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This up-to-date introduction to type theory and homotopy type theory will be essential reading for advanced undergraduate and graduate students interested in the foundations and formalization of mathematics. The book begins with a thorough and self-contained introduction to dependent type theory. No prior knowledge of type theory is required. The second part gradually introduces the key concepts of homotopy type theory: equivalences, the fundamental theorem of identity types, truncation levels, and the univalence axiom. This prepares the reader to study a variety of subjects from a univalent point of view, including sets, groups, combinatorics, and well-founded trees. The final part introduces the idea of higher inductive type by discussing the circle and its universal cover. Each part is structured into bite-size chapters, each the length of a lecture, and over 200 exercises provide ample practice material.
List of contents
Preface; Introduction; Part I. Martin-Löf's Dependent Type Theory: 1. Dependent type theory; 2. Dependent function types; 3. The natural numbers; 4. More inductive types; 5. Identity types; 6. Universes; 7. Modular arithmetic via the Curry-Howard interpretation; 8. Decidability in elementary number theory; Part II. The Univalent Foundations of Mathematics: 9. Equivalences; 10. Contractible types and contractible maps; 11. The fundamental theorem of identity types; 12. Propositions, sets, and the higher truncation levels; 13. Function extensionality; 14. Propositional truncations; 15. Image factorizations; 16. Finite types; 17. The univalence axiom; 18. Set quotients; 19. Groups in univalent mathematics; 20. General inductive types; Part III. The Circle: 21. The circle; 22. The universal cover of the circle; Index; Bibliography.
About the author
Egbert Rijke is Postdoctoral Research Fellow at Johns Hopkins University and is a pioneering figure in homotopy type theory. As one of the co-authors of the influential book 'Homotopy Type Theory: Univalent Foundations of Mathematics' (2013), he has played a pivotal role in shaping the field. He is also a founder and lead developer of the agda-unimath library, which stands as the largest library of formalized mathematics written in the Agda proof assistant.
