[2607.12207] Rzk: a Proof Assistant for Synthetic $\infty$-Categories

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arXiv:2607.12207 (cs)

[Submitted on 13 Jul 2026]

Title:Rzk: a Proof Assistant for Synthetic $\infty$-Categories

Authors:Nikolai Kudasov, Violetta Sim, Benedikt Ahrens<br>View a PDF of the paper titled Rzk: a Proof Assistant for Synthetic $\infty$-Categories, by Nikolai Kudasov and 2 other authors

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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.

Comments:<br>54 pages, including appendices. Describes Rzk v0.7.8. Ancillary files include the code of every example in the paper and the scripts and trace behind the evaluation

Subjects:

Logic in Computer Science (cs.LO); Programming Languages (cs.PL); Category Theory (math.CT)

Cite as:<br>arXiv:2607.12207 [cs.LO]

(or<br>arXiv:2607.12207v1 [cs.LO] for this version)

https://doi.org/10.48550/arXiv.2607.12207

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arXiv-issued DOI via DataCite (pending registration)

Submission history<br>From: Nikolai Kudasov [view email]<br>[v1]<br>Mon, 13 Jul 2026 23:16:49 UTC (189 KB)

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