An enriched category is a category with generalized morphisms.

1 Enriched category

A category \(\cC = (\Obj, \Hom, \circ, \id)\) 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 \(\circ: \Hom(B, C) \otimes \Hom(A, B) \to \Hom(A, C)\)
  • an identity element for each object \(\id_A: I \to \Hom(A, A)\)

subject to

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

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: \(F_{A, C} \circ_\cC = \circ_\cD (F_{B, C} \otimes F_{A, B})\)
  • 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: \(\circ_\cD (\alpha_B \otimes F_{A, B}) \lambda\inv = \circ_\cD (G_{A, B} \otimes \alpha_A) \rho\inv\)

4 Examples

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