Enriched category theory

Enriched category theory is a branch of category theory in which the hom-sets of a category — the collections of morphisms between objects — are replaced by objects from some other monoidal category. Instead of asking 'how many morphisms are there from A to B?', enriched category theory asks 'what is the *structure* of morphisms from A to B?' — where the answer might be a topological space, a metric space, a poset, or an abelian group.

The simplest example is a category enriched over sets: this is just an ordinary category. A category enriched over the category of vector spaces is a linear category, where morphism composition is bilinear. A category enriched over the poset of truth values is a preorder. Each choice of enriching category reveals different structural features.

Enriched categories are the natural setting for algebraic effects and other computational phenomena where the 'distance' or 'cost' between programs matters. In this reading, the enrichment captures not just whether one program can transform into another, but *how* — with what resources, under what constraints, at what cost.