nLab category over a category

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 $Cat/C$ of functors with fixed target category $C$. Objects in $Cat/C$ are thus called categories over $C$ and the morphisms are called functors over $C$. Standard passages between different slice categories are via the base change and the cobase change 2-functors.

An important sub-2-category of $Cat/C$ is the 2-category of fibered categories over $C$; this is not a full sub-2-category: one restricts only to cartesian functors among fibered categories. The 2-categories $Cat/C$ are fibers in a codomain 2-fibration $Cat^2$ over $Cat$.

Mike Shulman: Is that really a 2-fibration? You can pull back along $Cat/C \to Cat/D$ along a functor $f\colon D\to C$, but can you do anything with a natural transformation $f\to f'$?

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 $Fib$ of fibered categories is then also a Grothendieck 2-fibration $Fib\to Cat$ 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 $Fib$ 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)