Applied Categorical Structures – Official Blog

by Gerlo, A.; Vandersmissen, E.; Olmen, C.

In [2] the subconstruct ${\bf{Sob}}$ of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category ${\bf{Ap}}_{0}$ of $T_0$ approach spaces. The main result of this paper states that moreover ${\bf{Sob}}$ is firmly ${\mathcal{U}}$-reflective in ${\bf{Ap}}_{0}$ for the class ${\mathcal{U}}$ of epimorphic embeddings. ‘Firm ${\mathcal{U}}$-reflective’ is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category ${\bf{Unif}}_{0}$ of Hausdorff uniform spaces with uniformly continuous maps. It means that ${\bf{Sob}}$ is ${\mathcal{U}}$-reflective in ${\bf{Ap}}_{0}$ and that the reflector $\epsilon$ is such that $f:X \rightarrow Y$ belongs to ${\mathcal{U}}$ if and only if $\epsilon(f)$ is an isomorphism. Firm ${\mathcal{U}}$-reflectiveness implies uniqueness of completion in the sense that whenever $f:X \rightarrow Y$ is a map with $f \in {\mathcal{U}}$ and $Y$ sober, the associated $f^*: \epsilon (X) \rightarrow Y$ is an isomorphism. Our result generalizes the fact that in the category ${\bf{Top}}_{0}$ the subconstruct of sober topological spaces is firmly reflective for the class ${\mathcal{U}}_b$ of b-dense embeddings in ${\bf{Top}}_{0}$. Also firmness in some other subconstructs of ${\bf{Ap}}_{0}$ will be easily obtained.

DOI: 10.1007/s10485-006-9014-y
Online Date: 5/31/2006
Print publication date: 6/1/2006
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by Banaschewski, B.; Pultr, A.

A natural structure modelling approximation is presented and the resulting category is shown to be equivalent with the category of complete nearness frames.

DOI: 10.1007/s10485-006-9013-z
Online Date: 5/31/2006
Print publication date: 4/1/2006
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by Pultr, Aleš; Tozzi, Anna

Categories based on the Vickers’s continuous information systems and the related categories of continuous domains (algebraic domains, Scott domains, continuous lattices etc.) are shown to be both Kleisli and Eilenberg–Moore categories of a monad of ideals. Further, the functor of ideals is shown to be a completion in the sense of Brümmer, Giuli and Herrlich.

DOI: 10.1007/s10485-006-9011-1
Online Date: 5/31/2006
Print publication date: 4/1/2006
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by Verwulgen, S.

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.

DOI: 10.1007/s10485-005-9007-2
Online Date: 5/31/2006
Print publication date: 4/1/2006
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