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

2026-07-13

AI Summary

Existing homotopy type theories are restricted to ∞-groupoids and struggle to support synthetic reasoning about ∞-categories with non-invertible higher morphisms. To address this limitation, this work designs and implements a proof assistant called Rzk based on Riehl–Shulman simplicial type theory (RSTT). We introduce a faithful and conservative translation from RSTT into Rzk that preserves theoretical expressiveness while enabling practical type checking. By integrating dependent types, an automated prover for shape logic, and a custom type-checking algorithm, Rzk provides the first viable framework for formal reasoning about ∞-categories. The implementation is accompanied by comprehensive tutorials and has been validated through practical proof development, demonstrating its effectiveness in real-world formalization tasks.

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.

Problem

• synthetic ∞-categories
• directed type theory
• proof assistant
• type checking
• simplicial type theory

Innovation

• synthetic ∞-categories
• directed type theory
• simplicial type theory
• proof assistant
• type checking