Preliminary Proceedings of the 17 th Annual Conference on Mathematical Foundations of Programming Semantics MFPS ’ 01

In this paper I compare two well studied approaches to topological semantics| the domain-theoretic approach, exempli ed by the category of countably based equilogical spaces, Equ, and Type Two E ectivity, exempli ed 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 core ective 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 core ection 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 di erent. 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.