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 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; if you add these, then you have the ‘cartesian cube category’. This is done in (Cohen-Coquand-Huber-Moertberg 17), called the “CCHM model”. This model has a much richer cube category, the free De Morgan algebra generated by an interval. In addition to diagonals, this includes what are called ‘reversals’ and ‘connections’.
The CCHM model validates both univalence and can be used to model a variety of HITs.
One thing to be cautious about is that while it is possible to model the Martin-Löf identity type in both the BCH model and the CCHM model, it does not coincide with the paths in the model. But it is equivalent to the path types.
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:
Original articles:
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)
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)
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:
Last revised on May 21, 2020 at 14:56:44. See the history of this page for a list of all contributions to it.