Theorem 6.133.
There are equivalences
\[\Topos^{\coh,b} \simeq \Topos^{\pre,b}, \qquad T \mapsto T^{\coh}_{< \infty}, \qquad C \mapsto \Shv(C)\]
and
\[\Topos^{\coh,\text{loc coh}, \hyp} \simeq \Topos^{\pre,\hyp}, \qquad T \mapsto T^{\coh}, \qquad C \mapsto \Shv(C)^{\hyp}.\]
These equivalences are induced by the effective epimorphism topology and are proved in [Lurie 2018, Sections A.6.5--A.7.4]. In the second equivalence, local coherence ensures that \(T^{\coh}\) is essentially small; hypercompleteness is exactly the additional descent condition needed to recover \(T\) from this pretopos.
References
- Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.