{"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/categories-of-optics","title":"Categories of Optics","arxiv_id":"1809.00738","date":"2018-09-03","proceeding":null,"authors":["Mitchell Riley"],"abstract":"Bidirectional data accessors such as lenses, prisms and traversals are all instances of the same general 'optic' construction. We give a careful account of this construction and show that it extends to a functor from the category of symmetric monoidal categories to itself. We also show that this construction enjoys a universal property: it freely adds counit morphisms to a symmetric monoidal category. Missing in the folklore is a general definition of 'lawfulness' that applies directly to any optic category. We provide such a definition and show that it is equivalent to the folklore profunctor optic laws.","url_abs":"https://arxiv.org/abs/1809.00738v2","url_pdf":"https://arxiv.org/pdf/1809.00738v2.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":"categories-of-optics","repo_url":"https://github.com/masaeedu/newlenses","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"categories-of-optics","repo_url":"https://github.com/mroman42/vitrea","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1809.00738","atlas_url":"https://app.syntology.ai/?focus=1809.00738","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}