Open Horn Type Theory: Extending Type Theory with Coherence and Gap
Analysis
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.
Key Takeaways
- •OHTT extends dependent type theory with 'coherence' and 'gap' judgments.
- •Gap is a primitive witness of non-coherence, unlike negation.
- •OHTT can model obstructions that HoTT cannot, like transport failures.
- •The semantics are based on ruptured simplicial sets and Kan complexes.
- •Applications include modeling topological, semantic, and logical obstructions.
“The central construction is the transport horn: a configuration where a term and a path both cohere, but transport along the path is witnessed as gapped.”