Augusti Roig wrote:
I'm not an expert on the field and I don't know if for split fibrations you would have a better result, but as far as I know, if you want the total category A of a fibered category \pi : A ---> E to be complete, you need the fiber categories A_x to be complete, but also the base category E .
If you want also A to be cocomplete, then you need that the fibered category is cofibered and the base and the fibers cocomplete.
You do not need that \pi is also cofibred, it suffices that the reindexing functors \alpha^* (for \alpha in the base) preserve limits. Consider e.g. the category CC of all ordinals with reverse order. Then Fam(CC) fibred over Set is a complete fibration of complete categories but not a cofibration as otherwise each fibre would contain an initial object (the fibre over 0 has an initial object and thus \coprod_\alpha 0 were initial in Fam(CC)(I) for \alpha : 0 -> I). Thomas