Type theory and homotopy

The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Martin-L"of into homotopy theory, resulting in new examples of higher-dimensional categories.

💡 Research Summary

The survey paper “Type Theory and Homotopy” provides a comprehensive overview of the deep and now‑well‑established connection between Martin‑Löf’s constructive type theory (MLTT) and modern homotopy theory, a relationship that has given rise to the field known as Homotopy Type Theory (HoTT). The authors begin by recalling the syntax and semantics of MLTT, emphasizing its dependent types, Σ‑types (dependent sums), Π‑types (dependent products), and especially the identity type Id_A(a, b), which internalizes the notion of equality within the theory. They then introduce the essential concepts of homotopy theory—topological spaces, paths, homotopies, and higher homotopies—showing how these give rise to ∞‑groupoid structures.

The core of the paper is the homotopical interpretation of type theory. Under this interpretation, an element of the identity type Id_A(a, b) is interpreted as a continuous path between points a and b in a space that models the type A. Higher identity types correspond to homotopies between paths, homotopies between homotopies, and so on, yielding an infinite tower of higher‑dimensional equalities. This perspective naturally equips every type with the structure of an ∞‑groupoid. To make the interpretation mathematically robust, the authors discuss two crucial axioms: Function Extensionality (Funext) and the Univalence Axiom. Funext asserts that pointwise equal functions are equal, while Univalence states that equivalent types can be identified as equal. Together, these axioms turn the type‑theoretic notion of equivalence into a genuine homotopical equivalence, allowing the theory to model spaces up to homotopy rather than up to strict equality.

The paper proceeds to describe concrete models of HoTT, focusing on the simplicial set (Kan complex) model and the more recent cubical set models. In these models, types are interpreted as fibrant objects, terms as sections, and identity types as path objects. The authors outline the proof of consistency for HoTT relative to classical set theory, showing that adding Univalence does not introduce contradictions. They also discuss normalization and canonicity results, indicating that despite the higher‑dimensional content, the computational behavior of terms remains well‑behaved.

A substantial portion of the survey is devoted to the implications for higher‑category theory. Because the tower of identity types mirrors the hierarchy of n‑morphisms in an n‑category, HoTT provides a syntax for describing ∞‑categories directly inside type theory. The authors illustrate how categorical constructions such as limits, colimits, adjunctions, and monads can be expressed and reasoned about within HoTT, and they point out that many classical results about (∞,1)‑categories can be recovered from the type‑theoretic setting.

Finally, the authors highlight several emerging research directions. These include the development of synthetic homotopy theory (doing homotopy theory inside HoTT without reference to external topological spaces), the exploration of higher‑inductive types for constructing spaces like spheres and torus directly in the type theory, and the application of HoTT to formalized mathematics and computer‑assisted proof verification. The survey concludes that the bridge between logic, geometry, and algebra forged by the homotopical interpretation of type theory not only enriches each field individually but also opens a unified framework for studying higher‑dimensional mathematical structures.