natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Cubical type theory is a version of homotopy type theory in which univalence is not just an axiom but a theorem, hence, since this is constructive, has “computational content”. Cubical type theory models the infinity-groupoid-structure implied by Martin-Löf identity types on constructive cubical sets, whence the name.
The first constructive account of the univalence axiom was given in (Coquand 13, Bezem-Coquand-Huber 17), called the “BCH-model”.
The BCH model, unfortunately, has some problems that appear to make it unsuitable for general HoTT. In particular, it is not known how to model higher inductive types. The problem is that the BCH model is based on presheaves on the ‘symmetric monoidal cube category’, which is basically the free PROP generated by an interval. In particular, the base category’s maps are generated by face maps and permutative renamings of dimension variables (this is where the ‘symmetric monoidal’ part comes in). For somewhat technical reasons, this doesn’t work out when you want to define the elimination rules for higher inductive types (like for the circle).
To account for HITs, you seem to need diagonals in the base category. There is more than one choice in addition as to what other morphisms to add.
If you add only diagonals, then you have the ‘cartesian cube category’. This was used as a basis for a cubical type theory by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata.
The first cubical type theory, due to Cohen-Coquand-Huber-Moertberg 17, used an even richer cube category, the free De Morgan algebra generated by an interval. In addition to diagonals, this includes what are called ‘reversals’ and ‘connections’.
Both models validate univalence (like the BCH model) and can be used to model a variety of HITs as well as supporting a syntactic type theory based on ‘dimension variables’. However, while it is possible to interpret the Martin-Löf identity type in all of these models, it is only equivalent to, not definitionally isomorphic to, the native cubical path-types in the model. Thus, the latter support the Martin-Lof eliminator, but only with a typal computation rule.
Cubical type theory can be modeled in a number of varieties of cubical sets, for example in a type-theoretic model structure.
Introductory lecture notes:
Exposition in view of synthetic homotopy theory:
Original articles on the BCH model:
Thierry Coquand (with Marc Bezem and Simon Huber), Computational content of the Axiom of Univalence, September 2013 (pdf)
Marc Bezem, Thierry Coquand, Simon Huber, The univalence axiom in cubical sets (arXiv:1710.10941)
On the CCHM model and type theory:
Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg, Cubical Type Theory: a constructive interpretation of the univalence axiom (pdf)
Simon Huber, Canonicity for Cubical Type Theory (arXiv:1607.04156)
Thierry Coquand, Simon Huber, Anders Mörtberg, On Higher Inductive Types in Cubical Type Theory (arXiv:1802.01170)
Evan Cavallo, Robert Harper, Internal parametricity for cubical type theory (PDF)
On the cartesian cubical model and type theory:
Carlo Angiuli, Guillaume Brunerie, Thierry Coquand, Kuen-Bang Hou (Favonia), Robert Harper, and Daniel R. Licata, Cartesian cubical type theory, https://www.cs.cmu.edu/~rwh/papers/uniform/uniform.pdf
Carlo Angiuli, Computational Semantics of CartesianCubical Type Theory, Ph.D. Thesis, https://www.cs.cmu.edu/~cangiuli/thesis/thesis.pdf
On comparing the models:
Evan Cavallo, Anders Mörtberg, Andrew W Swan, Unifying Cubical Models of Univalent Type Theory, 2019, PDF
Thierry Coquand, Simon Huber, Christian Sattler Homotopy canonicity for cubical type theory, 2019 PDF
Bruno Bentzen, Naive cubical type theory, 2019 (pdf)
Discussion of implementation in the proof assistant Cubical Agda:
On normalization for cubical type theory:
Last revised on June 20, 2022 at 18:00:41. See the history of this page for a list of all contributions to it.