Dear Michael, I remind you that it was Benabou who observed (and proved) the equivalence between anafunctors and representable distributors, reproduced in the nLab as "folklore", from which I concluded (without any knowledge of anafunctors) that the stack completion of a category C in S represents anafunctors with target C. I am glad that Makkai is now aware of this fact, which gives a universal flavor to his subject, whatever "morally" means. I know nothing about the "axiom of cardinal selection". As for there being an example of an elementaru topos which does not satisfy the "axiom of stack compleitons", Joyal gave one lomg 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. Also, so far as I know, it is not yet known (Hyland, Robinson, and Rosolini,"The discrete objects in the effective topos", Proc. London Math. Soc. (3) 60 (1990, 1-36)) whether the full internal subcategory Q on the subquotients of N in Eff (the effective topos) has an internal stack completion. The stack completion is identified as Orth(Delta 2), families of discrete objects. Regards,Marta
Date: Mon, 11 Jul 2011 18:20:42 -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
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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