Natural dualities in partnership
Priestley duality for the variety \(\mathcal{D}\) of bounded distributive lattices sets up a dual equivalence between \(\mathcal{D}\) and the category of Priestley spaces. Banaschewski's duality sets up a dual equivalence between the category of Boolean topological distributive lattices and the category of ordered sets. These partner dualities are set up by hom-functors into a pair of structures: a structure (with no topology) and a topological structure acting as its \textit{alter ego}. Thus the two dualities are related to one another by topology swapping: each can be obtained from its partner by removing the topology from the \textit{alter ego} and applying it to the untopologised structure. In their paper the authors demonstrate that the above examples illustrate a very general procedure that takes a duality and swaps the topology from the structure side to the algebra side to obtain a new partner duality paired with the original one. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences. The theorem is illustrated via a variety of natural dualities, some classic and some less familiar.
scientific article; zbMATH DE number 6121298
scientific article