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The category of well-filtered dcpos is not $Γ$-faithful

Hualin Miao, Huijun Hou, Xiaodong Jia, Qingguo Li

Abstract

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 $\mathbf{DOMI}$.

The category of well-filtered dcpos is not $Γ$-faithful

Abstract

The Ho-Zhao problem asks whether any two dcpo's with isomorphic Scott closed set lattices are themselves isomorphic, that is, whether the category of dcpo's and Scott-continuous maps is -faithful. In 2018, Ho, Goubault-Larrecq, Jung and Xi answered this question in the negative, and they introduced the category of dominated dcpo's and proved that it is {-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 of well-filtered dcpo's is not -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 -compact dcpo's is -faithful. Moreover, we locate a class of dcpo's which we call weakly dominated dcpo's, and show that this class is -faithful and strictly larger than .
Paper Structure (11 sections, 32 theorems, 3 figures)

This paper contains 11 sections, 32 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\Omega^{*}$-compact dcpo's are dominated
  4. The category of well-filtered dcpo's is not $\Gamma$-faithful
  5. The definition of the counterexample $\mathcal{Z}$
  6. $\mathcal{Z}$ is a dcpo
  7. The Scott closed irreducible subsets of $\mathcal{Z}$
  8. $\mathcal{Z}$ is well-filtered
  9. $\mathbf{WF}$ is not $\Gamma$-faithful
  10. The category of weak dominated dcpo's is $\Gamma$-faithful
  11. Weakly dominated dcpo's may fail to be dominated

Key Result

Lemma 3.3

Let $L$ be an $\Omega^{*}$-compact dcpo. Then $L$ is a dominated dcpo.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (70)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Remark 4.1
  • Lemma 4.2
  • Proposition 4.3
  • Definition 4.4
  • Proposition 4.5
  • ...and 60 more