Lindelöf tightness and the Dedekind-MacNeille completion of a regular σ-frame

Abstract

Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which sub-objects of an object in the subcategory reflect to it - those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis of λ-tightness, tightness with respect to the localic Lindelöf reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular σ-frame A is a frame. Moreover, the embedding of A in its normal completion is the Bruns-Lakser injective hull of A in the category of meet semilattices and semilattice homomorphisms.

Since every regular  σ-frame is the cozero part of a regular Lindelöf  frame, this result points towards λ-tightness. For any regular  Lindelöf  frame L, the normal completion of CozL embeds in L as the sublocale generated by CozL. Although this completion is clearly contained in every sublocale having the same cozero part as L, we show by example that its cozero part need not be the same as the cozero part as L. We prove that a sublocale S is λ-tight in L iff S has the same cozero part as L. The aforementioned counterexample shows that the completion of CozL is not always λ-tight in L; on the other hand, we present a large class of locales for which this is the case.

Keywords: Lindelöf  frame, coreflection, normal completion, completely regular frame