Provide categorical semantics for OHTT

Construct categorical semantics for Open Horn Type Theory—such as ruptured (∞,1)-categories, stratified fibration categories augmented with gap structure, or bilattice‑enriched categories—and prove that these semantics validate the OHTT calculus (soundness) and characterize its derivations (completeness) relative to the ruptured simplicial model.

Background

Beyond the ruptured simplicial model, the authors expect categorical semantics that mirror HoTT’s ∞‑toposes and fibration categories, adapted to track gaps.

They propose several candidate frameworks and note that identifying and developing the right categorical semantics remains to be done.