Bas Spitters (Université Paris-Sud/INRIA Saclay) |
Steven Vickers (School of Computer Science, University of Birmingham) |
Sander Wolters (Radboud University Nijmegen, IMAPP) |

In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra, A, a topos T(A) of sheaves on a locale and a commutative C*-algebra, a, within that topos. The Gelfand spectrum of a is a locale S in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale S, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise.
As a by-product of the geometricity of Gelfand spectra, we find an explicit external description of the spectrum whenever the topos is a functor category. As an intermediate result we show that locally perfect maps compose, so that the externalization of a locally compact locale in a topos of sheaves over a locally compact locale is locally compact, too. |

Published: 29th July 2014.

ArXived at: http://dx.doi.org/10.4204/EPTCS.158.7 | bibtex | |

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