Open Horn Type Theory: Extending Type Theory with Coherence and Gap

This paper introduces Open Horn Type Theory (OHTT), a novel extension of dependent type theory. The core innovation is the introduction of 'gap' as a primitive judgment, distinct from negation, to represent non-coherence. This allows OHTT to model obstructions that Homotopy Type Theory (HoTT) cannot, particularly in areas like topology and semantics. The paper's significance lies in its potential to capture nuanced situations where transport fails, offering a richer framework for reasoning about mathematical and computational structures. The use of ruptured simplicial sets and Kan complexes provides a solid semantic foundation.