  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.
