Exponentiability and single universes

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-Conduché fibrations over B is a (model generated) topos, (2) characterize locally closed inclusions in the category Cat of small categories, and (3) investigate “generalized coverings” in topos theory, as one of several possible single universes for local homeomorphisms and complete spreads over a topos.