****************************************** Alexey Cherchago escribió:
Dear Cat Community,
I have the following question about preservation of properties of fiber categories in the total category:
Given a contravariant indexed category F : B^Op -> Cat, and the Grothendieck translation from the indexed category to fibrations Flat(F). Then, the first projection P : Flat(F) -> B forms a (split) fibration, called the flattening of F.
Now assume that the fiber categories are complete/cocomplete/adhesive. Would it be the case that the total category Flat(F) of the described (split) fibration is also complete/cocomplete/adhesive?
Any references would be appreciated. Thank you kindly.
Best regards, Alexey Cherchago
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 categoy \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. So, if you have a bifibered category (fibered and cofibered), with base and fibers complete and cocomplete, then the total category is complete and cocomplete (and I don't know anything about "adhesive"). Computations are easy and you can find them, for instance, in section 3 of Model category structures in bifibred categories JPAA 95, (1994), 203 - 223 But, as I said, I'm not an expert: only a user of fibered categories. So I would also appreciate some references for Flat(F) : is this the fibered category associated to F in the sense of SGA1, or am I missing something? Also another question about fibered categories: has someone developped the notion of fibered 2-category? 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? -- Agustí Roig Martí Universitat Politècnica de Catalunya Dept. Matemàtica Aplicada I, ETSEIB - FME Diagonal 647 08028 Barcelona