Proof theory and higher categorical semantics of homotopy type theory

Research Project, 2020 – 2023

Homotopy type theory is a young and fast-moving research area at the intersection of computer science, mathematical logic, and algebraic topology. The area was jump-started by Fields medalist Vladimir Voevodsky’s discovery of deep ideas connecting these areas and his resultant introduction of the univalence axiom to constructive type theory, resulting in a practically usable language for machine-checked formalization of mathematics called univalent foundations. This project aims to make progress on several major and well-known open questions in this area:The first of these questions is Voevodsky’s famous homotopy canonicity conjecture, asserting that homotopy type theory has computational meaning. I have found a proof of the conjecture, obtained in collaboration with Chris Kapulkin, and want to explore the consequences of this result.The second question is the internal language conjecture, asking for a characterization of those theories that homotopy type theory can serve as an internal language for. I have an angle on this problem that simplifies previous work and allows extensions in new directions.A third aspect of the proposal, related to the previous questions, is concerned with giving computational meaning to higher category theory, the natural semantic domain for homotopy type theory. I have a new model for higher categories and want to develop their basic theory in a constructive fashion.