Enriched Category Theory

Enriched Category Theory extends traditional category theory by allowing the relationships (or hom-sets) between objects to carry additional structure beyond simple mappings. Instead of just having a set of morphisms from one object to another, the "hom-objects" can be elements of more complex mathematical structures, such as vector spaces or metric spaces. This framework enables deeper analysis of systems where relationships have internal structure—for example, quantities, weights, or probabilities—giving a more nuanced and flexible way to model interconnected systems across mathematics and theoretical computer science.