Homotopy Type Theory:

Univalent Foundations of Mathematics

The Univalent Foundations Program

Institute for Advanced Study

About the book



Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky’s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, 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. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.

Feedback

We have released the book under a permissive Creative Commons licence which allows everyone to participate and improve it. We would love to hear your comments, suggestions, and corrections. The best way to provide feedback is by creating an issue on the github.com book repository. Git users may also fork the book and make pull requests. (Would you like to understand the last sentence? Learn git in 15 minutes!)

Updates

This is the first (and to date, only) edition of the book. For the benefit of all readers, the available PDF and printed copies are being updated on a rolling basis with minor corrections and clarifications as we receive them. Every copy has a version marker that can be found on the title page and is of the form “first-edition-XX-gYYYYYYY”, where XX is a natural number and YYYYYYY is the git commit hash that uniquely identifies the exact version. Higher values of XX indicate more recent copies. The date and version marker of the most recent update can be found on the nightly builds page.

A list of corrections and clarifications that have been made so far (except for trivial formatting and spacing changes), along with the version marker in which they were first made, can be found in the errata file. While the page numbering may differ between copies with different version markers (and indeed, already differs between the letter/A4 and printed/ebook copies with the same version marker), we promise that the numbering of chapters, sections, theorems, and equations will remain constant, and no new mathematical content will be added, unless and until there is a second edition.

Citing the book

Since the book has no “publisher” in the traditional sense, it may not be obvious how to cite it. Here is one possible BibTeX entry:

@Book{hottbook, author = {The {Univalent Foundations Program}}, title = {Homotopy Type Theory: Univalent Foundations of Mathematics}, publisher = {\url{https://homotopytypetheory.org/book}}, address = {Institute for Advanced Study}, year = 2013}