# Cubical Type Theory: a constructive interpretation of the univalence axiom

arXiv: https://arxiv.org/abs/1611.02108 · HTML: https://mlsanity.com/paper/1611.02108/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2016.

## Built on by (most cited first, 1 of 1)

- [Functorial Neural Architectures from Higher Inductive Types](https://mlsanity.com/paper/2603.16123/) (2026)

## Abstract (arXiv)

This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.