Re: Not invariant but good

[Richard Garner <richard.garner@mq.edu.au>]

> 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.)

Though we can always make such a simultaneous choice as soon as we
have it in one particular case. Given the Grothendieck fibration p:
E->B in Cat, and letting X denote the comma object (B,p), it is enough
to choose a lifting for the X-indexed family of morphisms of B
corresponding to the projection X --> B^2 at the X-indexed family of
objects of E corresponding to the projection X --> E; for then to give
a Y-indexed family of morphisms in B and a Y-indexed family of objects
of E over their codomains is to give a morphism Y -> X, and so the
chosen lifting for the latter induces a chosen lifting for the former.

Richard


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