Dear Mike, Yes, AC is elementary. Of course, so is ASC by the same token. Quantifying over all epis is not a problem. Thanks for questioning me on that point. I remembered incorrectly what the problem was. 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. What is still open is whether Eff satisfies ASC. All the best, Marta
Date: Tue, 12 Jul 2011 19:24:25 -0700 Subject: Re: categories: RE: stacks (was: size_question_encore) From: mshulman@ucsd.edu To: marta.bunge@mcgill.ca CC: categories@mta.ca
I guess I misunderstood what you meant by "elementary". You wanted a single statement that can be expressed in the internal logic of the topos? Many other properties that people refer to as "elementary", such as the existence of finite limits or power objects, are defined by quantifying over all objects and morphisms of the category in question. Is AC "elementary"?
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