Ultrapowers as sheaves on a category of ultrafilters
type: Item
instance of: scholarly article
title: Ultrapowers as sheaves on a category of ultrafilters
zbMATH Open document ID: 1072.03040
author: Jonas Eliasson
published in: Archive for Mathematical Logic
publication date: 2005-02-11T00:00:00Z
review text: The category whose objects are sets equipped with a filter, and whose morphisms are equivalence classes (modulo agreement on a set in the filter) of partial functions `respecting' the given filters, has been studied by several authors including V.~Koubek, J.~Reiterman and A. R.~Blass. I.~Moerdijk observed that the category carries a subcanonical `finite covering' topology, so that, provided we impose a suitable size restriction (a complication which is never mentioned in the present paper), we may study the topos of sheaves on it. In the present paper, the author restricts his attention to the full subcategory whose objects are ultrafilters: he observes that the induced Grothendieck topology makes this into an atomic site in the sense of Barr and Diaconescu, and he studies the logical properties of the resulting atomic topos. (In general, the proofs given are rather long-winded: for example, the author gives a detailed proof that the axiom of choice does not hold in the topos, when all that is needed is to observe that -- being two-valued and not equivalent to the category of sets -- it cannot be localic over sets.)
reviewed by: Peter Johnstone
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 03C20
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18F10
zbMATH DE Number: 2133510
zbMATH Keywords: topos of sheaves
zbMATH Keywords: Grothendieck topology
zbMATH Keywords: atomic site
zbMATH Keywords: category of ultrafilters
MaRDI profile type: Publication
full work available at URL: s00153-004-0228-0
OpenAlex ID: W2023562444
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DOI: 10.1007/S00153-004-0228-0
description: scientific article; zbMATH DE number 2133510