The reflective hull of the two-element chain in DCPO: properness, maximal Γ-faithfulness, and an internal reflection formula

Xulong He; Zhenchao Lyu; Yuxu Chen; Hui Kou

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arXiv · 2608.23128

The reflective hull of the two-element chain in DCPO: properness, maximal Γ-faithfulness, and an internal reflection formula

Xulong He·Zhenchao Lyu·Yuxu Chen·Hui Kou

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Abstract

Let $\DCPO$ be the category of dcpos and Scott-continuous maps, and let $\Rtwo$ be the reflective hull of the two-element chain $2$. We prove that $\Rtwo$ is the least non-discrete proper reflective full subcategory of $\DCPO$. It is closed under limits and Skula-closed sub-dcpos and contains every sober dcpo. We show that $\Rtwo$ is the maximal $Γ$-faithful full subcategory of $\DCPO$ that strictly contains weakly dominated dcpos. Finally, we give the concrete construction of $2$-reflection internally by a transfinite iteration and a criterion for reflective full subcategories of $\DCPO$, analogous to the Keimel--Lawson conditions for $T_0$ topological spaces.

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Xulong He, Zhenchao Lyu, Yuxu Chen, Hui Kou. 2026-08-24. The reflective hull of the two-element chain in DCPO: properness, maximal Γ-faithfulness, and an internal reflection formula. https://arxiv.org/abs/2608.23128

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