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.