Agusti Roig wrote:
The reason for my question is this: I've encountered the following situation at least four times recently. I have a 2-functor
F : E ---> Cat
and I am interested in the (co)fibered category associated to F . So I forget the 2-structure and I take it:
\pi : A ---> E
Then I realize (and I need, at least in one of the examples) that A has also a "natural" (at least in 3 of the four examples I have in mind) structure of 2-category.
So my question is: Is there any canonical way to build a fibered 2-category from a 2-functor?
I suspect that, in general, the answer should be "no", because in my fourth example the 2-structure of A seems very ad hoc, and has nothing to do with F .
So maybe the right questions are:
- What should be a "fibered 2-category"? - Which extra conditions do you have to impose on F in order to obtain a natural, canoncial, fibered 2-category from it?
The notion of fibered 2-category was introduced in (1) below (2-fibration). The indexed version is a "homomorphism of Gray-categories" F: K^coop -> 2-Cat and is used implicitly as such in (2), which includes a brief discussion of related formulations of 2-fibrations in the groupoidal context. Given a mere 2-functor F::C -> Cat, one surely produces a 'covariant' version of 2-fibration if one does not forget anything. define \int(F) with: objects: (X,x) with X in C and x in FX morphs: (f,g):(X,x) -> (X',x') is f::X ->X' and g:Ff(x) -> x' (in FX') 2-cells: a :(f,g) => (f´,g') is a: f => f' such that g' o Fa(c) = g with the evident forgetful \pi: \int(F) -> C. References: (available from http://slc.math.ist.utl.pt/claudio/publications.html) (1) C. Hermida, {\em Some Properties of Fib as a fibred 2-category\/}, in {\it Journal of Pure and Applied Algebra\/} 134 (1), 83-109, 1999. (2) C. Hermida, {\em Descent on 2-fibrations and 2-regular 2-categories}, to appear in special issue of {\it Applied Categorical Structures} on {\em Descent} (coproceedings of Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, Fields Institute, Toronto, September 23-28, 2002).(\textbf{in print}) Claudio