Gelfand spectra in Grothendieck toposes using geometric mathematics

TY - GEN

T1 - Gelfand spectra in Grothendieck toposes using geometric mathematics

AU - Spitters, Bas

AU - Vickers, Steven

AU - Wolters, Sander

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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 Σ in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale Σ, 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.

AB - 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 Σ in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale Σ, 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.

UR - http://www.scopus.com/inward/record.url?scp=84938570031&partnerID=8YFLogxK

U2 - 10.4204/EPTCS.158.7

DO - 10.4204/EPTCS.158.7

M3 - Conference article

AN - SCOPUS:84938570031

SN - 2075-2180

VL - 158

SP - 77

EP - 107

JO - Electronic Proceedings in Theoretical Computer Science, EPTCS

JF - Electronic Proceedings in Theoretical Computer Science, EPTCS

T2 - 9th Workshop on Quantum Physics and Logic, QPL 2012

Y2 - 10 October 2012 through 12 October 2012

ER -