Re: Fibrations in a 2-category

[David Roberts <droberts@maths.adelaide.edu.au>]

Hi Michal,

it is not *always* false. Consider the topoi Set and Set_choice, where
the first is the category of sets without choice and the second is
with choice. Then the bicategory of categories, anafunctors and
transformations in Set is equivalent (assuming choice in the
metalogic) to the 2-category of categories, functors and natural
transformations in Set_choice. This is (essentially) shown by Makkai
in his original anafunctors paper.

However, I doubt that it is always true (only a hunch). Also, one does
not need a topos as an ambient category in which to define
anafunctors, only a site where the Grothendieck pretopology is
subcanonical and singleton (single maps as covering families). The
topos case is when you take the regular pretopology.

And although you did not ask for a reference, here's one:

http://arxiv.org/abs/1101.2363

which builds on internal anafunctors introduced here

http://arxiv.org/abs/math.CT/0410328

and Makkai's original paper is available in parts from here:

http://www.math.mcgill.ca/makkai/anafun/

David

On 15 January 2011 09:14, Michal Przybylek <michal.przybylek@gmail.com> wrote:
> On Fri, Jan 14, 2011 at 12:02 AM, Michael Shulman <mshulman@ucsd.edu> wrote:
>
>> One way to deal with the difficulty you mention is by using
>> "anafunctors," which were introduced by Makkai precisely in order to
>> avoid the use of AC in category theory.
>
> [...]
>
> Interesting. But before I ask for references on ``anafunctors'' I
> would like to know the following - is it false that for any (say)
> topos T there exists a category C whose 2-category of internal
> categories, functors, and natural transformations is (weakly)
> equivalent to the bicategory Cat_ana(T)?
>
>
> Best,
> MRP


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