Epicompletion in Frames with Skeletal Maps, I: Compact Regular Frames

Abstract

A frame homomorphism h : A ⟶ B is skeletal if x ^⊥⊥ = 1 in A implies that h(x)^⊥⊥ = 1 in B. It is shown that, in $\mathfrak{KRegS}$, the category of compact regular frames with skeletal maps, the subcategory $\mathfrak{SPRegS}$, consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in $\mathfrak{KRegS}$. Further, $\mathfrak{SPRegS}$ is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame ε A of $\mathcal{P} A$, the boolean algebra of polars of A.