{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-hurewicz-theorem-in-homotopy-type-theory","title":"The Hurewicz theorem in Homotopy Type Theory","arxiv_id":"2007.05833","date":"2020-07-11","proceeding":null,"authors":["J. Daniel Christensen","Luis Scoccola"],"abstract":"We prove the Hurewicz theorem in homotopy type theory, i.e., that for $X$ a pointed, $(n-1)$-connected type $(n \\geq 1)$ and $A$ an abelian group, there is a natural isomorphism $\\pi_n(X)^{ab} \\otimes A \\cong \\tilde{H}_n(X; A)$ relating the abelianization of the homotopy groups with the homology. We also compute the connectivity of a smash product of types and express the lowest non-trivial homotopy group as a tensor product. Along the way, we study magmas, loop spaces, connected covers and prespectra, and we use $1$-coherent categories to express naturality and for the Yoneda lemma. As homotopy type theory has models in all $\\infty$-toposes, our results can be viewed as extending known results about spaces to all other $\\infty$-toposes.","url_abs":"https://arxiv.org/abs/2007.05833v3","url_pdf":"https://arxiv.org/pdf/2007.05833v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-hurewicz-theorem-in-homotopy-type-theory","repo_url":"https://github.com/jdchristensen/HoTT","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}