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I don’t understand the definition at locally infinity-connected (infinity,1)-site. Why does it refer to hypercompletion? It seems to me that it should clearly be about the ordinary $\infty$-topos of sheaves, not its hypercompletion.
Also, the definition should really be an explicit one in terms of coverages, like that of a locally connected site. The obvious corresponding condition is that all covering sieves have contractible nerves; is that sufficient?
@Mike It is indeed necessary and sufficient, since a constant presheaf with value $X$ is a sheaf iff $X=lim_{R} X=Map(N(R),X)$ for every covering sieve R.
I’m having trouble typechecking your equation; can you elaborate it for me?
Thanks for the alert. I forget why I mentioned hypercompleteness when I created the entry. I suppose one may as well mention it, if it is what one is after, but since the remainder of the entry does not refer back to it anyway, we can as well leave it open. So I have removed the adjective.
Then I noticed that in prop. 3.1 the “$X$” suddenly turned into a “$C$”. I have fixed that now, too.
In #3, the first equivalence is the sheaf condition, and the second its unwinding, where due to the constancy of the presheaf, on the left “$X$” denotes the presheaf while on the right it denotes the $\infty$-groupoid that it is constant on.
Thanks, I have added this other characterization to the page.
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