A 2-adjunction between representations and preorder morphisms

TL;DR

This paper establishes a 2-adjunction between the recently introduced model of representations and the classical model of preorder morphisms, linking modern and traditional approaches in order theory.

Contribution

It demonstrates a formal 2-adjunction connecting representations to preorder morphisms, providing new insights and potential classical results applicable to representations.

Findings

Establishes a 2-adjunction between representations and preorder morphisms.

Provides a justification for the relevance of representations through classical constructs.

Suggests classical order-preserving results could influence the theory of representations.

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.