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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 byproduct 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 byproduct 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 

Title of host publication  Proceedings 9th Workshop on Quantum Physics and Logic (QPL2012) 
Editors  Ross Duncan, Prakash Panangaden 
Publisher  Open Publishing Association 
Pages  77107 
Number of pages  31 
Volume  158 
DOIs  
Publication status  Published  2014 
Publication series
Name  EPTCS 

Volume  158 
ISSN (Electronic)  20752180 
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

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