Homotopy Type Theory: Univalent Foundations of Mathematics

TL;DR

Homotopy type theory introduces a new foundation for mathematics that integrates homotopy theory and type theory, enabling a more invariant and computationally friendly approach to mathematical reasoning.

Contribution

It presents the univalence axiom and higher inductive types as foundational tools, establishing a new logical framework for mathematics based on homotopical content.

Findings

Univalence axiom allows identification of isomorphic structures.

Higher inductive types describe basic spaces and constructions.

Provides a new, invariant foundation for mathematics.

Abstract

Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.