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  • A ralationship between equilogical spaces and Type Two Effectivity
    Bauer, Andrej
    In this paper we compare two well studied approaches to topological semantics - the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, ▫$\mathbf{Equ}$▫ and ... Type Two Effectivity, exemplified by the category of Baire space representations ▫$\mathbf{Rep}(\mathbb{B}$▫. These two categories are both locally cartesian closed extensions of countably based ▫$T_0$▫-spaces. A natural question to ask is how they are related. First, we show that ▫$\mathbf{Rep}(\mathbb{B}$▫ is equivalent to a full coreflective subcategory of ▫$\mathbf{Equ}$▫, consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between ▫$\mathbf{Rep}(\mathbb{B}$▫ and ▫$\mathbf{Equ}$▫. The inclusion ▫$\mathbf{Rep}(\mathbb{B} \to \mathbf{Equ}$▫ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of ▫$\mathbf{Rep}(\mathbb{B}$▫ and ▫$\mathbf{Equ}$▫ are essentially different. However, in a second comparison we show that ▫$\mathbf{Rep}(\mathbb{B}$▫ and ▫$\mathbf{Equ}$▫ do share a common cartesian closed subcategory that contains all countably based ▫$T_0$▫-spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based ▫$T_0$▫-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.
    Vir: Mathematical logic quarterly. - ISSN 0942-5616 (Vol. 48, suppl. 1, 2002, str. 1-15)
    Vrsta gradiva - članek, sestavni del
    Leto - 2002
    Jezik - angleški
    COBISS.SI-ID - 12033369

vir: Mathematical logic quarterly. - ISSN 0942-5616 (Vol. 48, suppl. 1, 2002, str. 1-15)
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