2013/10/31 by Bas Spitters, Steven Vickers, Sander Wolters
Mathematics · Physics and Astronomy · #math.CT #math-ph #math.MP
paper · pdf · doi:10.4204/eptcs.158.7
published as EPTCS 158, 2014, pp. 77-107 · In Proceedings QPL 2012, arXiv:1407.8427
arxiv created 2014/08/01 · arxiv updated 2014/08/04
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.