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"? On Tue, Jul 12, 2011 at 12:56 PM, Marta Bunge <martabunge@hotmail.com> wrote:

Dear Mike,

...

Does it not work to say that every internal category admits a weak equivalence functor to an internal category which is a stack?

Sure. This is so by Corollary 2.11 in Bunge-Pare. No problem with internal categories or internal weak equivalence functors of course. But how does one internalize the notion of a stack? It comes down to parametrizing all epimorphisms in the topos itself.

All the best,Marta

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