Contents

# Contents

## Idea

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.

## Models

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

## References

Introductory lecture notes:

Original articles:

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.