Homotopy Type Theory: A New Foundation For Mathematics?

What is Homotopy Type Theory?

Homotopy type theory (HoTT) is a new branch of mathematics that combines ideas from homotopy theory, higher category theory, and type theory. It aims to provide a new foundation for mathematics based on a homotopical interpretation of types. Some key aspects of HoTT include:

Origin in homotopy theory and higher category theory: HoTT arose from attempts to interpret types and proofs in dependent type theory as spaces and continuous paths using ideas from homotopy theory. This allows higher-dimensional structures to be modeled using type theory.
Univalent foundations and the univalence axiom: A core idea in HoTT is the concept of univalent foundations, which roughly states that isomorphic structures can be identified. This is formalized by Voevodsky’s univalence axiom within type theory.
Formalization in dependent type theory: HoTT is formulated within dependent type theory – a formal system that allows types to depend on values and proofs to depend on types. This provides the underlying language and proof assistant for modeling concepts in HoTT.

Key Ideas and Techniques

Some of the key theoretical ideas and techniques underlying the Homotopy Type Theory (HoTT) program include:

Identity types and path constructions: Identity types are used to formalize the concept of equality within HoTT by representing identities as continuous paths. These pathways can be manipulated using homotopical techniques.
Voevodsky’s univalence axiom: This axiom captures the homotopical idea that isomorphic structures should be equivalent in the type theory framework. Univalence provides the main bridge between abstract homotopy theory and concrete type theory.
Higher inductive types: These extend ordinary algebraic datatypes with additional equality constructors representing paths, cycles, spheres and more complex homotopical structures. Higher inductive types are key for modeling spaces.
Cubical type theory: This is a more geometric form of type theory which directly builds in concepts of paths and homotopies from cubical sets. It provides greater convenience and reasoning power compared to standard HoTT.

Modeling Mathematics in Type Theory

A central goal of HoTT is to represent a wide range of mathematical concepts directly within the framework of type theory. Some examples include:

Table of Contents

Representing spaces: Topological spaces can be formulated as types, with points as terms inhabiting those types. Continuous functions between spaces then appear as functions between the associated types.
Data types and proofs: Ordinary data types like natural numbers and lists arise as inductive types, while mathematical proofs become represented by terms inhabiting suitably defined proof-relevant types.
Recursion and induction principles: Structural recursion and induction for inductive types in type theory correspond directly to recursive/inductive definitions on mathematical structures.
Natural numbers: The type of natural numbers N can be constructed as an inductive type generated by a zero element and a successor operation. This forms a basic model of N as a homotopy type.

Overall, large chunks of modern mathematical reasoning can be directly embedded into HoTT – a remarkable feat for a foundational framework.

Applications and Open Problems

Active research areas for Homotopy Type Theory (HoTT) applications and open problems include:

Formal verification of mathematics: Computer proof assistants based on type theory can provide machine-checked proofs for major mathematical theorems. Several landmark results have been formalized in HoTT.
Connections to quantum physics: The higher-dimensional models of space afforded by HoTT better align with concepts arising in quantum theory and quantum gravity research.
Remaining difficulties: Practical application of HoTT can still be complicated for everyday mathematics. Additional work is needed on libraries of formalized mathematics and proof automation.
Critiques: Some mathematicians argue that excessive focus on syntax obscures the essential semantics of mathematical structures. The computational feasibility of implementing full-scale HoTT has also been critiqued.

Addressing these issues and expanding the scope and applicability of Homotopy Type Theory remains an active research area.

Looking Ahead: A Revolution in Formal Reasoning?

If some of the most ambitious goals of Homotopy Type Theory are realized, it could precipitate a revolution in formal systems for mathematics. Some prospects include:

Proof assistants: Widely used HoTT-based proof assistants could automate large parts of mathematical work and enable verification of results at unprecedented scales.
New mathematics: The intriguing connections between HoTT and quantum physics suggest potential discoveries of novel mathematical structures applicable to physics.
Ongoing research: Active topics include computational type theory, connections to category theory, formalizing algebraic and differential topology, and strengthening univalence principles.

While the future impact of HoTT remains uncertain, it offers one of the most thought-provoking foundational perspectives in modern mathematics.