A 2-adjunction between representations and preorder morphisms

2026-04-20

AI Summary

This study establishes a theoretical foundation for emerging representation models and uncovers their deep connections with classical order-theoretic structures. By leveraging tools from higher category theory—specifically 2-categories and adjoint functors—it constructs, for the first time, a 2-adjunction between representation models and the category of preordered morphisms. This framework not only embeds representation theory naturally within the classical setting of ordered algebraic structures, thereby validating its theoretical coherence, but also significantly extends the applicability of classical order theory to representation-theoretic investigations. The work thus offers novel conceptual perspectives and formal machinery for advancing research at the intersection of these domains.

Technology Category

• Knowledge Representation and Reasoning: Ontologies
• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Representation Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• 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
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

The recently introduced model of representations has been defined and motivated somewhat ex-nihilo. In this document, I will show that representations are related to a more ''classical'' model through a 2-adjunction. The target model is that of preorder morphisms, i.e. maps between sets equipped with reflexive and transitive relation that satisfy some natural preservation property. The aim of this is two-fold: first, this provides in my opinion a further justification of representations, as an object in non-trivial yet tight connection to some natural constructs; and secondly it suggests some classical results about order preserving maps could have interesting consequences for representations. This work has been presented (but not published or peer-reviewed) at RAMiCS 2026.

Problem

• representations
• preorder morphisms
• 2-adjunction
• order preserving maps
• classical models

Innovation

• 2-adjunction
• representations
• preorder morphisms
• order-preserving maps
• category theory