RE: stacks (was: size_question_encore)

[Marta Bunge <martabunge@hotmail.com>]

Dear Mike,

Thank you for the information. It would take me a while to check them all.   "Weakly initial covers" sounds a bit like "RC" or like one of its variations. 
All the best,
Marta

> Date: Thu, 14 Jul 2011 23:51:46 -0700
> Subject: Re: categories: RE: stacks (was: size_question_encore)
> From: mshulman@ucsd.edu
> To: marta.bunge@mcgill.ca
> CC: categories@mta.ca
> 
> On Wed, Jul 13, 2011 at 2:16 AM, Marta Bunge <martabunge@hotmail.com> wrote:
>> Bob Pare and I wanted to posit something like the existence of a "representative (or universal) cover" as an axiom (RC), such that GT => RC => ASC, and also (ET + AC) => RC. We hoped to show that RC had good properties, and more importantly, that Eff satisfied it but this did not  work.
> 
> That's a very interesting question!  There seem to be a lot of axioms
> of this flavor, which say in various different ways that AC fails "in
> only a small way".  Some that I am aware of include:
> 
> * Small Violations of Choice (Blass):
>   http://nlab.mathforge.org/nlab/show/small+violations+of+choice
> * Small Cardinality Selection (Makkai):
>   http://nlab.mathforge.org/nlab/show/small+cardinality+selection+axiom
> * Axiom of Multiple Choice (Moerdijk & Palmgren):
>   http://nlab.mathforge.org/nlab/show/axiom+of+multiple+choice
> * Weakly Initial Sets of Covers (Roberts):
>   http://nlab.mathforge.org/nlab/show/WISC
> * The ex/lex completion of Set (or the topos in question) is well-powered
> * The ex/lex completion of Set is a topos, i.e. Set has a generic proof
> 
> Some of these hold in any Grothendieck topos, but others apparently
> need not, and I have no idea which of them might hold in Eff.  It
> would be interesting to know if any of them imply ASC (or André's
> proposed strengthening thereof).
> 
> Mike


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