Topological representations for frame-valued domains via $L$-sobriety

TL;DR

This paper develops a topological framework for frame-valued domains using $L$-sobriety, establishing categorical isomorphisms between $L$-dcpos and $L$-sober spaces, and applies this to continuous $L$-posets and their completions.

Contribution

It introduces new notions of locally super-compact $L$-topological spaces and demonstrates their role in representing $L$-dcpos via $L$-sobriety, establishing categorical equivalences.

Findings

Categorical isomorphism between continuous $L$-dcpos and $L$-sober spaces.

Representation of algebraic $L$-dcpos via $L$-sobriety.

Equivalence between directed completions and $L$-sobrifications.

Abstract

With a frame $L$ as the truth value table, we study the topological representations for frame-valued domains. We introduce the notions of locally super-compact $L$-topological space and strong locally super-compact $L$-topological space. Using these concepts, continuous $L$-dcpos and algebraic $L$-dcpos are successfully represented via $L$-sobriety. By means of Scott $L$-topology and specialization $L$-order, we establish a categorical isomorphism between the category of the continuous (resp., algebraic) $L$-dcpos with Scott continuous maps and that of the locally super-compact (resp., strong locally super-compact) $L$-sober spaces with continuous maps. As an application, for a continuous $L$-poset $P$, we obtain a categorical isomorphism between the category of directed completions of $P$ with Scott continuous maps and that of the $L$-sobrifications of $(P, \sigma_{L}(P))$ with continuous maps.