22. Cubical Type Theory

Cubical Type Theory

Cubical Type Theory is an extension of traditional type theory that introduces new features to better handle higher-dimensional structures and reasoning.

In Cubical Type Theory, the key innovation is the use of cubical sets, which provide a geometric interpretation of types and allow for the direct manipulation of paths and higher-dimensional structures.

Key features of Cubical Type Theory include:

• Cubical sets: Geometric objects used to model types and paths in the theory.
• Cubical operations: Operations that allow for the manipulation of paths and higher-dimensional structures, such as transport and composition.
• Cubical type constructors: Constructors that enable the creation and manipulation of higher-dimensional types, such as interval types and higher inductive types.
• Cubical type checking: Algorithms for checking the validity of cubical proofs and ensuring coherence of higher-dimensional structures.

Cubical Type Theory provides a powerful framework for formalizing and reasoning about mathematical and computational concepts involving higher-dimensional structures.

Lab Activity: Implementing Interval Types in Agda

Let's implement interval types in Agda, a dependently-typed programming language, to demonstrate Cubical Type Theory.

module Interval where

data I : Set where
  zero one : I
  seg : zero ≡ one

_∨_ : I → I → I
zero ∨ i = i
one ∨ _ = one

infixr 5 _∨_

seg_suc : ∀ (i : I) → (i ∨ i) ≡ i
seg_suc zero = refl
seg_suc one = refl
seg_suc seg = refl
  

In this lab activity, we define an interval type `I` in Agda, which represents the unit interval [0,1].

We then define the operation `_∨_`, which represents concatenation of intervals.

Finally, we define a proof `seg_suc` that shows the segment concatenated with itself is equal to the segment.