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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.
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 language | English |
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Title of host publication | Proceedings 9th Workshop on Quantum Physics and Logic (QPL2012) |
Editors | Ross Duncan, Prakash Panangaden |
Publisher | Open Publishing Association |
Pages | 77-107 |
Number of pages | 31 |
Volume | 158 |
DOIs | |
Publication status | Published - 2014 |
Publication series
Name | EPTCS |
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Volume | 158 |
ISSN (Electronic) | 2075-2180 |
Bibliographical note
See arXiv:1310.0705Fingerprint
Dive into the research topics of 'Gelfand Spectra in Grothendieck Toposes using Geometric Mathematics'. Together they form a unique fingerprint.Projects
- 1 Finished
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Applications of geometric logic to topos approaches to quantum theory
Vickers, S.
Engineering & Physical Science Research Council
1/09/09 → 31/08/12
Project: Research Councils