Gelfand Spectra in Grothendieck Toposes using Geometric Mathematics

Bas Spitters, Steven Vickers, Sander Wolters

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)
123 Downloads (Pure)

Abstract

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.
Original languageEnglish
Title of host publicationProceedings 9th Workshop on Quantum Physics and Logic (QPL2012)
EditorsRoss Duncan, Prakash Panangaden
PublisherOpen Publishing Association
Pages77-107
Number of pages31
Volume158
DOIs
Publication statusPublished - 2014

Publication series

NameEPTCS
Volume158
ISSN (Electronic)2075-2180

Bibliographical note

See arXiv:1310.0705

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