preprint available

preprint available

[Susan Niefield]

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. 

withdrawal of preprint

[Susan Niefield]

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temporarily withdrawn.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.

revised preprint available

[Susan Niefield]

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.