On Quasi-Modular Pseudometric Spaces and Asymmetric Uniformities
Philani Rodney Majozi
TL;DR
The paper develops a comprehensive theory of quasi-modular pseudometric spaces by relaxing symmetry in Luxemburg-type gauges to produce left/right (bi-directional) quasi-uniform structures. It shows how forward and backward topologies and entourages arise from one-sided modulars, and how a canonical symmetrization recovers a symmetric uniform framework that aligns with classical uniform completion results. Using enriched category theory (Lawvere), Kan extensions, and Isbell completion, the authors establish bicompletion, compactness criteria, and universal properties linking enriched and uniform viewpoints. Applications to weighted Musielak–Orlicz spaces, directed graphs, and asymmetric Sobolev-type spaces illustrate the practicality of the framework for asymmetric variational problems and directional analyses. Overall, the work unifies analytic, topological, and categorical perspectives on non-symmetric variational problems and provides robust criteria for completeness, compactness, and well-posedness in asymmetric settings.
Abstract
We introduce and analyze quasi-modular pseudometric spaces: structures generated by left/right (generally non-symmetric) Luxemburg-type gauges coming from one-sided modulars. This relaxes symmetry in Chistyakov's modular metric setting and places it within the context of quasi-metrics and quasi-uniform spaces. We develop the basic theory: induced (quasi-)topologies and quasi-uniformities; unit-ball characterizations; and Cauchy notions tailored to asymmetry. We give criteria for completeness, precompactness, and compactness, and prove bicompletion results. A symmetrization principle relates the asymmetric gauges to a canonical modular space; under mild $Δ_2/\nabla_2$-type hypotheses, symmetrized and bicomplete constructions agree. Categorically, we treat these spaces as Lawvere-enriched objects and describe symmetrization as a Kan-style construction in the enriched setting; the induced entourages admit an Isbell-style uniform interpretation. Applications cover weighted and generalized Orlicz models (including variable-exponent and double-phase growth), asymmetric function spaces built from positive/negative parts, and directed-graph energies with time-dependent costs. These examples yield modular Sobolev-type estimates and demonstrate fixed-point and stability principles for directional media. Overall, this setting unifies analytic, topological, and categorical perspectives for non-symmetric variational problems and provides practical criteria for compactness, duality, and well-posedness in asymmetric settings.