A Relationship between Equilogical Spaces and Type Two Effectivity

Abstract

In this paper I compare two well studied approaches to topological semanticsthe 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.

Quotients of countably based spaces are not closed under sobrification

“…The same holds for function realisability since ExPer(N N ) ExPer(Pω), as shown in Bauer (2002). Whether our counterexample can be adapted to number realisability remains a task for future investigations.…”

Effectivity in Spaces with Admissible Multirepresentations

“…[2,3]). Such a representation η can be found in [13,14,2,3] or constructed from one for the Cantor space (e.g. the one in [16]).…”

“…The main idea of the proofs of the equalities PQ = AdmSeq and PQ L = AdmLim is due to A. Bauer (cf. [2,3]). M. Menni and A. Simpson characterize PQ as the largest common full subcategory of Seq and ωEqu, the category of countably-based equilogical spaces, and PQ L as the largest common full subcategory of ωEqu and Lim (cf.…”

“…If γ is an admissible multirepresentation of Y and h : X → Y reflects convergent sequences3) , then h −1 • γ is 4) an admissible multirepresentation of X.P r o o f. (1) Clearly, δ ⊗ γ is continuous. For every continuous correspondence φ :⊆ B ⇒ X × Y , the projections pr 1 • φ and pr 2 • φ are continuous as well.…”