RE: stacks (was: size_question_encore)

[Marta Bunge <martabunge@hotmail.com>]

Dear Mike,

As for stacks being the primary motivation Makkai had for anafunctors, that is not the impression I got from people attending his course, as they not only expressed interest in this comment of mine, but also someone suggested reducing the theory of stacks to that of anafunctors. In fact, I was told by another that the introduction of anafunctors was quite different  from that of stacks, as it had more to do with expressing anafunctors in  FOLDS. Since you are interested in this, why don't you try getting it directly from the horse's mouth? 

The answer to your other question (Are there any interesting non-Grothendieck elementary toposes whichare known to satisfy the axiom of stack completions?) is still open. Bob Pare and I investigated this question a long time ago (more than 30 years ago, actually) and in our attempts to formulate  this axiom, all we could say was that every Grothedieck topos satisfied it. I forgot to mention that an elementary formulation of ASC ("axiom of stack completions") is till missing. All I mean by it at the moment is that it holds for an elementary topos S if, for any category C in S, the stack completion \tilde([C]) of the fibration [C] over S which is the externalization of C, which is explicilty constructed, is equivalent to the externalization [\tilde(C)] of a category \tilde(C) in S. 

Regards,Marta 




> Date: Tue, 12 Jul 2011 07:33:46 -0700
> Subject: Re: categories: RE: stacks (was: size_question_encore)
> From: mshulman@ucsd.edu
> To: martabunge@hotmail.com
> CC: david.roberts@adelaide.edu.au; joyal.andre@uqam.ca; categories@mta.ca
> 
> 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
  		 	   		  

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