An enriched category is a category with generalized morphisms.

1 Enriched category

A category $$\cC = (\Obj, \Hom, \compL, \id)$$ enriched in a monoidal category $$(\cV, \otimes, I, \alpha, \lambda, \rho)$$ ($$\cV$$-enriched category) consists of

• a collection of objects $$\Obj$$
• a hom-object between objects $$\Hom(A, B) \in \Obj_\cV$$
• a composition morphism $$\compL: \Hom(B, C) \otimes \Hom(A, B) \to \Hom(A, C)$$
• an identity element for each object $$\id_A: I \to \Hom(A, A)$$

subject to

• associativity: $$\compL (\id \otimes \compL) \alpha = \compL (\compL \otimes \id)$$
• identity: $$\compL (\id_B \otimes \id) = \lambda$$ and $$\compL (\id \otimes \id_A) = \rho$$

2 Enriched functor

For $$\cV$$-enriched categories $$\cC$$ and $$\cD$$, a $$\cV$$-enriched functor $$F: \cC \to \cD$$ consists of

• a function sending $$\cC$$-objects to $$\cD$$-objects $$F: \Obj_\cC \to \Obj_\cD:= A \mapsto FA$$
• a collection of $$\cV$$-morphisms indexed by pairs of $$\cC$$-objects $$F_{A, B}: \Hom_\cC(A, B) \to \Hom_\cD(FA, FB)$$

in such a way that $$F$$ preserves

• composition: $$F_{A, C} \compL_\cC = \compL_\cD (F_{B, C} \otimes F_{A, B})$$
• identity: $$F_{A, A} \id_{\cC:A} = \id_{\cD:FA}$$

3 Enriched natural transformation

For $$\cV$$-enriched functors $$F, G: \cC \to \cD$$, a $$\cV$$-enriched natural transformation $$\alpha: F \nat G$$ is a collection of $$\cV$$-morphisms indexed by $$\cC$$-objects $$\alpha_A: I \to \Hom_\cD(FA, GA)$$ that satisfies

• naturality: $$\compL_\cD (\alpha_B \otimes F_{A, B}) \lambda\inv = \compL_\cD (G_{A, B} \otimes \alpha_A) \rho\inv$$

4 Examples

• locally small category: $$(\cSet, \times, \set{*})$$
• preorder: $$(\cTwo, \land, \top)$$
• strict 2-category: $$\cCat$$
• Lawvere metric space: $$\overline{R}_+ = [0, \infty]$$