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

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

Mike Shulman: Is that really a 2-fibration? You can pull back along Cat/CCat/DCat/C \to Cat/D along a functor f:DCf\colon D\to C, but can you do anything with a natural transformation fff\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 FibFib of fibered categories is then also a Grothendieck 2-fibration FibCatFib\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 FibFib as a fibred 2-category, Journal of Pure and Applied Algebra 134 (1), 83–109, 1999 (earlier draft version can be found at

Revised on November 14, 2017 15:59:56 by Tim Porter (