The $\infty$-category of $\infty$-categories in simplicial type theory

Daniel Gratzer; Jonathan Weinberger; Ulrik Buchholtz

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arXiv · 2602.02218

The $\infty$-category of $\infty$-categories in simplicial type theory

Daniel Gratzer·Jonathan Weinberger·Ulrik Buchholtz

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Abstract

Simplicial type theory (STT) was introduced by Riehl and Shulman to leverage homotopy type theory to prove results about $(\infty,1)$-categories. Initial work on simplicial type theory focused on "formal" arguments in higher category theory and, in particular, no non-trivial examples of $\infty$-category theory were constructible within STT. More recent work has changed this state of affairs by applying techniques developed initial for cubical type theory to construct the $\infty$-category of spaces. We complete this process by constructing the $\infty$-category of $\infty$-categories, recovering one of the main foundational results of $\infty$-category theory (straightening--unstraightening) purely type-theoretically. We also show how this construction enables new examples of the directed version of the structure identity principle, the structure homomorphism principle.

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Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz. 2026-02-02. The $\infty$-category of $\infty$-categories in simplicial type theory. https://arxiv.org/abs/2602.02218

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