On Tue, Jul 12, 2011 at 5:30 AM, Marta Bunge <martabunge@hotmail.com> wrote:
I am glad that Makkai is now aware of this fact, which gives a universal flavor to his subject, whatever "morally" means.
The paper I was referring to is the one that first introduced anafunctors, so I think he's been aware of it since the beginning (I suspect it was a primary motivation, even). The word "morally" was my own weasel word, to cover the fact that I didn't have time to look up the paper and remind myself what precisely he actually wrote. (-:
As for there being an example of an elementary topos which does not satisfy the "axiom of stack completions", Joyal gave one long ago and Lawvere mentioned it in his 1974 Montreal lectures. Take a group G with a proper class of subgroups having a small index in G. The topos [G, Sets] is an example.
Ah, thanks. That makes sense. The question about the effective topos is also intriguing! Are there any interesting non-Grothendieck elementary toposes which are known to satisfy the axiom of stack completions? (By "interesting" I mean to exclude toposes such as the category of sets smaller than some strong limit cardinal -- not to say that such toposes are not interesting for other purposes.) Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]