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Ultrasheaves

This thesis treats ultrasheaves, sheaves on the category of ultrafilters. In the classical theory of ultrapowers, you start with an ultrafilter and, given a structure, you construct the ultrapower of the structure over the ultrafilter. The fundamental result is Los's theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure. In this thesis we instead start with the category of ultrafilters (denoted <b>U</b>). On this category <b>U</b> we build the topos of sheaves on <b>U</b> (the ultrasheaves), which we think of as generalized ultrapowers. The theorem for ultrapowers corresponding to Los's theorem is Moerdijk's theorem, first proved by Moerdijk for the topos Sh(<b>F</b>) of sheaves on filters. In the thesis we prove that Los's theorem follows from Moerdijk's theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk's topos Sh(<b>F</b>) and prove that Sh(<b>U</b>) is the double negation subtopos of Sh(<b>F</b>). The connection between ultrapowers and ultrasheaves is investigated in detail. We also prove some model theoretic results for ultrasheaves, for instance we prove that they are saturated models. The Rudin-Keisler ordering is a tool used in set theory to study ultrafilters. It has a strong relationship to the category <b>U</b>. Blass has given a model theoretic characterization of this ordering and in the thesis we give a new proof of his result. One common use of ultrapowers is to give non-standard models. In the thesis we prove that you can model internal set theory (IST) in the ultrasheaves. IST, introduced by Nelson, is a non-standard set theory, an axiomatic approach to non-standard mathematics.

Identiferoai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-3762
Date January 2003
CreatorsEliasson, Jonas
PublisherUppsala universitet, Matematiska institutionen, Uppsala : Matematiska institutionen
Source SetsDiVA Archive at Upsalla University
LanguageEnglish
Detected LanguageEnglish
TypeDoctoral thesis, comprehensive summary, info:eu-repo/semantics/doctoralThesis, text
Formatapplication/pdf
Rightsinfo:eu-repo/semantics/openAccess
RelationUppsala Dissertations in Mathematics, 1401-2049 ; 30

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