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

2026-02-02

AI Summary

This work addresses the long-standing absence of a formal construction of nontrivial ∞-categories—particularly the ∞-category of ∞-categories—within simply typed type theory (STT). By integrating techniques from cubical type theory, the paper presents the first complete internal construction of the ∞-category of ∞-categories in STT and formally develops, in purely type-theoretic terms, the foundational straightening–unstraightening theorem of ∞-category theory. This achievement not only enables the intrinsic representation of higher-directed structures within type theory but also advances new instances of the principle of identity for directed structures and the principle of structural homomorphism. The results provide novel tools for the formalization and computational treatment of ∞-categories, thereby bridging a significant gap between higher category theory and constructive type-theoretic foundations.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Structured Learning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing

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.

Problem

• ∞-categories
• simplicial type theory
• homotopy type theory
• straightening-unstraightening
• structure identity principle

Innovation

• simplicial type theory
• ∞-categories
• straightening–unstraightening
• structure homomorphism principle
• homotopy type theory