nLab Homotopy Type Theory -- Univalent Foundations of Mathematics

An introductory textbook on homotopy type theory and its use as a univalent foundations for mathematics:

• Univalent Foundations Project

  Homotopy Type Theory – Univalent Foundations of Mathematics

  2013

  (web, pdf)

About the book (from the cover)

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.

• A more detailed description of the contents here:

  http://golem.ph.utexas.edu/category/2013/06/the_hott_book.html#more

• And a description of the “sociology” of the writing of the book is here:

  Andrej Bauer, The HoTT book (web, alternative)

• The source code for the book is here:

  https://github.com/hott/book

• It can be cited as:

Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013.

category: reference