The category of probabilistic metric spaces

TL;DR

This paper provides a categorical analysis of probabilistic metric spaces, describing their structure, morphisms, and relationships with other metric space categories, including closure properties and embeddings.

Contribution

It offers an isomorphic description of probabilistic metric spaces and characterizes their categorical properties, such as epimorphisms, monomorphisms, and embeddings.

Findings

Category of probabilistic metric spaces is monotopological over sets.

Regular closure coincides with strong topology closure.

Extended metric spaces embed both reflectively and coreflectively.

Abstract

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and is in terms of objects that are sets endowed with a collection of distances, where the distances involved do not satisfy the triangle inequality but fulfil a mixed triangle condition instead. The morphisms are levelwise non-expansive maps. We show that the category of probabilistic metric spaces is a monotopological category over the category of sets. We describe the regular closure on a probabilistic space and prove that it coincides with the closure in the underlying strong topology. This enables us to characterize the class of all epimorphisms as the dense maps and the class of all regular monomorphisms as the closed embeddings in terms of the closure operator. We prove that the category of extended metric spaces with non-expansive maps is both coreflectively and reflectively embedded in the category of probabilistic metric spaces.