Metrically generated theories

Abstract

Abstract. 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.

Epimorphisms and cowellpoweredness for separated metrically generated theories

“…In this section we gather all the preliminary material that is needed from [4] to explain what constructs we will be dealing with. We use categorical terminology as developed in [1] or [12].…”

“…The main result of [4] states that M C provides a model for all C-metrically generated theories in the sense that a topological construct is C-metrically generated if and only if it is concretely isomorphic to M C ξ for some expander ξ on M C .…”

“…In this paper we will make one extra technical assumption on the expanders which was not needed in [4].…”

“…These examples and in fact also many others, can be described as the T 0 objects in some C-metrically generated construct in the sense of [4] and this observation will allow us to treat the epimorphism problem for all the examples at the same time. C-metrically generated theories were introduced as categories that are naturally described by means of a suitable functor defined on a certain construct C consisting of quasi-metric spaces.…”

“…a topology T d , a (quasi-)uniformity U d , a totally bounded uniformity or an approach structure A d [10] i.e. to "the metrizable" objects of the category, satisfy the conditions formulated in [4]. The extra structure that arises is provided by giving an isomorphic description of the constructs.…”