Theorem 7.21.
Let \((E,E^{\corp})\) be a complete fractured topos. Consider the functor
\[E \to \Fun(\Topos^{\loc}_{//E}, \An), \qquad X \mapsto \Gamma^X : (f\colon T \to E) \mapsto \Gamma(f^*(X)).\]
If \(X\) is corporeal, then \(\Gamma^X\) is representable by the pair \((E^{\ad}_{/X}, E^{\ad}_{/X} \hookrightarrow E_{/X} \to E)\).
The resulting functor
\[E^{\corp} \xrightarrow{\mathrm{Re}} \Topos^{\loc}_{//E}\]is fully faithful.
A morphism \(f\colon Y \to X\) in \(E\) is admissible if and only if, after every base change \(f'\colon Y'\to X'\) for which \(\Gamma^{X'}\) is represented by an object of \(\Topos^{\loc}_{//E}\), the functor \(\Gamma^{Y'}\) is represented by an object étale over it.
Combining these statements, it follows that the subcategory \(\Topos^{\loc}_{//E}\subseteq \Topos_{//E}\) determines the fracture structure on \(E\).
Proof
These assertions summarize the reconstruction results of [Lurie 2018, Chapter 20]. The third assertion uses the usual identification of étale objects over a topos with objects of that topos.
References
- Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.