Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

Abstract

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.