A Relationship between Equilogical Spaces and Type Two Effectivity

In this paper I compare two well studied approaches to topological semantics— the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ, and Type Two Effectivity, exemplified by the category of Baire space representations, Rep(B). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related. First, we show that Rep(B) is equivalent to a full coreflective subcategory of Equ, consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep(B) and Equ. The inclusion Rep(B) → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep(B) and Equ are essentially different. However, in a second comparison we show that Rep(B) and Equ do share a common cartesian closed subcategory that contains all countably based T0-spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another.