Categorical foundations of variety-based bornology

TL;DR

This paper introduces a new framework called variety-based bornology, extending lattice-valued bornology through bornological theories, and establishes a categorical equivalence with bornological systems inspired by topological concepts.

Contribution

It develops the concept of variety-based bornology and demonstrates its categorical relationship with bornological systems, generalizing previous topological theories.

Findings

Category of variety-based bornological spaces is isomorphic to a full reflective subcategory of bornological systems.

Introduces the concept of variety-based bornological system.

Extends topological theories to lattice-valued bornology.

Abstract

Following the concept of topological theory of S.~E.~Rodabaugh, this paper introduces a new approach to (lattice-valued) bornology, which is based in bornological theories, and which is called variety-based bornology. In particular, motivated by the notion of topological system of S.~Vickers, we introduce the concept of variety-based bornological system, and show that the category of variety-based bornological spaces is isomorphic to a full reflective subcategory of the category of variety-based bornological systems.