ebook img

Homotopy Type Theory: Univalent Foundations of Mathematics PDF

599 Pages·2013·3.17 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Homotopy Type Theory: Univalent Foundations of Mathematics

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

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.