Abstract
AbstractWe present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, suspension, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgmental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of generating trivial cofibrations and a different set of generating cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model of this type theory in Cartesian cubical sets. We give a mechanized proof, using Agda as the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, suspension types, and the universe itself are Kan in this model, and that the universe is univalent. An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category, as used in the CCHM model, and bicubical sets, as used in directed type theory.
Publisher
Cambridge University Press (CUP)
Subject
Computer Science Applications,Mathematics (miscellaneous)
Reference60 articles.
1. Awodey, S. (2016). Notes on cubical sets. Available from https://github.com/awodey/math/blob/master/Cubical/cubical.pdf.
2. Cavallo, E. and Harper, R. (2019). Higher inductive types in cubical computational type theory. Proceedings of the ACM on Programming Languages 3 (POPL) 1:1–1:27.
3. The Frobenius condition, right properness, and uniform fibrations
4. Hofmann, M. (1995). Extensional Concepts in Intensional Type Theory. Phd thesis, University of Edinburgh.
Cited by
7 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Formalizing the ∞-Categorical Yoneda Lemma;Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs;2024-01-09
2. Internal Parametricity, without an Interval;Proceedings of the ACM on Programming Languages;2024-01-05
3. Algebraic Effects Meet Hoare Logic in Cubical Agda;Proceedings of the ACM on Programming Languages;2024-01-05
4. Internal and Observational Parametricity for Cubical Agda;Proceedings of the ACM on Programming Languages;2024-01-05
5. A Formal Logic for Formal Category Theory;Lecture Notes in Computer Science;2023