****************************************** 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
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
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
participants (3)
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Agusti Roig -
Claudio Hermida -
Thomas Streicher