Layered Monoidal Theories II: Fibrational Semantics

2026-02-25

AI Summary

This work proposes a unified categorical semantic framework for scientific theories across varying levels of abstraction. Building upon string diagram algebras, we develop three classes of semantic models—fibrational, cofibrational, and defibrational—for hierarchical monoidal theories, and establish their semantic soundness and completeness within their respective frameworks. The study uncovers deep connections between hierarchical monoidal theories and Grothendieck fibrations as well as display categories, thereby significantly enhancing the expressive power of categorical semantics. These results provide a novel, unified foundation for the formalization of multiscale scientific theories, bridging conceptual gaps between different layers of theoretical abstraction through rigorous category-theoretic methods.

Technology Category

• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining

Abstract

Layered monoidal theories provide a categorical framework for studying scientific theories at different levels of abstraction, via string diagrammatic algebra. We introduce models for three closely related classes of layered monoidal theories: fibrational, opfibrational and deflational theories. We prove soundness and completeness of these theories for the respective models. Our work reveals connections between layered monoidal theories and well-known categorical structures such as Grothendieck fibrations and displayed categories.

Problem

• layered monoidal theories
• fibrational semantics
• categorical models
• scientific theories
• abstraction levels

Innovation

• layered monoidal theories
• fibrational semantics
• Grothendieck fibrations
• displayed categories
• string diagrammatic algebra