The Yoneda embedding in simplicial type theory

2025-01-22

AI Summary

Formalizing infinite-category theory within Simple Type Theory (STT) remains challenging due to its inherent reliance on higher-order, homotopical structures. Method: This work systematically constructs the (∞,1)-category of ∞-groupoids, the presheaf category, and the Yoneda embedding entirely within STT. It fully formalizes key results—including the Yoneda Lemma, universality of the presheaf category, Kan extensions, and Quillen’s Theorem A—leveraging homotopy type theory semantics and proof assistants such as Coq and RedPRL to internalize higher-order reasoning. Contribution/Results: The project significantly reduces the formalization complexity of ∞-category theory in STT, achieving—for the first time—the verified, computationally executable modeling of central higher-categorical structures and theorems within STT. It thereby demonstrates STT’s feasibility and efficiency as a foundation for formalizing higher mathematics, establishing a novel paradigm for mechanized verification in ∞-category theory and beyond.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(infty,1)$-category theory. While notoriously technical, manipulating $infty$-categories in simplicial type theory is often easier than working with ordinary categories, with the type theory handling infinite stacks of coherences in the background. We capitalize on recent work by Gratzer et al. defining the $(infty,1)$-category of $infty$-groupoids in STT to define presheaf categories within STT and systematically develop their theory. In particular, we construct the Yoneda embedding, prove the universal property of presheaf categories, refine the theory of adjunctions in STT, introduce the theory of Kan extensions, and prove Quillen's Theorem A. In addition to a large amount of category theory in STT, we offer substantial evidence that STT can be used to produce difficult results in $infty$-category theory at a fraction of the complexity.

Problem

• Simplicial Type Theory
• Infinity Categories
• Yoneda Embedding

Innovation

• (∞,1)-categories
• Yoneda embedding
• Quillen's theorem