{"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/an-equivalent-presentation-of-the-bezem","title":"An Equivalent Presentation of the Bezem-Coquand-Huber Category of Cubical Sets","arxiv_id":"1401.7807","date":"2014-01-30","proceeding":null,"authors":["Andrew M. Pitts"],"abstract":"Staton has shown that there is an equivalence between the category of presheaves on (the opposite of) finite sets and partial bijections and the category of nominal restriction sets: see [2, Exercise 9.7]. The aim here is to see that this extends to an equivalence between the category of cubical sets introduced in [1] and a category of nominal sets equipped with a \"01-substitution\" operation. It seems to me that presenting the topos in question equivalently as 01-substitution sets rather than cubical sets will make it easier (and more elegant) to carry out the constructions and calculations needed to build the intended univalent model of intentional constructive type theory.","url_abs":"http://arxiv.org/abs/1401.7807v1","url_pdf":"http://arxiv.org/pdf/1401.7807v1.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":"an-equivalent-presentation-of-the-bezem","repo_url":"https://github.com/hansbugge/cubicaltt","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"an-equivalent-presentation-of-the-bezem","repo_url":"https://github.com/mortberg/cubicaltt","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"an-equivalent-presentation-of-the-bezem","repo_url":"https://github.com/simhu/cubical","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}