On Thu, Sep 16, 2010 at 2:47 AM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:

For such a thing the key intuitions are lost as far as I can see (even if I and J are isomorphic the fibre over I may be inhabited whereas the fibre over J is inhabited).

The key necessary change to intuition is that one has to replace "fiber" (itself a non-kosher notion) with "essential fiber". http://ncatlab.org/nlab/show/essential+fiber With this change, all the usual intuitions and facts about fibrations still hold (in corresponding kosher ways).

I doubt that category theory over a base (topos) can be developed this way.

The 2-category of Street fibrations over a given category (such as a topos) is biequivalent to the 2-category of Grothendieck fibrations over that same category, and both are biequivalent to the 2-category of Cat-valued pseudofunctors. (In fact, any Street fibration is equivalent to a Grothendieck fibration, using the same construction which shows that any functor is equivalent to an isofibration; thus the second is a full biequivalent sub-2-category of the first. The second and third are actually strictly 2-equivalent.) Thus, anything kosher that can be done in one can equally be done in the others.

At least it would be very cumbersome. Has the generalised notion of fibration been used for something?

Indeed it would be cumbersome, and unnecessary for most purposes. The only use I know of for Street fibrations is when working internally to a bicategory. Both Street and Grothendieck fibrations can be defined internally to a strict 2-category, and I believe that if the 2-category has some simple strict 2-limits then every Street fibration will be equivalent to a Grothendieck one, just as in Cat. However, since Grothendieck fibrations are non-kosher, their internal definition involves equality of arrows, and hence is not really sensible in a bicategory rather than a strict 2-category. This was Street's original application. I didn't mean to say that Street's kosher fibrations *should* be used in any place where Grothendieck non-kosher ones suffice, or that the latter aren't easier, simpler, more common, and better to use in practice when possible. This is often the case with kosher and non-kosher things, like weak and strict 2-categories, or bilimits and pseudolimits. But in almost all cases where we use non-kosher things, there *exists* an equivalent kosher notion, and occasionally it happens that the equivalence breaks and in that case we have to use the kosher notion instead. The only thing I was objecting to was your conclusion that equality of objects is sometimes "absolutely necessary" -- in this case, as in many others, it's just very convenient. (There are a few situations where equality of objects -- or, in Toby's language, the use of "strict categories" -- does seem to be conceptually fundamental, such as Peter May's Galois theory example. But I don't think fibrations is one of them.) Why should we distinguish between "absolutely necessary" and "very convenient"? For the "working mathematician" perhaps there is no reason to. But I think that for a category theorist developing new categorical concepts, it is a useful heuristic guide -- if a non-kosher concept is not equivalent to some kosher one, then that is a reason to be suspicious of it, if nothing more. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]