Books & notes

Higher Topos Theory Section 7.3: Fractured topoi

Basic properties of fractured topoi

Proposition 7.13.

Let \((E,E^{\corp})\) be a fractured topos. Then:

  1. The category \(E^{\corp}\) is a topos.

  2. For every \(X\in E^{\corp}\), the functor

    \[j_!\colon E^{\corp}_{/X}\longrightarrow E_{/j_!X}\]

    is fully faithful.

  3. The topos \(E\) is generated under colimits by corporeal objects.

  4. The functor \(j^*\) admits a right adjoint \(j_*\colon E^{\corp}\to E\), and the adjunction \(j^*\dashv j_*\) is a morphism of topoi.

Proof
See [Lurie 2018, Propositions 20.1.3.1 and 20.1.3.3, Corollary 20.1.3.4, and Remark 20.1.3.6].

References

  1. Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.