Homotopy Type Theory: Univalent Foundations of Mathematics by The Univalent Foundations Program (PDF)

Description

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.

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Reviews from Amazon users which were colected at the time this book was published on the website:

⭐“Homotopy Type Theory: Univalent Foundations of Mathematics”c 2013 The Univalent Foundations ProgramBook version: first-edition-323-g28e4374MSC 2010 classification: 03-02, 55-02, 03B15This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy ofthis license, visit http://creativecommons.org/licenses/by-sa/3.0/.This book is freely available at http://homotopytypetheory.org/book/.And even so, note:About this bookWe did not set out to write a book. The present work has its origins in our collective attempts todevelop a new style of “informal type theory” that can be read and understood by a human being,as a complement to a formal proof that can be checked by a machine. Univalent foundationsis closely tied to the idea of a foundation of mathematics that can be implemented in a computerproof assistant. Although such a formalization is not part of this book, much of the materialpresented here was actually done first in the fully formalized setting inside a proof assistant,and only later “unformalized” to arrive at the presentation you find before you — a remarkableinversion of the usual state of affairs in formalized mathematics.Best to wait for a book authored by folk who actually set out to write a book on these topics. This one is not much of a book!

⭐This is a really interesting and well written book, the product of a yearlong collaboration at the Institute for Advanced Study. It describes (the beta version of) a new language for mathematics, which may some day replace set theory. I’ll be pleased if that happens, as homotopy type theory allows formal proofs which read much more like ordinary mathematical language.Potential buyers should be aware the hardcover’s list price is $26.98. (It is currently being sold for $79.99.)

⭐The book is self published at lulu and as of Jan 2019 is available for significantly less at lulu.

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