Quasi-pseudometric modular spaces as $\mathscr{Q}$-categories

TL;DR

This paper establishes an isomorphism between the category of quasi-pseudometric modular spaces with nonexpansive maps and a quantale enriched category, linking topology and categorical structures.

Contribution

It constructs a quantale of isotone functions to demonstrate the isomorphism and relates the topology of quasi-pseudometric modular spaces to quantale enriched categories.

Findings

The category of quasi-pseudometric modular spaces is isomorphic to a quantale enriched category.

The topology of a quasi-pseudometric modular space coincides with that generated by its quantale enriched category.

Quasi-pseudometrizable topological spaces are characterized by topologies induced by quasi-pseudometric modulars.

Abstract

We prove that the category of quasi-pseudometric modular spaces whose morphisms are the nonexpansive mappings is isomorphic to a quantale enriched category. To achieve this, we construct an appropriate quantale of isotone functions. We also show that, by means of this isomorphism, the topology associated with a quasi-pseudometric modular coincides with that generated by its corresponding quantale enriched category. Furthermore, we demonstrate that the class of quasi-pseudometrizable topological spaces coincides with the topological spaces whose topology is induced by a quasi-pseudometric modular.