Exponentiability and single universes
The phrase `single universe' refers to the idea that structures of `opposite handedness' on a given base (such as functions and distributions on a space, or covariant and contravariant set-valued functors on a category) can often be regarded as special cases of the objects of a single category; thus covariant and contravariant functors on a category \(\mathbb B\), if regarded as discrete opfibrations and fibrations over \(\mathbb B\), form full subcategories of \({\mathcal {C}at}/{\mathbb B}\). However, the latter is too large to be seen as a common generalization of \([{\mathbb B},{\mathcal {S}et}]\) and \([{\mathbb B}^{\text{op}},{\mathcal {S}et}]\), the authors observe that cutting down to a suitable subcategory of the exponentiable objects of \({\mathcal {C}at}/{\mathbb B}\), consisting of the discrete Conduché fibrations (here called unique factorization lifting functors, or UFL functors for short) yields a better `single universe'. In particular, they show that for a suitable class of categories \(\mathbb B\) the resulting category \({\mathcal {UFL}}/{\mathbb B}\) is a topos. The main thrust of the paper, however, is concerned with `lifting' these ideas to toposes and geometric morphisms: they introduce a notion of `\({\mathcal {UFL}}\) geometric morphism' and show that it serves as a single universe for local homeomorphisms and complete spreads. However, not all complete spreads over a topos \(\mathbb E\) are exponentiable as objects of \({\mathcal {T}op}/{\mathbb E}\); they also introduce a candidate for a single universe for local homeomorphisms and exponentiable complete spreads.
scientific article; zbMATH DE number 1463288