Metrically generated theories

Abstract

Many examples are known of natural functors K K describing the transition from categories C \mathcal {C} of generalized metric spaces to the “metrizable" objects in some given topological construct X \mathcal {X} . If K K preserves initial morphisms and if K ( C ) K(\mathcal {C}) is initially dense in X \mathcal {X} , then we say that X \mathcal {X} is C \mathcal {C} -metrically generated. Our main theorem proves that X \mathcal {X} is C \mathcal {C} -metrically generated if and only if X \mathcal {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 \mathcal {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.