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Exponentiability and single universes. (English) Zbl 0947.18006

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.

MSC:

18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18B25 Topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
54C35 Function spaces in general topology
54D45 Local compactness, \(\sigma\)-compactness
57M12 Low-dimensional topology of special (e.g., branched) coverings
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