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

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.