Homotopy Type Theory: Univalent Foundations of Mathematics

599 Pages·2013·3.17 MB·English

by  The Univalent Foundations Program

Homotopy
Type Theory
Univalent Foundations of Mathematics
THE UNIVALENT FOUNDATIONS PROGRAM
INSTITUTE FOR ADVANCED STUDY
Homotopy Type Theory
Univalent Foundations of Mathematics
The Univalent Foundations Program
Institute for Advanced Study
“Homotopy Type Theory: Univalent Foundations of Mathematics”
⃝c 2013 The Univalent Foundations Program
Book version: first-edition-0-ge0aa8e9
MSC 2010 classification: 03-02, 55-02, 03B15
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported
License. To view a copy of this license, visit http://creativecommons.org/licenses/by-
sa/3.0/.
This book is freely available at homotopytypetheory.org/book/.
Acknowledgment
Apart from the generous support from the Institute for Advanced Study, some contributors
to the book were partially or fully supported by the following agencies and grants:
• Association of Members of the Institute for Advanced Study: a grant to the Institute
for Advanced Study
• Agencija za raziskovalno dejavnost Republike Slovenije: P1–0294, N1–0011.
• Air Force Office of Scientific Research: FA9550-11-1-0143, and FA9550-12-1-0370.
This material is based in part upon work supported by the AFOSR under the
above awards. Any opinions, findings, and conclusions or recommendations ex-
pressed in this publication are those of the author(s) and do not necessarily reflect
the views of the AFOSR.
• Engineering and Physical Sciences Research Council: EP/G034109/1, EP/G03298X/1.
• European Union’s 7th Framework Programme under grant agreement nr. 243847
(ForMath).
• National Science Foundation: DMS-1001191, DMS-1100938, CCF-1116703, and DMS-
1128155.
This material is based in part upon work supported by the National Science Foun-
dation under the above awards. Any opinions, findings, and conclusions or rec-
ommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the National Science Foundation.
• The Simonyi Fund: a grant to the Institute for Advanced Study
Preface
IAS Special Year on Univalent Foundations
A Special Year on Univalent Foundations of Mathematics was held in
2012-13 at the Institute for Advanced Study, School of Mathematics, or-
ganized by Steve Awodey, Thierry Coquand, and Vladimir Voevodsky.
The following people were the official participants.
Peter Aczel Peter Lumsdaine
Benedikt Ahrens Assia Mahboubi
Thorsten Altenkirch Per Martin-Lo¨f
Steve Awodey Sergey Melikhov
Bruno Barras Alvaro Pelayo
Andrej Bauer
Andrew Polonsky
Yves Bertot
Michael Shulman
Marc Bezem
Matthieu Sozeau
Thierry Coquand
Eric Finster Bas Spitters
Daniel Grayson Benno van den Berg
Hugo Herbelin Vladimir Voevodsky
Andre´ Joyal Michael Warren
Dan Licata Noam Zeilberger
There were also the following students, whose participation was no less
valuable.
Carlo Angiuli Chris Kapulkin
Anthony Bordg Egbert Rijke
Guillaume Brunerie Kristina Sojakova
iv
In addition, there were the following short- and long-term visitors, in-
cluding student visitors, whose contributions to the Special Year were
also essential.
Jeremy Avigad Martin Hofmann
Cyril Cohen Pieter Hofstra
Robert Constable Joachim Kock
Pierre-Louis Curien Nicolai Kraus
Peter Dybjer Nuo Li
Mart´ın Escardo´ Zhaohui Luo
Kuen-Bang Hou Michael Nahas
Nicola Gambino Erik Palmgren
Richard Garner Emily Riehl
Georges Gonthier Dana Scott
Thomas Hales Philip Scott
Robert Harper Sergei Soloviev
About this book
We did not set out to write a book. The present work has its origins in
our collective attempts to develop 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 foundations
is closely tied to the idea of a foundation of mathematics that can be im-
plemented in a computer proof assistant. Although such a formalization
is not part of this book, much of the material presented here was actu-
ally 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 remarkable inversion of the usual state of affairs in formalized
mathematics.
Each of the above-named individuals contributed something to the
Special Year — and so to this book — in the form of ideas, words, or
deeds. The spirit of collaboration that prevailed throughout the year was
truly extraordinary.
Special thanks are due to the Institute for Advanced Study, without
which this book would obviously never have come to be. It proved to
be an ideal setting for the creation of this new branch of mathematics:
stimulating, congenial, and supportive. May some trace of this unique
v
atmosphere linger in the pages of this book, and in the future develop-
ment of this new field of study.
The Univalent Foundations Program
Institute for Advanced Study
Princeton, April 2013
Contents
Introduction 1
I Foundations 21
1 Type theory 23
1.1 Type theory versus set theory . . . . . . . . . . . . . . . . . 23
1.2 Function types . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3 Universes and families . . . . . . . . . . . . . . . . . . . . . 32
1.4 Dependent function types (Π-types) . . . . . . . . . . . . . 33
1.5 Product types . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6 Dependent pair types (Σ-types) . . . . . . . . . . . . . . . . 39
