Exponentiability and single universes

In this paper, we first consider known universes for pairs of opposite noti ons 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 fu nctions/distributions on a topos. Along the way, we (1) prove that, for a f actorization linearly ordered small category B, the category of discrete Gi raud-Conduche fibrations over B is a (model generated) topos, (2) character ize locally closed inclusions in the category Cat of small categories, and (3) investigate "generalized coverings" in topos theory as one of several p ossible single universes for local homeomorphisms and complete spreads over a topos. (C) 2000 Elsevier Science B.V. All rights reserved.