categories: Re: pullbacks of toposes

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Tue, 13 Nov 2001, Steve Lack wrote:

> Can anyone tell me what is known about
> the existence of pullbacks in the 2-category
> of elementary toposes, geometric morphisms,
> and natural transformations? I know (from
> Peter Johnstone's book) that pullbacks along
> bounded morphisms exist.
>
> (I presume that when I say pullback I really
> mean bipullback, but if I should mean something
> else then do please do let me know!)
>
> Steve Lack.
>
As far as I know, the position is this. The (bi)pullback of two
morphisms f and g exists if either f or g is bounded. It seems
very likely that some such restriction is necessary, but as far
as I'm aware nobody has constructed an example of a pair of
morphisms for which the pullback is definitely known not to exist
(though there are pairs for which the pullback is not known to exist).
If anyone has such an example, please let me know!

Peter Johnstone