Enriched category

Summary

A category enriched over a monoidal category $\mathcal{V}$ has, instead of a set of morphisms between two objects, an object of $\mathcal{V}$. Composition and identities are morphisms of $\mathcal{V}$ that satisfy associativity and unit laws. Ordinary locally small categories are the case $\mathcal{V} = \mathbf{Set}$; preadditive categories are the case of abelian groups; strict 2-categories are the case of small categories. Lawvere observed that metric spaces are categories enriched over the non-negative reals.