Dear Colleagues, A preprint, whose abstract follows, is available in compressed .dvi form (for DOS and for UNIX) from: http://www.emu.edu.tr/academic/facartsc/mathsdep/staffpic/jfunk.htm or if you are browsing the web, click on academics, teaching staff, Mathematics, Jonathon Funk, additional information, after you have reached the EMU homepage http://www.emu.edu.tr If you would like a copy, but are unable to retrieve the preprint, please don't hestitate to contact me, as I would be happy to send you the .dvi file personally. funk@mozart.emu.edu.tr ---------------------------------------------------------------- ``On branched covers in topos theory'' Abstract: We present some new findings conerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. We also have some new results concerning the general theory of KZ-doctrines, such as the the closure under composition of discrete fibrations for a KZ- doctrine (in the sense of Bunge/Funk, ``On a bicomma object condition for KZ-doctrines''). Regards, Jonathon Funk Jonathon Funk Department of Mathematics Eastern Mediterranean University Gazimagusa Turkish Republic of North Cyprus via Mersin 10, Turkey tel: (90) 392 366 6588, Ext: 1227, 1228, 1138 fax: (90) 392 366 1604
The following reprint is available at http://www1.union.edu/~niefiels/ESU.ps http://www1.union.edu/~niefiels/ESU.dvi EXPONENTIABILITY AND SINGLE UNIVERSES by Marta BUNGE and Susan NIEFIELD ABSTRACT - The search for suitable single universes for opposite or dual pairs of notions (such as those of discrete fibration and discrete opfibration, or of open and closed inclusions, or of functions and distributions on a Grothendieck topos) leads naturally to exponentiability. Using exponentiability techniques, such as model-generated categories and glueing, we settle a standing conjecture and an open problem. The conjecture, due to F. Lamarche, states that for a small category B, the category of unique factorization liftings (also known as discrete Conduche fibrations) over B is a topos. We also construct the smallest topos containing the local homeomorphisms (functions) and the complete spreads (distributions) over any given topos satisfying a certain condition (true of presheaf toposes). This solves a problem posed by F. W. Lawvere. Along the way, we introduce two new sorts of geometric morphisms, characterize locally closed inclusions in Cat, and investigate new features of generalized coverings in topos theory, such as branched coverings, cuts, and complete spreads.
The paper "Exponentiablity and Single Universes" by Marta Bunge and Susan Niefield, recently announced on the site ww1.union.edu/~niefiels has been temporarily withdrawn. A revised version will be posted soon. The paper contained an erroneous result - namely, that for an arbitrary small category B, the category UFL/B of Giraud-Conduche fibrations over B is a topos. A counterexample has been found by Peter Johnstone.
A revised version of the following reprint is available at http://www1.union.edu/~niefiels/ESU.ps http://www1.union.edu/~niefiels/ESU.dvi EXPONENTIABILITY AND SINGLE UNIVERSES by Marta BUNGE and Susan NIEFIELD ABSTRACT - In this paper, we first consider known universes for pairs of opposite notions such as those of discrete fibrations/discrete opfibrations and of open/closed locale inclusions, and then extrapolate these in order to introduce new single universes for open/closed inclusions of subcategories and for functions/distributions on a topos. A key factor that these notions have in common is exponentiability in the ambient category. Along the way, we (1) prove that, for a factorization linearly ordered small category B, the category of discrete Giraud-Conduche fibrations over B is a (model generated) topos, (2) characterize locally closed inclusions in the category Cat of small categories, (3) investigate ``generalized coverings'' in topos theory, including branched coverings, cuts, and complete spreads, and (4) examine the preservations of exponetiability under the passage from Cat/B to the category of Grothendieck toposes over the presheaves PB.
participants (2)
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JONATHON FUNK -
Susan Niefield