{"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/a-type-theoretical-definition-of-weak-o","title":"A Type-Theoretical Definition of Weak ω-Categories","arxiv_id":"1706.02866","date":"2017-06-09","proceeding":null,"authors":["Eric Finster","Samuel Mimram"],"abstract":"We introduce a dependent type theory whose models are weak {\\omega}-categories, generalizing Brunerie's definition of {\\omega}-groupoids. Our type theory is based on the definition of {\\omega}-categories given by Maltsiniotis, himself inspired by Grothendieck's approach to the definition of {\\omega}-groupoids. In this setup, {\\omega}-categories are defined as presheaves preserving globular colimits over a certain category, called a coherator. The coherator encodes all operations required to be present in an {\\omega}-category: both the compositions of pasting schemes as well as their coherences. Our main contribution is to provide a canonical type-theoretical characterization of pasting schemes as contexts which can be derived from inference rules. Finally, we present an implementation of a corresponding proof system.","url_abs":"https://arxiv.org/abs/1706.02866v1","url_pdf":"https://arxiv.org/pdf/1706.02866v1.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":"a-type-theoretical-definition-of-weak-o","repo_url":"https://github.com/ThiBen/catt","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"a-type-theoretical-definition-of-weak-o","repo_url":"https://github.com/thibautbenjamin/catt","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"a-type-theoretical-definition-of-weak-o","repo_url":"https://github.com/thibautbenjamin/catt-formalization","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}