Bicompletion of metrically generated constructs

Abstract

We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms L(R) is known. This functor will permit to "lift" the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory R F for the construct X0 of T0 objects and that the associated firm class of morphisms can be derived from L(R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones.