preprint available

[Susan Niefield <niefiels@union.edu>]

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.