This updated introduction to type theory and homotopy type theory is an essential read for advanced undergraduate and graduate students interested in the foundations and formalization of mathematics.

The book begins with a detailed and self-contained introduction to dependent type theory. prior knowledge of type theory is not required.

The second section gradually introduces the fundamental concepts of homotopy type theory: equivalences, the fundamental theorem of equality types, propositional truncation, and the univalence axiom. This prepares the reader to study various topics from a unified perspective, including sets, groups, combinatorics, and well-founded trees.

The final section introduces the idea of the higher inductive type, discussing its circle and universal properties. Each section is structured into small chapters, each roughly the length of a lecture, and over 200 exercises provide abundant practice material.

Pages: 386