research-article

Public Access

Free Higher Groups in Homotopy Type Theory

Authors:

Thorsten AltenkirchAuthors Info & Claims

LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Pages 599 - 608

https://doi.org/10.1145/3209108.3209183

Published: 09 July 2018 Publication History

Abstract

Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit: F(A), cons: A~F(A)~F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [22, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ||F(A)||1 is a set.

References

[1]

John Frank Adams. 1978. Infinite loop spaces. Annals of Mathematics Studies 90 (1978).

[2]

Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, Nicolai Kraus, and Fredrik Nordvall Forsberg. 2018. Quotient inductive-inductive types. In Foundations of Software Science and Computation Structures (FoSSaCS 2018), Christel Baier and Ugo Dal Lago (Eds.). Springer International Publishing, 293--310.

[3]

Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus. 2016. Extending Homotopy Type Theory with Strict Equality. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016), Jean-Marc Talbot and Laurent Regnier (Eds.), Vol. 62. Dagstuhl, Germany, 21:1--21:17.

[4]

Thorsten Altenkirch, Nils Anders Danielsson, and Nicolai Kraus. 2017. Partiality, Revisited. In Foundations of Software Science and Computation Structures (FoSSaCS 2017), Javier Esparza and Andrzej S. Murawski (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 534--549.

Digital Library

[5]

Thorsten Altenkirch and Ambrus Kaposi. 2016. Type Theory in Type Theory Using Quotient Inductive Types. SIGPLAN Not. 51, 1 (Jan. 2016), 18--29.

Digital Library

[6]

C. Angiuli, K.-B. Hou, and R. Harper. 2017. Computational Higher Type Theory III: Univalent Universes and Exact Equality. ArXiv e-prints (Dec. 2017). arXiv:cs.LO/1712.01800

[7]

Danil Annenkov, Paolo Capriotti, and Nicolai Kraus. 2017. Two-Level Type Theory and Applications. Arxiv e-prints (2017). http://arxiv.org/abs/1705.03307

[8]

Steve Awodey, Nicola Gambino, and Kristina Sojakova. 2012. Inductive Types in Homotopy Type Theory. In Logic in Computer Science (LICS). IEEE Computer Society, Washington, DC, USA, 95--104.

Digital Library

[9]

Guillaume Brunerie. 2017. The James construction and π4 (S3) in homotopy type theory. CoRR (2017). http://arxiv.org/abs/1710.10307

[10]

Ulrik Buchholtz, Floris van Doorn, and Egbert Rijke. 2018. Higher Groups in Homotopy Type Theory. (2018). https://arxiv.org/abs/1802.04315

[11]

Paolo Capriotti. 2016. Models of Type Theory with Strict Equality. Ph.D. Dissertation. School of Computer Science, University of Nottingham, Nottingham, UK. Available online at https://arxiv.org/abs/1702.04912.

[12]

Paolo Capriotti and Nicolai Kraus. 2017. Univalent Higher Categories via Complete Semi-Segal Types. Proc. ACM Program. Lang. 2, POPL, Article 44 (Dec. 2017), 29 pages.

Digital Library

[13]

Paolo Capriotti, Nicolai Kraus, and Andrea Vezzosi. 2015. Functions out of Higher Truncations. In 24th EACSL Annual Conference on Computer Science Logic (CSL) 2015) (Leibniz International Proceedings in Informatics (LIPIcs)), Stephan Kreutzer (Ed.), Vol. 41. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 359--373.

[14]

Nicolai Kraus. 2015. The General Universal Property of the Propositional Truncation. In 20th International Conference on Types for Proofs and Programs (TYPES 2014) (Leibniz International Proceedings in Informatics (LIPIcs)), Hugo Herbelin, Pierre Letouzey, and Matthieu Sozeau (Eds.), Vol. 39. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 111--145.

[15]

Nicolai Kraus. 2015. Truncation Levels in Homotopy Type Theory. Ph.D. Dissertation. School of Computer Science, University of Nottingham, Nottingham, UK.

[16]

Nicolai Kraus, Martín H. Escardó, Thierry Coquand, and Thorsten Altenkirch. 2017. Notions of Anonymous Existence in Martin-Löf Type Theory. Logical Methods in Computer Science Volume 13, Issue 1 (mar 2017). In the special issue of TLCA'13.

