Construction 7.22.
Consider the fractured topos \(E = \Shv_{\tau}(C)\), \(E^{\corp} := \Shv_{\tau}(C^{\ad})\). By Theorem 7.21, we obtain a functor
\[\Gamma\colon \Topos^{\loc}_{//E} \to \Pt(\PSh(C))\catop \simeq \Pro(C),\]
An object \(f\colon T\to E\) is sent to the pro-object classified by the left exact functor
\[C \hookrightarrow \PSh(C) \twoheadrightarrow \Shv_{\tau}(C) = E \xrightarrow{f^*} T \xrightarrow{\Gamma} \An.\]
The functor \(\Gamma\) admits a right adjoint
\[\Spec_{\Gg}\colon \Pro(C) \to \Topos^{\loc}_{//E}\]
called the spectrum functor. If \(\tau\) is subcanonical, then \(\Spec_{\Gg}\) is fully faithful. See [Lurie 2011, Chapter 2] for this structured-topos formulation of the spectrum construction.
References
- Jacob Lurie. Derived algebraic geometry V: Structured spaces. 2011.