Gelfand spectra in Grothendieck toposes using geometric mathematics

• Main Authors: Spitters, Bas, Vickers, Steven, Wolters, Sander
• Format: Journal Article
• Language: English
• Published: 01-08-2014
• Subjects: Mathematics - Category Theory; Mathematics - Mathematical Physics; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1310.0705

EPTCS 158, 2014, pp. 77-107 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.