In SGA I.6 Grothendieck and Pierre Gabriel presented the formalism of fibered categories and descent; the exposition and the terminology emphasises a version of the relative point of view for categories. Thus they start with preliminaries over the slice 2-category of functors with fixed target category . Objects in are thus called categories over and the morphisms are called functors over . Standard passages between different slice categories are via the base change and the cobase change 2-functors.
An important sub-2-category of is the 2-category of fibered categories over ; this is not a full sub-2-category: one restricts only to cartesian functors among fibered categories. The 2-categories are fibers in a codomain 2-fibration over .
Mike Shulman: Is that really a 2-fibration? You can pull back along along a functor , but can you do anything with a natural transformation ?
Zoran: sorry there is indeed an error there, but I will not yet erase it, as there is (in my memory) a partial repair of the statement for which I need to sit down to formulate precisely.
Its restriction to the sub-2-category of fibered categories is then also a Grothendieck 2-fibration which has been studied by Claudio Hermida.
A. Grothendieck, M. Raynaud et al. Revêtements étales et groupe fondamental (SGA I), Lecture Notes in Mathematics 224, Springer 1971 (retyped as math.AG/0206203)
C. Hermida, Some properties of as a fibred 2-category, Journal of Pure and Applied Algebra 134 (1), 83–109, 1999 (earlier draft version can be found at jpaa.ps.gz)
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