1.7 Coproduct types . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.8 The type of booleans . . . . . . . . . . . . . . . . . . . . . . 45
1.9 The natural numbers . . . . . . . . . . . . . . . . . . . . . . 47
1.10 Pattern matching and recursion . . . . . . . . . . . . . . . . 51
1.11 Propositions as types . . . . . . . . . . . . . . . . . . . . . . 52
1.12 Identity types . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2 Homotopy type theory 73
2.1 Types are higher groupoids . . . . . . . . . . . . . . . . . . 77
2.2 Functions are functors . . . . . . . . . . . . . . . . . . . . . 86
2.3 Type families are fibrations . . . . . . . . . . . . . . . . . . 87
2.4 Homotopies and equivalences . . . . . . . . . . . . . . . . . 92
2.5 The higher groupoid structure of type formers . . . . . . . 96
2.6 Cartesian product types . . . . . . . . . . . . . . . . . . . . 98
2.7 Σ-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.8 The unit type . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
viii Contents
2.9 Π-types and the function extensionality axiom . . . . . . . 104
2.10 Universes and the univalence axiom . . . . . . . . . . . . . 107
2.11 Identity type . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.12 Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.13 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . 115
2.14 Example: equality of structures . . . . . . . . . . . . . . . . 117
2.15 Universal properties . . . . . . . . . . . . . . . . . . . . . . 120
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3 Sets and logic 129
3.1 Sets and n-types . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2 Propositions as types? . . . . . . . . . . . . . . . . . . . . . 132
3.3 Mere propositions . . . . . . . . . . . . . . . . . . . . . . . . 135
3.4 Classical vs. intuitionistic logic . . . . . . . . . . . . . . . . 137
3.5 Subsets and propositional resizing . . . . . . . . . . . . . . 139
3.6 The logic of mere propositions . . . . . . . . . . . . . . . . 141
3.7 Propositional truncation . . . . . . . . . . . . . . . . . . . . 142
3.8 The axiom of choice . . . . . . . . . . . . . . . . . . . . . . . 144
3.9 The principle of unique choice . . . . . . . . . . . . . . . . . 147
3.10 When are propositions truncated? . . . . . . . . . . . . . . 148
3.11 Contractibility . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4 Equivalences 159
4.1 Quasi-inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.2 Half adjoint equivalences . . . . . . . . . . . . . . . . . . . 163
4.3 Bi-invertible maps . . . . . . . . . . . . . . . . . . . . . . . . 168
4.4 Contractible fibers . . . . . . . . . . . . . . . . . . . . . . . . 169
4.5 On the definition of equivalences . . . . . . . . . . . . . . . 170
4.6 Surjections and embeddings . . . . . . . . . . . . . . . . . . 171
4.7 Closure properties of equivalences . . . . . . . . . . . . . . 172
4.8 The object classifier . . . . . . . . . . . . . . . . . . . . . . . 175
4.9 Univalence implies function extensionality . . . . . . . . . 178
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Contents ix
5 Induction 183
5.1 Introduction to inductive types . . . . . . . . . . . . . . . . 183
5.2 Uniqueness of inductive types . . . . . . . . . . . . . . . . . 186
5.3 W-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.4 Inductive types are initial algebras . . . . . . . . . . . . . . 194
5.5 Homotopy-inductive types . . . . . . . . . . . . . . . . . . 197
5.6 The general syntax of inductive definitions . . . . . . . . . 202
5.7 Generalizations of inductive types . . . . . . . . . . . . . . 207
5.8 Identity types and identity systems . . . . . . . . . . . . . . 210
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6 Higher inductive types 219
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.2 Induction principles and dependent paths . . . . . . . . . . 222
6.3 The interval . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.4 Circles and spheres . . . . . . . . . . . . . . . . . . . . . . . 229
6.5 Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.6 Cell complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 237
6.7 Hubs and spokes . . . . . . . . . . . . . . . . . . . . . . . . 238
6.8 Pushouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
6.9 Truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.10 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.11 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.12 The flattening lemma . . . . . . . . . . . . . . . . . . . . . . 259
6.13 The general syntax of higher inductive definitions . . . . . 266
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
7 Homotopy n-types 271
7.1 Definition of n-types . . . . . . . . . . . . . . . . . . . . . . 272
7.2 Uniqueness of identity proofs and Hedberg’s theorem . . . 276
7.3 Truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
7.4 Colimits of n-types . . . . . . . . . . . . . . . . . . . . . . . 287
7.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 292
7.6 Orthogonal factorization . . . . . . . . . . . . . . . . . . . . 298
7.7 Modalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311