良滤 dcpo 的范畴不是 Γ-忠实的
The category of well-filtered dcpos is not $Γ$-faithful
AI总结：
本文证明良滤 dcpo 的范畴不是 Γ-忠实的，从而说明良滤 dcpo 一般不必受控，同时确认 Ω*-紧 dcpo 范畴是 Γ-忠实的，并提出严格大于 DOMI 的弱受控 dcpo 类并证明其 Γ-忠实性。
摘要：
Ho-Zhao 问题询问：任意两个具有同构 Scott 闭集格的 dcpo 是否自身同构，即 dcpo 与 Scott 连续映射构成的范畴 $\mathbf{DCPO}$ 是否为 $Γ$-忠实的。2018 年，Ho、Goubault-Larrecq、Jung 和 Xi 对此问题给出了否定回答，并引入了受控 dcpo 的范畴 $\mathbf{DOMI}$，证明其是 $Γ$-忠实的。受控 dcpo 涵盖域理论中许多常见的 dcpo 族，例如有界完备 dcpo 的范畴和 sober dcpo 的范畴等。然而，受控 dcpo 的范畴是否涵盖所有良滤 dcpo 仍是未知的，而良滤 dcpo 是严格大于有界完备格和 sober dcpo 的类。本文针对这一非常自然的问题展开研究，证明良滤 dcpo 的范畴 $\mathbf{WF}$ 不是 $Γ$-忠实的，由此可知良滤 dcpo 一般不必是受控的。由于并非所有 dcpo 都是良滤的，我们的工作细化了 Ho、Goubault-Larrecq、Jung 和 Xi 的结果。作为第二个贡献，我们确认了 Lawson 的 $Ω^{*}$-紧 dcpo 范畴是 $Γ$-忠实的。此外，我们定位了一类我们称之为弱受控 dcpo 的 dcpo，并证明该类是 $Γ$-忠实的，且严格大于 $\mathbf{DOMI}$。
英文摘要：
The Ho-Zhao problem asks whether any two dcpo's with isomorphic Scott closed set lattices are themselves isomorphic, that is, whether the category $\mathbf{DCPO}$ of dcpo's and Scott-continuous maps is $Γ$-faithful. In 2018, Ho, Goubault-Larrecq, Jung and Xi answered this question in the negative, and they introduced the category $\mathbf{DOMI}$ of dominated dcpo's and proved that it is {$Γ$-faithful}. Dominated dcpo's subsume many familiar families of dcpo's in domain theory, such as the category of bounded-complete dcpo's and that of sober dcpo's, among others. However, it is unknown whether the category of dominated dcpo's dominates all well-filtered dcpo's, a class strictly larger than that of bounded-complete lattices and that of sober dcpo's. In this paper, we address this very natural question and show that the category $\mathbf{WF}$ of well-filtered dcpo's is not $Γ$-faithful, and as a result of it, well-filtered dcpo's need not be dominated in general. Since not all dcpo's are well-filtered, our work refines the results of Ho, Goubault-Larrecq, Jung and Xi. As a second contribution, we confirm that the Lawson's category of $Ω^{*}$-compact dcpo's is $Γ$-faithful. Moreover, we locate a class of dcpo's which we call weakly dominated dcpo's, and show that this class is $Γ$-faithful and strictly larger than $\mathbf{DOMI}$.
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