Re: reverting religious terminology

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

On Mon, Oct 4, 2010 at 3:36 AM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> But for such weak fibrations one looses the important property that for every
> u : J -> I one can transport X over I to u^*X over J along u.

As I pointed out in my message to the list on September 16, all that
one has to do to remedy this situation is consider "essential fibers"
rather than strict fibers.  In other words, the notion of "X over I"
is itself evil and needs to be replaced by "X equipped with an
isomorphism from P(X) to I".  The category of all so-equipped Xs is
called the "essential fiber" of P over I, and in a weak fibration
there is indeed a functor u^* from the essential fiber over I to the
essential fiber over J.  In this way, any weak fibration also gives
rise to an indexed category, and the 2-category of weak fibrations is
biequivalent to that of indexed categories (whereas the 2-category of
strict fibrations is strictly 2-equivalent to that of indexed
categories).  Also, if P is a strict Grothendieck fibration (indeed,
an isofibration suffices), then its essential fibers are equivalent to
its strict fibers, so the two constructions are compatible.

Mike


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