Gelfand spectra in Grothendieck toposes using geometric mathematics

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.

In Ross Duncan and Prakash Panangaden: Proceedings 9th Workshop on Quantum Physics and Logic (QPL 2012), Brussels, Belgium, 10-12 October 2012, Electronic Proceedings in Theoretical Computer Science 158, pp. 77–107.
Published: 29th July 2014.

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