Re: Not invariant but good

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

On Tue, Sep 28, 2010 at 3:18 AM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> The notion of Grothendieck fibration is a property
> of functors and not an additional structure. However, the notion of fibration
> in a(n abstract) 2-category can be formulated only postulating a certain kind
> of structure which, however, is unique up to canonical isomorphism. But this
> amounts to defining Grothendieck fibrations in terms of cleavages (which
> certainly are all canonically isomorphic). But choosing cleavages amounts
> to accepting very strong choice principles which is maybe no real problem
> but at least aesthetically moderately pleasing.

I'm not sure what you mean here.  The notion of fibration in a
2-category can be defined as a property if you like: a morphism E -->
B in a 2-category K is a fibration if all the induced functors K(X,E)
--> K(X,B) are fibrations and all commutative squares induced by
morphisms X --> X' are morphisms of fibrations (preserve cartesian
arrows).  This is equivalent to giving some structure on E --> B, but
that structure is unique up to unique isomorphism when it exists.

(One can argue that both ordinary fibrations and fibrations in a
2-category are actually "property-like structures," or "properties
that are not necessarily preserved by morphisms," since their
forgetful functors are pseudomonic but not full.  But that applies
equally to both.)

It is true that if one has a Grothendieck fibration between categories
in the ordinary sense, then it only becomes an internal fibration in
the naively defined 2-category Cat if we have the axiom of choice,
since the latter amounts to saying that we can simultaneously choose
cartesian liftings for any families of objects of E and morphisms of B
we might want to pull them back along.  (I don't think any "global
choice" is necessary, since the definition doesn't require us to make
such a choice simultaneously for every possible family--only that for
any particular family, we -could- make such a choice.)

However, in the absence of the axiom of choice, the naive definition
of "functor" is not very well-behaved; it's better to use
"anafunctors," or equivalently to invert the weak equivalences
(fully-faithful and essentially surjective functors) in the naively
defined 2-category Cat.  In the resulting bicategory, I think any
ordinary Grothendieck fibration will indeed be an internal fibration,
without any need for choice.  (Of course, "internal fibration" should
probably now be interpreted in the looser sense of Street, as
appropriate when working in a non-strict 2-category.)

Mike


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