On Mon, Oct 4, 2010 at 12:38 PM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:

From http://ncatlab.org/nlab/show/Grothendieck+fibration I take that weak fibrations are necessary only when considering fibrations in bicategories that are not 2-categories.

Yes, the only reasons I know of for caring about weak fibrations are (1) if you're in a bicategory, or even just in a strict 2-category which lacks the property that weak fibrations can be strictified to strict ones, and (2) to assuage any worries (ideological or otherwise) one might have about the notion of strict fibration not being covariant under equivalence. One bicategory which comes to mind which is not a strict 2-category, and in which I would certainly want to think about internal fibrations, is the bicategory of internal categories and anafunctors in some topos.

Why not take the paradigmatic case of the bicategory Dist of distributors. Has this exampe been worked out in detail.

I don't know; I certainly haven't seen it done. If it hasn't been done, someone should work it out; it might be interesting. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]