Simplicial HoTT and synthetic ∞-categories

:information_source: This project originated as a fork of https://github.com/emilyriehl/yoneda.

This is a formalization library for simplicial Homotopy Type Theory (sHoTT) with the aim of proving results in synthetic ∞-category theory, starting with the results from the following papers:

• "A type theory for synthetic ∞-categories" [1]
• "Synthetic fibered (∞,1)-category theory" [2]
• "Limits and colimits of synthetic ∞-categories" [3]

This formalization project follows the philosophy laid out in the article "Could ∞-category theory be taught to undergraduates?" [4].

The formalizations are implemented using rzk, an experimental proof assistant for a variant of type theory with shapes. See the list of contributors at src/CONTRIBUTORS.md.

The formalizations can be viewed as markdown files rendered at rzk-lang.github.io/sHoTT/ using syntax highlighting supplied by MkDocs plugin for Rzk.

Checking the formalisations locally

Install the rzk proof assistant. Then run the following command from the root of this repository:

rzk typecheck

Discussing sHoTT, Rzk, and getting help

A Zulip chat is available for all to join and chat about Rzk, including formalization projects such as this one, development of Rzk, and related projects: https://rzk-lang.zulipchat.com/register/

Style Guide

Please also have a look at our style guide before submitting your pull request.

References

1. Emily Riehl & Michael Shulman. A type theory for synthetic ∞-categories. Higher Structures 1(1), 147-224. 2017. https://arxiv.org/abs/1705.07442

2. Ulrik Buchholtz and Jonathan Weinberger. 2023. Synthetic fibered (∞, 1)-category theory. Higher Structures 7 (2023), 74–165. Issue 1. https://doi.org/10.21136/HS.2023.04

3. César Bardomiano Martínez. Limits and colimits of synthetic ∞-categories. Mathematical Structures in Computer Science, vol. 35, p. e24, 2025. https://doi.org/10.1017/S0960129525100248

4. Emily Riehl. Could ∞-category theory be taught to undergraduates? Notices of the AMS. May 2023. https://www.ams.org/journals/notices/202305/noti2692/noti2692.html