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