[17]

Daniel R. Licata and Michael Shulman. 2013. Calculating the Fundamental Group of the Circle in Homotopy Type Theory. In Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS '13). IEEE Computer Society, Washington, DC, USA, 223--232.

Digital Library

[18]

Peter LeFanu Lumsdaine. 2009. Weak omega-Categories from Intensional Type Theory. In Typed Lambda Calculi and Applications (TLCA). Springer-Verlag, 172---187.

Digital Library

[19]

Emily Riehl and Dominic Verity. 2016. Homotopy coherent adjunctions and the formal theory of monads. Advances in Mathematics 286 (2016), 802--888.

[20]

Kristina Sojakova. 2015. Higher Inductive Types As Homotopy-Initial Algebras. In Principles of Programming Languages (POPL). ACM, New York, NY, USA, 31--42.

Digital Library

[21]

Jim Stasheff. 1963. Homotopy associativity of H-spaces. Trans. Amer. Math. Soc. 108 (1963), 275--292, 293--312.

[22]

The Univalent Foundations Program. 2013. Homotopy Type Theory: Univalent Foundations of Mathematics. http://homotopytypetheory.org/book/, Institute for Advanced Study.

[23]

The Univalent Foundations Program. 2013. Semi-simplicial types. (2013). Wiki page of the Univalent Foundations project at the Institute for Advanced Studies, https://uf-ias-2012.wikispaces.com/Semi-simplicial+types.

[24]

Benno van den Berg and Richard Garner. 2011. Types are Weak ω-Groupoids. Proceedings of the London Mathematical Society 102, 2 (2011), 370--394.

[25]

Vladimir Voevodsky. 2013. A simple type system with two identity types. (2013). Unpublished note.

Cited By

Kraus NGorla D(2021)Internal ∞-categorical models of dependent type theoryProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470667(1-14)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470667
Kraus Nvon Raumer JHermanns HZhang LKobayashi NMiller D(2020)Coherence via Well-FoundednessProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394800(662-675)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394800
Altenkirch TScoccola LHermanns HZhang LKobayashi NMiller D(2020)The Integers as a Higher Inductive TypeProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394760(67-73)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394760
Show More Cited By

Index Terms

Free Higher Groups in Homotopy Type Theory
1. Theory of computation
  1. Logic
    1. Type theory

Recommendations

Higher Groups in Homotopy Type Theory
LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure ...
Eilenberg-MacLane spaces in homotopy type theory
CSL-LICS '14: Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very ...
Type theory in type theory using quotient inductive types
POPL '16: Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

We present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

July 2018

960 pages

ISBN:9781450355834

DOI:10.1145/3209108

Copyright © 2018 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGLOG: ACM Special Interest Group on Logic and Computation
EACSL: European Association for Computer Science Logic
IEEE-CS\DATC: IEEE Computer Society

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 July 2018

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed limited

Funding Sources

Conference

LICS '18

Sponsor:

SIGLOG
EACSL
IEEE-CS\DATC

LICS '18: 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

July 9 - 12, 2018

Oxford, United Kingdom

Acceptance Rates

Overall Acceptance Rate 215 of 622 submissions, 35%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
329
Total Downloads

Downloads (Last 12 months)90
Downloads (Last 6 weeks)14

Reflects downloads up to 03 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Kraus NGorla D(2021)Internal ∞-categorical models of dependent type theoryProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470667(1-14)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470667
Kraus Nvon Raumer JHermanns HZhang LKobayashi NMiller D(2020)Coherence via Well-FoundednessProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394800(662-675)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394800
Altenkirch TScoccola LHermanns HZhang LKobayashi NMiller D(2020)The Integers as a Higher Inductive TypeProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394760(67-73)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394760
Kraus Nvon Raumer JBouyer P(2019)Path spaces of higher inductive types in homotopy type theoryProceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3470152.3470159(1-13)Online publication date: 24-Jun-2019
https://dl.acm.org/doi/10.5555/3470152.3470159
Kraus Nvon Raumer J(2019)Path Spaces of Higher Inductive Types in Homotopy Type Theory2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2019.8785661(1-13)Online publication date: Jun-2019
https://doi.org/10.1109/LICS.2019.8785661

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Figures

Tables

Media

View Table of Conten