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

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 subobjects 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 Lindel¨of 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 Lindel¨of frame, this result points towards λ-tightness. For any regular Lindel¨of frame L, the normal completion of Coz L embeds in L as the sublocale generated by Coz L. Although this completion is clearly contained in every sublocale having the same coz...