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 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 lambda-tightness, tightness with respect to the localic Lindeldof reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular sigma-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 sigma-frame is the cozero part of a regular Lindeldof frame, this result points towards lambda-tightness. For any regular Lindeldof 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 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 lambda-tight in L iff S has the same cozero part as L.
The aforementioned counterexample shows that the completion of Coz L is not always -tight in L; on the other hand, we present a large class of locales for which this is the case.