The answer to the question is obviously NO unless one makes much stronger assumptions on F and B. Take for F the constant functor with value the terminal category 1, the category denoted by Flat(f) is then (canonically isomorphic to) B, whereas the fibers have all completeness properties one can desire. Le jeudi, 25 nov 2004, à 11:35 Europe/Paris, Alexey Cherchago a écrit :
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
26-Nov-2004 14:26:49 -0400,3309;000000000000-00000000