Enriched Category

A -enriched category over a monoidal category replaces hom-sets by hom-objects of . Each pair of objects carries a hom-object , and composition together with the unit are morphisms of satisfying the associativity and unit laws as -diagrams. An ordinary category is a -enriched one.

Lawvere's metric spaces as enriched categories

Enriching over makes a generalised metric space. The hom-object is a distance, composition is the triangle inequality , and the unit is . Enriching over truth values instead gives a preorder.

Potentials via the -enrichment

Set and for a potential . Composition is addition and the identity is , and the hom depends only on the endpoints, so it is a state function. A functor into such a assigns potentials, and a natural transformation between two of them is a family of differences whose naturality square is an equation in .

References

• G. Max Kelly, “Basic Concepts of Enriched Category Theory” (1982)
• F. William Lawvere, “Metric spaces, generalized logic, and closed categories” (1973)