一句话：否定 Ho–Zhao 问题：证明 dcpo 范畴并非 Γ-faithful，并引入涵盖已知情形的 Γ-faithful dcpo 子范畴。
论文研究 Scott 闭集格 Γ(P) 能否决定偏序集 P 的范畴论问题。给定偏序集 P，其所有 Scott 闭集按包含序构成完备格 Γ(P)；若子范畴 C 中 Γ(P)≅Γ(Q) 蕴含 P≅Q，则称 C 为 Γ-faithful。已知连续 dcpo 与有界完备 dcpo 具有该性质，而 Pos_d 不具有；Ho 与 Zhao (2009) 询问 dcpo 范畴 DCPO 是否 Γ-faithful。本文给出否定回答：通过构造反例证明 DCPO 不是 Γ-faithful。为构造反例，作者引入一个新的 dcpo 子范畴，证明其为 Γ-faithful，且涵盖当时所有已知的 Γ-faithful 子范畴；反例构造关键依赖于 Johnstone 那个在 Scott 拓扑下非 sober 的 dcpo。
Given a poset $P$, the set, $\Gamma(P)$, 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 $\Gamma$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $\Gamma(P) \cong \Gamma(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $\Gamma$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $\Gamma$-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 $\Gamma$-faithful. This subcategory subsumes all currently known $\Gamma$-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.Comment: 19 pages, 4 figures