38, 39]. One of the most interesting features of their approach is, in our opinion, the so-called Daseinisation map, which should play an important role in determining the empirical content of the formalism.

Over roughly the same period, in an independent development, Bernhard Banaschewski and Chris Mulvey published a series of papers on the extension of Gelfand duality (which in its usual form establishes a categorical duality between unital commutative C*-algebras and compact Hausdorff spaces; see, e.g., [57], [65]) to arbitrary toposes (with natural numbers object) [6–8]. One of the main features of this extension is that the Gelfand spectrum of a commutative C*-algebra is no longer defined as a space but rather as a locale (i.e., a lattice satisfying an infinite distributive law [57]; see also Section 6.2). Briefly, locales describe spaces through their topologies instead of through their points, and the notion of a locale continues to make sense even in the absence of points (whence the alternative name of pointfree topology for the theory of locales).

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  • Online ISBN: 9780511976971
  • Book DOI: https://doi.org/10.1017/CBO9780511976971
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