The Ho-Zhao Problem

TL;DR

This paper investigates the $ ext{Gamma}$-faithfulness of categories of dcpos, providing a counterexample for $ extbf{DCPO}$ and introducing a new $ extbf{Gamma}$-faithful subcategory that includes all known ones.

Contribution

It answers an open question by Ho & Zhao (2009) negatively and introduces a new $ extbf{Gamma}$-faithful subcategory of dcpos.

Findings

The category $ extbf{DCPO}$ is not $ extbf{Gamma}$-faithful.

A new $ extbf{Gamma}$-faithful subcategory of dcpos is constructed.

Counterexample relies on Johnstone's non-sober dcpo in Scott topology.

Abstract

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.