Category Archives: Higher Inductive Types

Eilenberg-MacLane Spaces in HoTT

For those of you who have been waiting with bated breath to find out what happened to your favorite characters after the end of Chapter 8 of the HoTT book, there is now a new installment: Eilenberg-MacLane Spaces in Homotopy Type Theory Dan … Continue reading

Posted in Applications, Code, Higher Inductive Types, Paper, Univalence | 6 Comments

Composition is not what you think it is! Why “nearly invertible” isn’t.

A few months ago, Nicolai Kraus posted an interesting and surprising result: the truncation map |_| : ℕ → ‖ℕ‖ is nearly invertible. This post attempts to explain why “nearly invertible” is a misnomer. I, like many others, was very … Continue reading

Posted in Code, Foundations, Higher Inductive Types | Tagged | 16 Comments

The surreals contain the plump ordinals

The very last section of the book presents a constructive definition of Conway’s surreal numbers as a higher inductive-inductive type. Conway’s classical surreals include the ordinal numbers; so it’s natural to wonder whether, or in what sense, this may be … Continue reading

Posted in Higher Inductive Types | 3 Comments

New writeup of πn(Sn)

I’m giving a talk this week at CPP.  While I’m going to talk more broadly about applications of higher inductive types, for the proceedings, Guillaume Brunerie and I put together an “informalization” of πn(Sn), which you can find here. This is … Continue reading

Posted in Higher Inductive Types, Homotopy Theory | 1 Comment

The Truncation Map |_| : ℕ -> ‖ℕ‖ is nearly Invertible

Magic tricks are often entertaining, sometimes boring, and in some rarer cases astonishing. For me, the following trick belongs to the third type: the magician asks a volunteer in the audience to think of a natural number. The volunteer secretly … Continue reading

Posted in Code, Higher Inductive Types | 42 Comments

The HoTT Book

This posting is the official announcement of The HoTT Book, or more formally: Homotopy Type Theory: Univalent Foundations of Mathematics The Univalent Foundations Program, Institute for Advanced Study The book is the result of an amazing collaboration between virtually everyone involved … Continue reading

Posted in Foundations, Higher Inductive Types, Homotopy Theory, Paper, Univalence | 3 Comments

Homotopy Theory in Type Theory: Progress Report

A little while ago, we gave an overview of the kinds of results in homotopy theory that we might try to prove in homotopy type theory (such as calculating homotopy groups of spheres), and the basic tools used in our synthetic approach … Continue reading

Posted in Higher Inductive Types, Homotopy Theory | 13 Comments