RE: stacks (was: size_question_encore)

[Michael Shulman <mshulman@ucsd.edu>]

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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