A 2-adjunction between representations and preorder morphisms
For researchers in category theory and theoretical computer science, this work offers a theoretical link between a new model and a classical one, but it is incremental and not yet peer-reviewed.
The paper establishes a 2-adjunction between representations and preorder morphisms, providing a formal connection that justifies representations as a natural construct and suggests classical results on order-preserving maps may apply to representations.
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.