Rzk: A Proof Assistant for Synthetic ∞-Categories

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.

Computer Science Logic in Computer Science Submitted on 13 Jul 2026 Title:Rzk: a Proof Assistant for Synthetic $\infty$-Categories View PDF /pdf/2607.12207 Abstract: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. Directed 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. We 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. We 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. Current browse context: cs.LO References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both 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. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .