Hello !

If I a have a grothendieck Site (C,J), I can consider :

• The Stacks on (C,J) : category fibered in groupoid over C which statisfy suitable descent condition with respect to the covering sieve of J...

• Internal Groupoid in the topos Sh(C,J).

Clearly those two notions are very close. but not exactly equivalent. I found a lot of book/article/thesis who define stacks but none of them were in term of the internal logic of the topos and it appears to me that some properties would be a lot more natural if they were stated in terms of an internal groupoid and the internal logic.

My feeling is that stacks corresponds to internal groupoids "up to weak categorical equivalences" (we invert the functor between groupoid which are internally fully faithfull and essentially surjective). But that this equivalence rely on the (external) axiom of choice. Am I right ? (and if I am, how can we make this more explicit ? )

Oh, and What about Higher Stacks ?

Thank you !