Dear Michael, You wrote:

Are there any interesting non-Grothendieck elementary toposes which are known to satisfy the axiom of stack completions?

I do not know if you will find the following example interesting. It is using MacLane's functor ML:Groups--->Sets, where Groups is the category of groups and Sets is the category of sets. The functor was constructed by Mac Lane as an example of a continuous functor which is not representable. (I learned about it from Jiri Adamek) By construction, ML is a product of representable functors Hom(G_n,-):Groups--->Sets where G_n is a simple group of cardinality n for each infinite cardinal n. That G_n exists was proved by group theorists. It is a good exercise to show that the functor ML is taking its values in Sets. I wish to regard ML as a kind of progroup and construct its classifying topos BML. Let me denote by elML the category of elements of the functor ML and let P:elML--->Groups be the canonical functor. The category elML is a complete, since the functor ML is continuous. It is thus alpha-cofiltered for every cardinal alpha. Let B:Groups--->CAT be the contravariant functor which associates to a group G its category of presheaves on G, where CAT is the category of locally small categories. The pseudo-colimit of the composite BP:elML--->CAT is a topos BML. I believe that BML is an example of a non-Grothendieck elementary topos which satisfies the axiom of stack completion. Best, André -------- Message d'origine-------- De: viritrilbia@gmail.com de la part de Michael Shulman Date: mar. 12/07/2011 10:33 À: Marta Bunge Cc: David Roberts; Joyal, André; categories@mta.ca Objet : Re: categories: RE: stacks (was: size_question_encore) 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/ ]