By now many of you have heard of the development of Homotopy Type Theory (HoTT), an extension of intuitionistic type theory that provides a natural foundation for doing synthetic homotopy theory. Last year the Institute for Advanced Study at Princeton sponsored a program on the Univalent Foundations of Mathematics, which was concerned with developing these ideas. One important outcome of the year-long program is a full-scale book presenting the main ideas of Homotopy Type Theory itself and showing how to apply them to various branches of mathematics, including homotopy theory, category theory, set theory, and constructive analysis. The book is the product of a joint effort by dozens of participants in the program, and is intended to document the state of the art as it is known today, and to encourage its further development by the participation of others interested in the topic (i.e., you!). Among the many directions in which one may take these ideas, the most important (to me) is to develop a constructive (computational) interpretation of HoTT. Some partial results in this direction have already been obtained, including fascinating work by Thierry Coquand on developing a constructive version of Kan complexes in ITT, by Mike Shulman on proving homotopy canonicity for the natural numbers in a two-dimensional version of HoTT, and by Dan Licata and me on a weak definitional canonicity theorem for a similar two-dimensional theory. Much work remains to be done to arrive at a fully satisfactory constructive interpretation, which is essential for application of these ideas to computer science. Meanwhile, though, great progress has been made on using HoTT to formulate and formalize significant pieces of mathematics in a new, and strikingly beautiful, style, that are well-documented in the book.
The book is freely available on the web in various formats, including a PDF version with active references, an ebook version suitable for your reading device, and may be purchased in hard- or soft-cover from Lulu. The book itself is open source, and is available at the Hott Book Git Hub. The book is under the Creative Commons CC BY-SA license, and will be freely available in perpetuity.