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
Given a fibration P: Flat(F) -> B, assuming B is complete then (E is complete and P continuous) iff P is fibred complete (complete fibres and continuous reindexing) . Dually for cofbirations (coming from 'covariant' indexed categories) and cocompleteness. I am clueless about 'adhesive categories' but Google points out to the following def: C adhesive means it has pullbacks, pushout along monos and these latter satisfy a certain exactness condition involving pullback stability (a so-called "VK square" which is actually a cube(?)). Assuming P: Flat(F) -> B preserves monos, an arrow in Flat(F) is monic iff its image in B and its vertical factor are monic. Ergo, assume: B adhesive P has direct images along monos (m^* have left adjoints) Then, (P has fibrewise pb/po along monos) implies (Flat(F) has same and P preserves that).. The VK condition is a little obtruse, and at first sight requires the following additional assumption: the ' po along monos' in Flat(F) preserves cartesian morphisms and reindexing functors preserve po along monos. Then it seems that one gets (if one cares to do the relevant calculations) P fibrewise adhesive iff Flat(P) adhesive (for the only if, given a stack of 2 cubes where the bottom one has all side faces pb, the total cube is a VK square iff the top one is such). References: On the correspondence of limits/colimits between fibres and total Flat(F) see: - Gray, J. W., Fibred and Cofibred categories, Proceedings of the Conference on Categorical Algebra,1966. There are more recent references which treat this subject (fibred vs. global properties) from a more abstract point of view (and it a broader context), but they are probably not the first place to look if one is not acquainted with the basics. 26-Nov-2004 14:46:11 -0400,2911;000000000001-00000000