Theorem 7.19.
Let \(E\) be a topos. The assignments \((E,E^{\corp}) \mapsto (E,E^{\ad})\) and \((E,E^{\ad}) \mapsto (E,E^{\corp})\) determine an equivalence of posets
\[\{\text{Complete fractured structures on $E$}\} \simeq \{\text{Geometric admissibility structures on $E$}\}.\]
Proof
This is [Lurie 2018, Theorem 20.3.4.4 and Remark 20.3.4.6]. The completeness hypothesis is essential: without it, the admissible morphisms recover the closure of the corporeal objects under retracts, rather than necessarily recovering the original fracture.
References
- Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.