Wondering if anyone here who knows some topos theory can help me out with something that's come up in my research lately. For (p : E \to B) a fibration, I'm interested in defining some additional structure on (p) which is equivalent to a pseudofunctor (P : B^{op} \to \mathbf{Site}), where (\mathbf{Site}) is the 2-category of categories equipped with coverages (i.e. sites) and morphisms of sites (i.e. functors which are covering-flat and preserve covering families) between them, such that postcomposing (P) with the forgetful 2-functor (\mathbf{Site} \to \mathbf{Cat}) yields the usual pseudofunctor given by taking fibres of (p). Essentially, such a structure should equip each fibre of (p) with a coverage in a manner compatible with the structure of (p) as a fibration. I'm wondering if anyone else has considered these sorts of things before, and if they have a name.
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