Ho-Zhao问题
The Ho-Zhao Problem
AI总结：
本文否定回答了Ho-Zhao问题：通过引入新的Γ-忠实dcpo子范畴，并借助Johnstone非sober的dcpo构造反例，证明dcpo范畴DCPO不是Γ-忠实的。
摘要：
给定一个偏序集$P$，所有Scott闭集按包含序构成的集合$\Gamma(P)$形成一个完全格。$\mathbf{Pos}_d$（偏序集与Scott连续映射构成的范畴）的一个子范畴$\mathbf{C}$被称为$\Gamma$-忠实的，如果对于$\mathbf{C}$中任意偏序集$P$和$Q$，$\Gamma(P) \cong \Gamma(Q)$蕴含$P \cong Q$。已知所有连续dcpo构成的范畴和有界完全dcpo构成的范畴是$\Gamma$-忠实的，而$\mathbf{Pos}_d$不是。Ho与Zhao（2009）问dcpo构成的范畴$\mathbf{DCPO}$是否是$\Gamma$-忠实的。在本文中，我们通过构造一个反例否定地回答了这个问题。为此，我们引入了一个新的dcpo子范畴，它是$\Gamma$-忠实的。该子范畴包含了所有目前已知的$\Gamma$-忠实子范畴。基于这一新概念，我们构造了所需的反例，该反例严重依赖于Johnstone著名的在其Scott拓扑下不sober的dcpo。
英文摘要：
Given a poset $P$, the set, $Γ(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of $\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $Γ$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $Γ(P) \cong Γ(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $Γ$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $Γ$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $Γ$-faithful. This subcategory subsumes all currently known $Γ$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
