-Metrically generated theories

-Many examples are known of natural functors K describing the transition from categories C of generalized metric spaces to the "metrizable" objects in some given topological construct X. If K preserves initial morphisms and if K(C) is initially dense in X then we say that X is C-metrically generated. Our main theorem proves that X is C- metrically generated if and only if X can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in C and natural morphisms. This theorem allows for a unifying treatment of many well known and varied theories. Moreover via suitable comparison functors the various relationships between these theories are studied.