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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