UPPSALA DISSERTATIONS IN MATHEMATICS 30 Ultrasheaves

Eliasson, J. 2003: Ultrasheaves. Uppsala dissertations in Mathematics 30. 54 pp. Uppsala. ISBN 91-506-1716-8. This thesis treats ultrasheaves, sheaves on the category of ultrafilters. In the classical theory of ultrapowers, you start with an ultrafilter (I,U) and, given a structure S, you construct the ultrapower S/U . The fundamental result is ! Loś’s theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure S. In this thesis we instead start with the category of ultrafilters (denoted U). On this category we build the topos Sh(U) of sheaves on U (the ultrasheaves), which we think of as generalized ultrapowers. The theorem for ultrasheaves corresponding to ! Loś’s theorem is Moerdijk’s theorem, first proved by Moerdijk for the topos Sh(F) of sheaves on filters. In the thesis we prove that ! Loś’s theorem follows from Moerdijk’s theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk’s topos Sh(F) and prove that Sh(U) is the double negation subtopos of Sh(F). 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 U. 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), a nonstandard set theory, in the ultrasheaves. IST, introduced by Nelson, is an axiomatic approach to nonstandard mathematics.