Dualities between finitely separated structures

TL;DR

This paper explores dualities in relational topological structures, revealing a universal 2-element structure with many relations that can reconstruct all similar structures, and applies this to convexity structures and known dualities.

Contribution

It introduces a universal 2-element structure with infinitely many relations that can generate all structures from a 2-element base, and applies this to various classes including convexity structures.

Findings

Existence of a universal 2-element structure with infinitely many relations.

Natural duality established for normal convexity structures.

Simplified proofs for existing dualities in 2-element generated classes.

Abstract

We study dualities between classes of relational topological structures, given by Hom-functors. We show that there exists a 2-element structure with infinitely many relations, which reconstructs all other structures generated by a 2-element one. As an application, we find a natural duality for the class of normal convexity structures. As another application, we give short proofs for several known dualities for classes of structures generated by a fixed 2-element structure.