[Paper Review] Introduction to Homotopy Type Theory

Egbert Rijke|arXiv (Cornell University)|Dec 21, 2022

History and Theory of Mathematics4 citations

TL;DR

This paper provides a comprehensive, pedagogical introduction to Homotopy Type Theory (HoTT), systematically developing Martin-Löf’s dependent type theory and extending it with univalent foundations. It establishes the connection between type theory, homotopy theory, and higher category theory, demonstrating how identity types model paths, and how the univalence axiom allows isomorphic types to be identified, thereby unifying logic, type theory, and topology in a constructive foundation for mathematics.

ABSTRACT

This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal. This textbook introduces the reader to Martin-Löf's dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. Over 200 exercises are included to train the reader in type theoretic reasoning. The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.

Motivation & Objective

• To provide a self-contained, accessible introduction to Homotopy Type Theory for researchers and students in logic, mathematics, and computer science.
• To systematically develop Martin-Löf’s dependent type theory as a foundation for constructive mathematics.
• To introduce and formalize the univalence axiom, showing how it equates isomorphic types and enables a synthetic approach to homotopy theory.
• To demonstrate how core mathematical structures—such as natural numbers, integers, finite types, and groups—can be constructed and reasoned about within the type-theoretic framework.
• To establish the logical and categorical foundations of univalent foundations, including propositional truncation, image factorizations, and set quotients.

Proposed method

• Uses Martin-Löf’s dependent type theory as a foundational system, with explicit inference rules, derivations, and judgments to formalize type constructions.
• Introduces identity types as path types, modeling homotopical structure through inductive definitions and transport operations.
• Applies the Curry-Howard isomorphism to interpret logical propositions as types and proofs as terms, enabling logical reasoning within type theory.
• Employs higher inductive types (HITs) to construct mathematical objects like the circle, integers, and propositional truncations with specified universal properties.
• Develops the univalence axiom as a key principle that identifies equivalent types, enabling function extensionality and propositional extensionality.
• Uses the concept of contractible types and singleton induction to characterize uniqueness and equivalence, forming the basis for higher truncation levels and set quotients.

Experimental results

Research questions

• RQ1 How can Martin-Löf’s dependent type theory be systematically developed to serve as a foundation for constructive mathematics?
• RQ2 How do identity types in type theory model paths and homotopies, and what algebraic structure do they form?
• RQ3 What is the role of the univalence axiom in identifying isomorphic types and unifying type theory with homotopy theory?
• RQ4 How can higher inductive types be used to construct mathematical objects like the circle, integers, and finite types in a synthetic way?
• RQ5 What is the relationship between propositional truncation, image factorizations, and the construction of set quotients in univalent type theory?

Key findings

• The identity type over a type X is isomorphic to the path space of X, endowing types with a groupoid-like structure where identifications are paths.
• The univalence axiom implies function extensionality and propositional extensionality, showing that isomorphic types are equal in the type-theoretic sense.
• The natural numbers are characterized by a universal property involving recursion and induction, with addition and well-ordering provable within the system.
• Propositional truncation provides a way to turn any type into a proposition, enabling the construction of logical disjunctions and existential quantifiers.
• Set quotients and image factorizations are constructed using higher inductive types, with the image of a map characterized as the type of all elements reachable via paths.
• The integers are constructed as a quotient of ℕ × ℕ under an equivalence relation, and are shown to be a group under addition with decidable equality.