Dear Thomas and Mike On 05/10/2010, at 9:42 PM, Thomas Streicher wrote:
Mike Shulman has clarified all my questions about weak fibrations. A useful account of weak fibrations can now be found at ncatlab.org/nlab/show/Street+fibration. It explains all issues raised and makes no exaggerated claims about the usefulness of the concept.
There is an aspect of this discussion which I do not think has been covered. There is a theorem about the weak fibrations which was a reason for bothering with them. It is about the construction of V-Mod (or V-Dist or V-Prof or V-Bimod) from V-Cat as described in "Fibrations in bicategories". Since we are constructing a bicategory V-Mod, it seems reasonable that we should only regard V-Cat as a bicategory even though it is a 2-category. Not only did I need to weaken the notion of fibration to include functorial equivalences (however the notion of cartesian arrow remains that of Grothendieck), I also needed the two-sided version (a span), and I needed the dual: bidiscrete 2- sided weak cofibrations in V-Cat. Remarkably to me at the time, these are precisely the two-sided modules between V-categories (up to appropriate equivalence). For the case V = Set, bidiscrete 2-sided fibrations in Cat give the modules too. Sorry if this is going over old ground. The introduction to "Fibrations in bicategories" was meant to explain this. This is an entirely different use of fibrations than doing parametrized (locally internal, indexed) category theory in a topos. Best wishes, Ross PS The paper with A. Carboni, S. Johnson and D. Verity called: Modulated bicategories, J. Pure Appl. Algebra 94 (1994) 229-282 continues the story started in "Fibrations in bicategories". [For admin and other information see: http://www.mta.ca/~cat-dist/ ]