RE: stacks (was: size_question_encore)

[Michael Shulman <mshulman@ucsd.edu>]

Is the "axiom of stack completions" related to the "axiom of small
cardinality selection" used by Makkai to prove that the bicategory of
anafunctors is cartesian closed?  I think I recall a remark in
Makkai's paper to the effect that the stack completion of a category C
is at least morally the same as the category Ana(1,C) of "ana-objects"
of C.

Are there known examples of elementary toposes which violate the axiom
of stack completions?

On Mon, Jul 11, 2011 at 5:32 AM, Marta Bunge <martabunge@hotmail.com> wrote:
> Concerning size matters, let me observe
> that my construction of the stack completion (Bunge, Cahiers 1979) is
> meaningful regardless of size questions, that is, for any base topos S.  The
> outcome, however, of applying it to an internal category need no longer  be
> internal. For this reason I introduce an "axiom of stack completions"
> which guarantees that stack completions of internal categories be again
> internal,and which is satisfied by any S a Grothehdieck topos. The question of
> stating such an axiom as an additional axiom to the ones for elementary toposes
> was proposed as a problem by Lawvere in his Montreal lectures in 1974.


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