[2405.03264] Delooping presented groups in homotopy type theory Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Logic in Computer Science arXiv:2405.03264 (cs) [Submitted on 6 May 2024 ( v1 ), last revised 30 Apr 2026 (this version, v3)] Title: Delooping presented groups in homotopy type theory Authors: Camil Champin , Samuel Mimram , Emile Oleon View a PDF of the paper titled Delooping presented groups in homotopy type theory, by Camil Champin and 2 other authors View PDF Abstract: Homotopy type theory is a logical setting based on Martin-Löf type theory in which geometric constructions and proofs can be carried out synthetically. Here, types can be interpreted as spaces up to homotopy, and proofs as homotopy-invariant constructions. In this context, the loop spaces of pointed connected groupoids provide a natural representation of groups, and every group can be realized as the loop space of such a type, which is then called a delooping of the group. There are two main methods for constructing a delooping of an arbitrary group G. The first describes it as a pointed higher inductive type, while the second takes the connected component of the principal G-torsor in the type of sets equipped with a G-action. We show that, when a presentation, or even just a generating set, is known for the group, simpler variants of these constructions can be used to build deloopings. The resulting types are more amenable to computation and lead to simpler metatheoretic reasoning. Finally, we develop a type-theoretic notion of 2-polygraph for manipulating higher inductive types such as those arising in the description of deloopings. This allows us to investigate a construction of the Cayley graph of a generated group and to show that it encodes the relations of the group, as well as a Cayley complex encoding relations between relations. Many of the developments in this article have been formalized in the cubical version of the Agda proof assistant. Subjects: Logic in Computer Science (cs.LO) ; Category Theory (math.CT) Cite as: arXiv:2405.03264 [cs.LO] (or arXiv:2405.03264v3 [cs.LO] for this version) https://doi.org/10.48550/arXiv.2405.03264 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Samuel Mimram [ view email ] [v1] Mon, 6 May 2024 08:36:21 UTC (241 KB) [v2] Tue, 1 Jul 2025 14:45:44 UTC (63 KB) [v3] Thu, 30 Apr 2026 07:47:02 UTC (60 KB) Full-text links: Access Paper: View a PDF of the paper titled Delooping presented groups in homotopy type theory, by Camil Champin and 2 other authors View PDF TeX Source view license Current browse context: cs.LO < prev | next > new | recent | 2024-05 Change to browse by: cs math math.CT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from