Equivalence between H and Sh(Esp, J_can)

Establish an equivalence between the sheaf topos H = Sh(P, J_can) — where P is the full subcategory of the category CPO of ω-cpos consisting of retracts of Scott’s graph model G and J_can is the canonical topology on P — and the sheaf topos Sh(Esp, J_can) built from the essential image Esp of the functor pt: Frm → Esp (sending countably presented σ-frames to their spaces of points) equipped with the canonical topology; in particular, prove that the fully faithful inclusion P ↪ Esp induces an equivalence of these sheaf categories.

Background

The paper compares established sheaf models for synthetic domain theory with new models derived from countably presented σ-frames. Fiore, Plotkin, and Rosolini constructed a sheaf topos H = Sh(P, J_can), where P is the full subcategory of ω-cpos whose objects are retracts of Scott’s graph model G and J_can is the canonical topology. The authors observe a fully faithful inclusion P ↪ Esp by relating CPO maps on G to Esp maps via dualities with countably presented σ-frames.

Motivated by this inclusion, the authors conjecture that H is equivalent to Sh(Esp, J_can). Proving this equivalence would unify the classical sheaf models with the quasi-coherence-based framework developed in the paper. They note that a detailed proof hinges on a deeper understanding of the categorical properties of Esp, indicating a concrete unresolved question connecting these two frameworks.