Epicompletion in Frames with Skeletal Maps, II: Compact Normal Joinfit Frames

Abstract

An epireflection ψ is constructed of the category KNArS of compact normal joinfit frames, with skeletal maps, in the subcategory SPArS consisting of strongly projectable KNArS-objects. The construction is achieved via a pushout in the category FrmS of frames with skeletal maps, and involves rather intimately the regular coreflection of the object to be reflected. Further, if the regular coreflection ρ is applied to the reflection map ψ A : A −→ ψ A one obtains the extension of ρ A to its absolute.Keywords Epireflection · Skeletal map · Coproduct · Pushout · Normal frame · Joinfitness · Strong projectability Mathematics Subject Classifications (2000) Primary 06D22; Secondary 18A20 · 18A30This article is the continuation of the work recorded in [20] and [15]; it continues in [16]. The first paper showed that the passage from a compact regular frame A to its absolute ε A is functorial, if one restricts to skeletal frame homomorphisms. In fact, the monoreflection ε is the functorial epicompletion on the category KRegS of all compact regular frames and skeletal frame maps. The goal of the research that went into [15] was to provide a bridge between [20] and this paper, by virtue of having gathered sufficient information about categories of archimedean frames and thereby having made this article easier to read. As indicated below, various

Epicompletion in Frames with Skeletal Maps, III: When Maps are Closed

“…This article is the continuation of the work recorded in [11,14,15]. The first paper showed that the passage from a compact regular frame A to its absolute ε A is functorial if one restricts to skeletal frame homomorphisms.…”

“…Details may be found in [15,Sections 4 and 5]. We remind the reader of the conventions set forth in Remark 1.3 on the naming of categories.…”

“…The work in [15] extends the functor ε to an epireflection ψ of the category KNArS of compact normal, joinfit frames with skeletal maps in the full subcategory consisting of frames which are strongly projectable-that is, in which every polar is complemented. The construction of ψ is independent of Choice Principles.…”

“…What [15] leaves unresolved is whether this functor is a monoreflection. By showing that the reflection maps ψ A are always closed-and hence one-to-one-we are finally able to prove that ψ is, indeed, a monoreflection (Theorem 3.3).…”

Patch-generated Frames and Projectable Hulls

“…In this work we have concentrated on the relationship between the structures of the two frames involved. Regarding the question of existence, we have made some progress on a related question, namely, the existence of a strongly projectable hull; let us refer the reader to [30].…”