Homotopy Type Theory: Univalent Foundations of Mathematics
Peter Aczel
(1)
,
Benedikt Ahrens
(2)
,
Thorsten Altenkirch
(3)
,
Steve Awodey
(4)
,
Bruno Barras
(5)
,
Andrej Bauer
(6)
,
Yves Bertot
(7)
,
Marc Bezem
(8)
,
Thierry Coquand
(9)
,
Eric Finster
(5)
,
Daniel Grayson
(10)
,
Hugo Herbelin
(11, 12)
,
André Joyal
(8)
,
Dan Licata
(4)
,
Peter Lumsdaine
(13)
,
Assia Mahboubi
(14)
,
Per Martin-Löf
(15)
,
Sergey Melikhov
(16)
,
Alvaro Pelayo
(17)
,
Andrew Polonsky
(8)
,
Michael Shulman
(18)
,
Matthieu Sozeau
(11, 12)
,
Bas Spitters
(19)
,
Benno van den Berg
(20)
,
Vladimir Voevodsky
(8)
,
Michael Warren
(8)
,
Carlo Angiuli
(4)
,
Anthony Bordg
(21)
,
Guillaume Brunerie
(22)
,
Chris Kapulkin
(23)
,
Egbert Rijke
(4)
,
Kristina Sojakova
(4)
,
Jeremy Avigad
(4)
,
Cyril Cohen
(9)
,
Robert Constable
(24)
,
Pierre-Louis Curien
(11, 12)
,
Peter Dybjer
(9)
,
Martín Escardó
(25)
,
Kuen-Bang Hou
(4)
,
Nicola Gambino
(26)
,
Richard Garner
(27)
,
Georges Gonthier
(28)
,
Thomas Hales
(4)
,
Robert Harper
(4)
,
Martin Hofmann
(29)
,
Pieter Hofstra
(30)
,
Joachim Koch
(31)
,
Nicolai Kraus
(32)
,
Nuo Li
(33)
,
Zhaohui Luo
(34)
,
Michael Nahas
(8)
,
Erik Palmgren
(35)
,
Emily Riehl
(35)
,
Dana Scott
(36)
,
Philip Scott
(30)
,
Sergei Soloviev
(2)
1
University of Manchester [Manchester]
2 IRIT-ACADIE - Assistance à la Certification d’Applications DIstribuées et Embarquées
3 Functional Programming Laboratory
4 CMU - Carnegie Mellon University [Pittsburgh]
5 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
6 University of Ljubljana
7 MARELLE - Mathematical, Reasoning and Software
8 IAS - Institute for Advanced Study [Princeton]
9 CSE - Department of Computer Science and Engineering [Göteborg]
10 NSF - National Science Foundation [Arlington]
11 PI.R2 - Design, study and implementation of languages for proofs and programs
12 PPS - Preuves, Programmes et Systèmes
13 Stockholm University
14 SPECFUN - Symbolic Special Functions : Fast and Certified
15 IHPST - Institute for the History and Philosophy of Science and Technology
16 RFBR - Russian Foundation for Basic Research
17 WUSTL - Washington University in Saint Louis
18 SDSU - San Diego State University
19 ICIS - Institute for Computing and Information Sciences [Nijmegen]
20 ILLC - Institute for Logic, Language and Computation
21 UK - Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic]
22 LJAD - Laboratoire Jean Alexandre Dieudonné
23 UWO - University of Western Ontario
24 Cornell University [New York]
25 University of Birmingham [Birmingham]
26 School of Mathematics [Leeds]
27 DPMMS - Department of Pure Mathematics and Mathematical Statistics
28 MSR - INRIA - Microsoft Research - Inria Joint Centre
29 LMU - Institut für Informatik [München/Munich]
30 Department of Mathematics and Statistics [Ottawa]
31 JGU - Johannes Gutenberg - Universität Mainz = Johannes Gutenberg University
32 UON - University of Nottingham, UK
33 BCM - Baylor College of Medicine
34 Royal Holloway University of London - Department of Computer Science
35 Department of Mathematics Stockholm University
36 Princeton University
2 IRIT-ACADIE - Assistance à la Certification d’Applications DIstribuées et Embarquées
3 Functional Programming Laboratory
4 CMU - Carnegie Mellon University [Pittsburgh]
5 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
6 University of Ljubljana
7 MARELLE - Mathematical, Reasoning and Software
8 IAS - Institute for Advanced Study [Princeton]
9 CSE - Department of Computer Science and Engineering [Göteborg]
10 NSF - National Science Foundation [Arlington]
11 PI.R2 - Design, study and implementation of languages for proofs and programs
12 PPS - Preuves, Programmes et Systèmes
13 Stockholm University
14 SPECFUN - Symbolic Special Functions : Fast and Certified
15 IHPST - Institute for the History and Philosophy of Science and Technology
16 RFBR - Russian Foundation for Basic Research
17 WUSTL - Washington University in Saint Louis
18 SDSU - San Diego State University
19 ICIS - Institute for Computing and Information Sciences [Nijmegen]
20 ILLC - Institute for Logic, Language and Computation
21 UK - Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic]
22 LJAD - Laboratoire Jean Alexandre Dieudonné
23 UWO - University of Western Ontario
24 Cornell University [New York]
25 University of Birmingham [Birmingham]
26 School of Mathematics [Leeds]
27 DPMMS - Department of Pure Mathematics and Mathematical Statistics
28 MSR - INRIA - Microsoft Research - Inria Joint Centre
29 LMU - Institut für Informatik [München/Munich]
30 Department of Mathematics and Statistics [Ottawa]
31 JGU - Johannes Gutenberg - Universität Mainz = Johannes Gutenberg University
32 UON - University of Nottingham, UK
33 BCM - Baylor College of Medicine
34 Royal Holloway University of London - Department of Computer Science
35 Department of Mathematics Stockholm University
36 Princeton University
Steve Awodey
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Yves Bertot
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Eric Finster
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Hugo Herbelin
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Guillaume Brunerie
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Cyril Cohen
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Peter Dybjer
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Sergei Soloviev
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Résumé
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 the- ory and type theory. Homotopy theory is an outgrowth of algebraic topology and homological algebra, with relationships to higher category theory; while type theory is a branch of mathematical logic and theoretical computer science. Although the connections between the two are currently the focus of intense investigation, it is increasingly clear that they are just the beginning of a subject that will take more time and more hard work to fully understand. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids.
