The category of well-filtered dcpos is not $\Gamma$-faithful
<|reference_start|>The category of well-filtered dcpos is not $\Gamma$-faithful: 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 $\Gamma$-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 {$\Gamma$-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 $\Gamma$-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 $\Omega^{*}$-compact dcpo's is $\Gamma$-faithful. Moreover, we locate a class of dcpo's which we call weakly dominated dcpo's, and show that this class is $\Gamma$-faithful and strictly larger than $\mathbf{DOMI}$.<|reference_end|>
@article{miao2024the,
    title={The category of well-filtered dcpos is not $\Gamma$-faithful},
    author={Hualin Miao, Huijun Hou, Xiaodong Jia, Qingguo Li},
    journal={arXiv preprint arXiv:2409.01546},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.01546},
    primaryClass={cs.LO}
}