Homotopy Type Theory

by The Univalent Foundations Program

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.

http://homotopytypetheory.org/book/

Genres: Mathematics, Computer Science, Textbooks, Logic, Programming, Nonfiction, Science

618 pages, ebook

First published January 1, 2013

Reviews

[Frank Ashe · Rating 4 out of 5]

I shouldn't have stayed up reading this, but it got exciting. Category Theory, Set Theory, the Integers, Reals, even Conway's Surreal numbers! Cauchy Reals are different to Dedekind Reals, unless we want to assume the Law of the Excluded Middle, and who wants to do that! But Homotopy Type Theory gives us another way to get Real numbers that are more fundamentally sound than Cauchy and Dedekind.

I like a book where the authors assume you're going to read it at least twice - "you can skip this section on the first reading, but it's essential on the second".

[Kaidi Pan]

Too hard, I’m dumb

[Crittens]

$21 on lulu.com vs. $50 on Amazon.