Quantale-valued Cauchy tower spaces and completeness

Abstract

Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.