research-article

Eilenberg-MacLane spaces in homotopy type theory

Authors:

Daniel R. Licata,

Eric FinsterAuthors Info & Claims

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)

Article No.: 66, Pages 1 - 9

https://doi.org/10.1145/2603088.2603153

Published: 14 July 2014 Publication History

Abstract

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 amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg-MacLane space K(G,n) is a space (type) whose n^th homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in G. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.

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Cited By

Cagne PBuchholtz UKraus NBezem MSobocinski PLago UEsparza J(2024)On symmetries of spheres in univalent foundationsProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662115(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662115
Mimram SOleon ESobocinski PLago UEsparza J(2024)Delooping cyclic groups with lens spaces in homotopy type theoryProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662077(1-15)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662077
Christensen J(2024)Non‐accessible localizationsJournal of Topology10.1112/topo.1233617:2Online publication date: 23-May-2024
https://doi.org/10.1112/topo.12336
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Index Terms

Eilenberg-MacLane spaces in homotopy type theory
1. Mathematics of computing
  1. Continuous mathematics
    1. Calculus
      1. Lambda calculus
  2. Mathematical analysis
    1. Calculus
      1. Lambda calculus
2. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Type structures

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cover image ACM Conferences

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)

July 2014

764 pages

ISBN:9781450328869

DOI:10.1145/2603088

Program Chairs:
Thomas Henzinger
Institute of Science and Technology (IST), Austria
,
Dale Miller
INRIA and LIX/Ecole Polytechnique, Palaiseau

Copyright © 2014 ACM.

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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New York, NY, United States

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Published: 14 July 2014

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CSL-LICS '14: JOINT MEETING OF the Twenty-Third EACSL Annual Conference on COMPUTER SCIENCE LOGIC

July 14 - 18, 2014

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CSL-LICS '14 Paper Acceptance Rate 74 of 212 submissions, 35%;

Overall Acceptance Rate 215 of 622 submissions, 35%

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Cited By

Cagne PBuchholtz UKraus NBezem MSobocinski PLago UEsparza J(2024)On symmetries of spheres in univalent foundationsProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662115(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662115
Mimram SOleon ESobocinski PLago UEsparza J(2024)Delooping cyclic groups with lens spaces in homotopy type theoryProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662077(1-15)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662077
Christensen J(2024)Non‐accessible localizationsJournal of Topology10.1112/topo.1233617:2Online publication date: 23-May-2024
https://doi.org/10.1112/topo.12336
Myers DSati HSchreiber U(2024)Topological Quantum Gates in Homotopy Type TheoryCommunications in Mathematical Physics10.1007/s00220-024-05020-8405:7Online publication date: 8-Jul-2024
https://doi.org/10.1007/s00220-024-05020-8
Christensen JScoccola L(2023)The Hurewicz theorem in homotopy type theoryAlgebraic & Geometric Topology10.2140/agt.2023.23.210723:5(2107-2140)Online publication date: 25-Jul-2023
https://doi.org/10.2140/agt.2023.23.2107
Lamiaux TLjungström AMörtberg AKrebbers RTraytel DPientka BZdancewic S(2023)Computing Cohomology Rings in Cubical AgdaProceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3573105.3575677(239-252)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3573105.3575677
Buchholtz URijke E(2023) The long exact sequence of homotopy n -groups Mathematical Structures in Computer Science10.1017/S0960129523000038(1-9)Online publication date: 7-Sep-2023
https://doi.org/10.1017/S0960129523000038
Wärn D(2023)Eilenberg–Maclane spaces and stabilisation in homotopy type theoryJournal of Homotopy and Related Structures10.1007/s40062-023-00330-518:2-3(357-368)Online publication date: 21-Sep-2023
https://doi.org/10.1007/s40062-023-00330-5
Finster EMimram SLucas MSeiller T(2021)A Cartesian Bicategory of Polynomial Functors in Homotopy Type TheoryElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.351.5351(67-83)Online publication date: 29-Dec-2021
https://doi.org/10.4204/EPTCS.351.5
Capriotti PDanielsson NVezzosi AGorla D(2021)Higher lensesProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470613(1-13)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470613
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