20. Homotopy Type Theory (HoTT)

Homotopy Type Theory (HoTT)

Homotopy Type Theory (HoTT) is a branch of mathematics and a formal system in type theory that establishes connections between types and homotopy theory.

In HoTT, types are seen as spaces that can be studied using the techniques of homotopy theory.

Key concepts in HoTT include:

• Identity types: Types that represent equality or paths between elements of a given type.
• Higher inductive types: Types that are generated not only by constructors but also by higher-dimensional paths, capturing more intricate topological structure.
• Univalence axiom: A principle that identifies isomorphic types as equal, capturing the notion of equivalence between spaces.
• Homotopy equivalence: A relation between types that captures the idea of them being "the same" up to continuous deformation.

HoTT provides a rich framework for reasoning about mathematical structures and has applications in various fields, including computer science, category theory, and topology.

Lab Activity: Demonstrating Identity Types in HoTT

Let's implement a simple example in a proof assistant like Coq to demonstrate identity types in HoTT.

Inductive Id {A : Type} (x : A) : A → Type :=
  | refl : Id x x.

Definition id_example : Id 3 3 := refl 3.
  

In this lab activity, we define the identity type `Id`, which represents equality or paths between elements of a given type.

We then demonstrate the identity type by constructing a proof that there exists a path between the number 3 and itself, using the reflexivity constructor.