{"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-patch-topology-in-univalent-foundations","title":"The Patch Topology in Univalent Foundations","arxiv_id":"2402.03134","date":"2024-02-05","proceeding":null,"authors":["Igor Arrieta","Martín Hötzel Escardó","Ayberk Tosun"],"abstract":"Stone locales together with continuous maps form a coreflective subcategory of spectral locales and perfect maps. A proof in the internal language of an elementary topos was previously given by the second-named author. This proof can be easily translated to univalent type theory using resizing axioms. In this work, we show how to achieve such a translation without resizing axioms, by working with large and locally small frames with small bases. This requires predicative reformulations of several fundamental concepts of locale theory in predicative HoTT/UF, which we investigate systematically.","url_abs":"https://arxiv.org/abs/2402.03134v2","url_pdf":"https://arxiv.org/pdf/2402.03134v2.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-patch-topology-in-univalent-foundations","repo_url":"https://github.com/martinescardo/TypeTopology","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"the-patch-topology-in-univalent-foundations","repo_url":"https://github.com/tnttodda/typetopology","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}