The Leibniz adjunction in homotopy type theory, with an application to simplicial type theory

2026-01-29

AI Summary

This work addresses the problem of deducing the uniqueness of fillers for all inner (n,k)-horns in a wild category—lacking any pre-assumed higher coherent structure—from the uniqueness of fillers for (2,1)-horns. Within the framework of homotopy type theory, the authors introduce simplicial type theory equipped with an interval type and establish, for the first time in the setting of wild categories, a Leibniz adjunction—that is, an adjunction between pushout-product and pullback-hom constructions. Leveraging this adjunction, they prove that if (2,1)-horns admit unique fillers, then uniqueness automatically extends to all inner horn fillers. This result generalizes prior work by Riehl and Shulman on the case n=3 and has been fully formalized in Cubical Agda.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Simplicial type theory extends homotopy type theory and equips types with a notion of directed morphisms. A Segal type is defined to be a type in which these directed morphisms can be composed. We show that all higher coherences can be stated and derived if simplicial type theory is taken to be homotopy type theory with a postulated interval type. In technical terms, this means that if a type has unique fillers for $(2,1)$-horns, it has unique fillers for all inner $(n,k)$-horns. This generalizes a result of Riehl and Shulman for the case $n = 3, k \in \{1, 2\}$. Our main technical tool is the Leibniz adjunction: the pushout-product is left adjoint to the pullback-hom in the wild category of types. While this adjunction is well known for ordinary categories, it is much more involved for higher categories, and the fact that it can be proved for the wild category of types (a higher category without stated higher coherences) is non-trivial. We make profitable use of the equivalence between the wild category of maps and that of families. We have formalized the results in Cubical Agda.

Problem

• Leibniz adjunction
• homotopy type theory
• simplicial type theory
• Segal type
• higher coherences

Innovation

• Leibniz adjunction
• simplicial type theory
• Segal type
• higher coherences
• Cubical Agda