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.