{"slug": "rzk-a-proof-assistant-for-synthetic-categories", "title": "Rzk: A Proof Assistant for Synthetic ∞-Categories", "summary": "Researchers have released Rzk, a proof assistant implementing a computational variant of Riehl and Shulman's simplicial type theory for synthetic reasoning about ∞-categories. The tool translates proofs from the original theory faithfully and conservatively, and includes a type-checking algorithm and automated prover for the logic of shapes.", "body_md": "# Computer Science > Logic in Computer Science\n\n[Submitted on 13 Jul 2026]\n\n# Title:Rzk: a Proof Assistant for Synthetic $\\infty$-Categories\n\n[View PDF](/pdf/2607.12207)\n\nAbstract:Homotopy type theory (HoTT) is a type theory that allows for synthetic reasoning about $\\infty$-groupoids. Several proof assistants (such as Rocq and Agda) implement variants of HoTT.\n\nDirected type theory is a type theory for synthetic reasoning about $\\infty$-categories, where morphisms (or paths) of dimension 1 are not necessarily invertible. Among the proposals for directed type theory, the most developed is Riehl and Shulman's simplicial type theory (RSTT), based on simplicial shapes such as directed intervals and triangles.\n\nWe present Rzk, a proof assistant implementing (a refinement of) RSTT for synthetic reasoning about $\\infty$-categories. Specifically, the type theory implemented by Rzk is a computational variant of RSTT adjusted to make type checking practical.\n\nWe define a translation from RSTT to Rzk and prove that it is sensible: every RSTT proof translates to an Rzk proof (faithfulness), and Rzk proves nothing new about RSTT types (conservativity). We also give a tutorial introduction to proving in Rzk, and describe its implementation, including the type-checking algorithm and the automated prover for the logic of shapes.\n\n### Current browse context:\n\ncs.LO\n\n### References & Citations\n\nLoading...\n\n# Bibliographic and Citation Tools\n\nBibliographic Explorer\n\n*(*[What is the Explorer?](https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer))\nConnected Papers\n\n*(*[What is Connected Papers?](https://www.connectedpapers.com/about))\nLitmaps\n\n*(*[What is Litmaps?](https://www.litmaps.co/))\nscite Smart Citations\n\n*(*[What are Smart Citations?](https://www.scite.ai/))# Code, Data and Media Associated with this Article\n\nalphaXiv\n\n*(*[What is alphaXiv?](https://alphaxiv.org/))\nCatalyzeX Code Finder for Papers\n\n*(*[What is CatalyzeX?](https://www.catalyzex.com))\nDagsHub\n\n*(*[What is DagsHub?](https://dagshub.com/))\nGotit.pub\n\n*(*[What is GotitPub?](http://gotit.pub/faq))\nHugging Face\n\n*(*[What is Huggingface?](https://huggingface.co/huggingface))\nScienceCast\n\n*(*[What is ScienceCast?](https://sciencecast.org/welcome))# Demos\n\n# Recommenders and Search Tools\n\nInfluence Flower\n\n*(*[What are Influence Flowers?](https://influencemap.cmlab.dev/))\nCORE Recommender\n\n*(*[What is CORE?](https://core.ac.uk/services/recommender))# arXivLabs: experimental projects with community collaborators\n\narXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.\n\nBoth individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.\n\nHave an idea for a project that will add value for arXiv's community? [ Learn more about arXivLabs](https://info.arxiv.org/labs/index.html).", "url": "https://wpnews.pro/news/rzk-a-proof-assistant-for-synthetic-categories", "canonical_source": "https://arxiv.org/abs/2607.12207", "published_at": "2026-07-15 21:49:07+00:00", "updated_at": "2026-07-15 22:25:39.180665+00:00", "lang": "en", "topics": ["artificial-intelligence", "machine-learning", "ai-research", "ai-tools", "developer-tools"], "entities": ["Rzk", "Riehl", "Shulman", "Rocq", "Agda"], "alternates": {"html": "https://wpnews.pro/news/rzk-a-proof-assistant-for-synthetic-categories", "markdown": "https://wpnews.pro/news/rzk-a-proof-assistant-for-synthetic-categories.md", "text": "https://wpnews.pro/news/rzk-a-proof-assistant-for-synthetic-categories.txt", "jsonld": "https://wpnews.pro/news/rzk-a-proof-assistant-for-synthetic-categories.jsonld"}}