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)
148 Downloads (Pure)


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
Number of pages31
Publication statusPublished - 2014

Publication series

ISSN (Electronic)2075-2180

Bibliographical note

See arXiv:1310.0705


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