@AdrianClough I have no idea if it works but I feel like it is worth looking at the category of sheaves on, say, BZ_p, with a trivialization after restricting to the circle. If i didn't mess things up this is an infinity topos in which hypersheaves are all constant (here I might have exchanged some functors and colimits e.t.c. im not sure) but BZ_p has no constant shape, so maybe it also don't have? just a shoot in the dark, not sure about anything here...

Oh sorry, what I really meant is sheaves on finite Z_p sets with trivialization of the Z_p action on the stalk

Suppose I have an inner fibration p: X \to S and f:x \to y, f':x \to y' are two arrows in X which maps to the same arrow in S. If f is equivalent to f' in X_{x/} \times_{S_{p(x)}/} p(f), is it true that f is cocartesian iff f' is?

I've been asked the following question: "Over what kind of base schemes is equivariant elliptic cohomology currently defined? Arbitrary Noetherian? Or something less general than that?" Has anybody thought about this?

I think I can show the above if X and S are quasi-categories, but is it true in general?

Also, if someone knows an easy proof of the above when X and S are quasi-categories, please do share it.

@S.carmeli Hmmm... so by Z_p do you mean the p-adic integers or $\mathbf{Z}/p\mathbf{Z}$? When you write "the stalk" do you have a particular stalk in mind, or do you mean a trivialisation on all stalks?

And the trivialisation on a/the stalk(s) is supposed to kill off hypercomplete sheaves of non-constant shape I suppose. It is this last step that I don't understand.

@AdrianClough by the "stalk" I mean, given a sheaf $F$, the colimit of $F(Z/p^kZ)$ over $k$. It is a point of the topos of continuous $Z_p$-sets. The addition of the trivialization is my atempt to force the shape after the hypercompletion to tirivalize. I have no idea if this construction works, I just wanted to share my first thought on this question because its cool.