Re: reverting religious terminology

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

On Mon, Oct 4, 2010 at 12:38 PM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> From http://ncatlab.org/nlab/show/Grothendieck+fibration I take that weak
> fibrations are necessary only when considering fibrations in bicategories
> that are not 2-categories.

Yes, the only reasons I know of for caring about weak fibrations are
(1) if you're in a bicategory, or even just in a strict 2-category
which lacks the property that weak fibrations can be strictified to
strict ones, and (2) to assuage any worries (ideological or otherwise)
one might have about the notion of strict fibration not being
covariant under equivalence.

One bicategory which comes to mind which is not a strict 2-category,
and in which I would certainly want to think about internal
fibrations, is the bicategory of internal categories and anafunctors
in some topos.

> Why not take the paradigmatic case of the bicategory Dist of
> distributors. Has this exampe been worked out in detail.

I don't know; I certainly haven't seen it done.  If it hasn't been
done, someone should work it out; it might be interesting.

Mike


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