{"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/diagrammatic-sets-and-rewriting-in-weak","title":"Diagrammatic sets and rewriting in weak higher categories","arxiv_id":"2007.14505","date":"2020-07-28","proceeding":null,"authors":["Amar Hadzihasanovic"],"abstract":"We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories. In the first part, we elaborate on Steiner's theory of directed complexes as a combinatorial foundation. We individuate convenient classes of directed complexes and develop the theory of diagrammatic sets relative to one such class. We study a notion of equivalence internal to a diagrammatic set, and single out as models of weak omega-categories those diagrammatic sets whose every composable diagram is connected by an equivalence to a single cell. We then define a semistrict model providing algebraic composites and study the embedding of strict omega-categories into this model. Finally, we prove a version of the homotopy hypothesis for the infinity-groupoids in the weak model, and exhibit a specific mistake in a proof by Kapranov and Voevodsky that had previously been refuted indirectly.","url_abs":"http://arxiv.org/abs/2007.14505v1","url_pdf":"http://arxiv.org/pdf/2007.14505v1.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":"diagrammatic-sets-and-rewriting-in-weak","repo_url":"https://github.com/ahadziha/rewalt","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}