Homotopy type theory and Voevodsky's univalent foundations

TL;DR

This paper introduces homotopy type theory and Voevodsky's univalent foundations, exploring their theoretical basis and practical implementation using the Coq proof assistant, bridging abstract homotopy theory with computational logic.

Contribution

It provides an accessible introduction to homotopy type theory and details Voevodsky's univalent foundations, including practical aspects of formalization in Coq.

Findings

Connection between homotopy theory and type theory established

Model of type theory using simplicial sets satisfies Univalence Axiom

Practical implementation demonstrated with Coq proof assistant

Abstract

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened "homotopy type theory". In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, called the Univalence Axiom, which has a number of striking consequences. He has subsequently advocated a program, which he calls univalent foundations, of developing mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated by the simplicial set model. Because type theory possesses good computational properties, this program can be carried out in a computer proof assistant. In this paper we give an introduction to homotopy type theory in Voevodsky's setting, paying attention to both theoretical and practical issues. In particular, the paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky's work using the well-known proof assistant Coq. The paper is written for a general audience of mathematicians with basic knowledge of algebraic topology; the paper does not assume any preliminary knowledge of type theory, logic, or computer science.