nLab cubical type theory

Idea

Cubical type theory is a flavor of dependent type theory in which maps out of an interval primitive is used to define cubical path types, rather than the inductive family of Martin-Löf identity types as in Martin-Löf type theory. Cubical type theory additionally differs from Martin-Löf type theory in that function extensionality is a theorem in cubical type theory, rather than an axiom as is the case in Martin-Löf type theory.

Similarly to Martin-Löf type theory, cubical type theory comes in extensional and intensional flavours as well. One could also add an equality reflection rule for path types to make the cubical type theory definitionally extensional.

Univalent cubical type theory models the $\infty$-groupoid-structure implied by Martin-Löf identity types on constructive cubical sets (whence the name) thus making it a form of homotopy type theory.

Syntax

The interval and face formulas

We begin with a dependent type theory which has rules for dependent product types, dependent sum types, the empty type, unit type, booleans type, and natural numbers type.

Cubical type theory adds to dependent type theory an interval primitive $I$, whose elements are called “dimensions” as well as face formulas $F$, which behave like propositions in a propositional logic. Both come with the following judgments for dimension variables and face formulae:

$$\frac{\Gamma \; ctx}{\Gamma, i:I \; ctx} \qquad \frac{\Gamma \; ctx \quad \Gamma \vdash \phi:F}{\Gamma, \phi \; ctx}$$

Similarly to types, dimension variables and face formulas come with their own substitution rules, weakening rules, and hypothesis rules?.

In addition, there are rules for forming the endpoints of the interval:

$$\frac{\Gamma}{\Gamma \vdash 0:I} \qquad \frac{\Gamma}{\Gamma \vdash 1:I}$$

Equality of dimensions in the interval is a face formula:

$$\frac{\Gamma \vdash r:I \quad \Gamma \vdash s:I}{\Gamma \vdash r = s:F}$$

and face formulas are closed under disjunction

$$\frac{\Gamma \vdash \phi:F \quad \Gamma \vdash \psi:F}{\Gamma \vdash \phi \vee \psi:F}$$

Since face formulae behave like propositions, it should be possible to judge them to be true. Thus, we have the following rules:

$$\frac{\Gamma \vdash \phi \; true}{\Gamma \vdash \phi \vee \psi \; true} \qquad \frac{\Gamma \vdash \psi \; true}{\Gamma \vdash \phi \vee \psi \; true} \qquad \frac{\Gamma \vdash \phi \vee \psi \; true \quad \Gamma, \phi \vdash \chi \; true \quad \Gamma, \psi \vdash \chi \; true}{\Gamma \vdash \chi \; true}$$

The interval primitive $I$ has more points than $0$ and $1$, so it is not the case that the sequent

$$r:I \vdash r = 0 \vee r = 1 \;\mathrm{true}$$

holds. Thus, there is a boundary formula

$$\delta(r) \coloneqq r = 0 \vee r = 1$$

Canonicity in cubical type theory

In contrast to Martin-Löf type theory, there exist cubical type theories, such as XTT, in which UIP is not just an axiom but a theorem. Similarily, there exist cubical type theories in which univalence is not just an axiom but a theorem. As a result, in those cubical type theories, canonicity still holds. This is useful for computational purposes, and for the use of cubical type theory in proof assistants.

However, it is equally valid to add an axiom to a cubical type theory that says all type universe satisfy UIP or univalence, so canonicity is not required to hold in all cubical type theories.

Univalent cubical type theories

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.

Models

Cubical type theory can be modeled in a number of varieties of cubical sets, for example in a type-theoretic model structure.

Related concepts

• synthetic Tait computability
• simplicial type theory
• type theory with shapes
• two-level type theory

References

Introductory lecture notes:

• Anders Mörtberg, Cubical methods in HoTT/UF, 2019 (pdf)

Exposition in view of synthetic homotopy theory:

• Anders Mörtberg, Loïc Pujet, Cubical synthetic homotopy theory, CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs January 2020, pp. 158–171, doi:10.1145/3372885.3373825, (pdf)

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 Cartesian Cubical 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 (arXiv:1911.05844)

Discussion of implementation in the proof assistant Cubical Agda:

• Andrea Vezzosi, Anders Mörtberg and Andreas Abel, Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types, 2019 (pdf)

On normalization for cubical type theory:

• Jonathan Sterling, Carlo Angiuli, Normalization for Cubical Type Theory, (arXiv:2101.11479)
• Jonathan Sterling, First Steps in Synthetic Tait Computability: The Objective Metatheory of Cubical Type Theory, (pdf)

On XTT:

• Jonathan Sterling, Carlo Angiuli, Daniel Gratzer, Cubical syntax for reflection-free extensional equality. In Herman Geuvers, editor, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019), volume 131 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:25. (arXiv:1904.08562, doi:10.4230/LIPIcs.FCSD.2019.31)

• Jonathan Sterling, Carlo Angiuli, Daniel Gratzer, A Cubical Language for Bishop Sets, Logical Methods in Computer Science, 18 (1), 2022. (arXiv:2003.01491).