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

In previous work it was shown that there is an epireflection ψ of the category of all compact normal, joinfit frames, with skeletal maps, in the full subcategory of frames which are also strongly projectable, and that ψ restricts to the epicompletion ε, which is the absolute reflection on compact regular frames. In the first part of this paper it is shown that ψ is a monoreflection and that the reflection map is, in fact, closed. Restricted to coherent frames and maps, ψA can then be characterized as the least strongly projectable, coherent, normal, joinfit frame in which A can be embedded as a closed, coherent, and skeletal subframe. The second part discusses the role of the nucleus d in this context. On algebraic frames with coherent skeletal maps d becomes an epireflection. Further, it is shown that e = d · ψ epireflects the category of coherent, normal, joinfit frames, with coherent skeletal maps, in the subcategory of those frames which are also regular and strongly projectable, which are epicomplete. The action of e is not monoreflective. This article is the continuation of the work recorded in [MZ08, M08, MZ09]. The first paper showed that the passage from a compact regular frame A to its absolute εA 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 [M08] was to provide a bridge between [MZ08] and subsequent papers on archimedean frames, with the dependence on “Choice” removed from consideration. Thus, as of [M08], we have the concept of a joinfit frame. The work in [MZ09] 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 [MZ09] leaves unresolved is whether this functor is a monoreflection. By showing that the reflection maps ψA are always closed – and hence one-to-one – we are finally able to prove that ψ is, indeed, a monoreflection (Theorem 3.3). In §1 we provide the necessary frame-theoretic background, as well as the highlights from [MZ08]. We revisit archimedean frames in §2 and recall some of the results of [M08]. The discussion of ψ takes place in §3 and §4; in the latter it is shown that a