Deligne's theorem for $n$-topos

Question

Deligne's theorem states that a coherent topos has enough points, i.e. that we can prove that a morphism of sheaves on a "nice" site is an isomorphism by showing that the induced morphism on stalks are isomorphisms.

I'm looking for a higher categorical analogue. Specifically, if I have a morphism of $n$-sheaves on a "nice" site, can I test if it is an isomorphism by testing it on points, i.e. do the fibre functors form a conservative family? I would be very grateful if someone could point me to a nice reference, thanks! PS: I assume that we need to assume $n$-coherent, but I might be wrong.

Answer 1

There are two cases:

1.) If your ∞-topos is locally coherent and hypercomplete, then you have Lurie's ∞-categorical version of Deligne's completeness theorem (SAG A.4.0.5).

2.) If your ∞-topos is bounded and coherent, Lurie shows another version of this theorem, similar to Makkai's conceptual completeness theorem (SAG A.9.0.6). Part of the proof of this theorem shows that you can test for equivalence of truncated objects on points.

Note: The ∞-categorical version of coherence is rather different from the 1-categorical version. In particular, coherence no longer implies compact generation (they are quite different, in fact).