Re: are fibrations evil?

[Michael Shulman <shulman@math.uchicago.edu>]

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


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