Warning 6.56.
The finite-limit hypothesis in the site presentation is essential. If \((C,\tau)\) is an \((n,1)\)-site whose underlying category does not have finite limits, then \(\Shv_{\tau}(C)\) need not be \(N\)-localic for any \(N\). For example, [Lurie 2018, Counterexample 20.4.0.1] constructs a basis \(C\) for the topology of the Hilbert cube
\[Q:=\prod_{k\in\N}[0,1]\]
such that the topos of sheaves on \(C\) for the induced topology is not \(N\)-localic for any \(N\). Its hypercompletion nevertheless agrees with the hypercompletion of the \(0\)-localic topos \(\Shv(Q)\).
References
- Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.