Investigations into Algebra and Topology over Nominal Sets

The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach lies the existence of an adjunction of descent type between nominal sets and a category of many-sorted sets. Hence nominal sets are a full reflective subcategory of a many-sorted variety. This is presented in Chapter 2. Chapter 3 introduces functors over many-sorted varieties that can be presented by operations and equations. These are precisely the functors that preserve sifted colimits. They play a central role in Chapter 4, which shows how one can systematically transfer results of universal algebra from a many-sorted variety to nominal sets. However, the equational logic obtained is more expressive than the nominal equational logic of Clouston and Pitts, respectively, the nominal algebra of Gabbay and Mathijssen. A uniform fragment of our logic with the same expressivity as nominal algebra is given. In the second part, we give an account of duality theory in the nominal setting. Chapter 5 shows that Stone’s representation theorem cannot be internalized in nominal sets. This is due to the fact that the adjunction between nominal Boolean algebras and nominal sets is no longer of descent type. We prove a duality theorem for nominal distributive lattices with a restriction operation in terms of nominal bitopological spaces. This duality restricts to duality between nominal Boolean algebras and a category of nominal topological spaces. Our notion of compactness allows for generalisation of Manes’ theorem to the nominal setting.