Enriched Category

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set is often a vector space of morphisms, or a topological space of morphisms, and thus, one wants to capture this structure. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an opaque object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, it must be a monoidal category.

Enriched category theory thus encompasses within the same framework a wide variety of structures including

• ordinary categories, in which the hom-set carries additional structure beyond being a set. That is, there are operations on, or properties of morphisms that need to be respected by composition (e.g., the existence of 2-cells between morphisms and horizontal composition thereof in a 2-category, or the addition operation on morphisms in an abelian category)
• category-like entities that don't themselves have any notion of individual morphism but whose hom-objects have similar compositional aspects (e.g., preorders where the composition rule ensures transitivity, or Lawvere's metric spaces, where the hom-objects are numerical distances and the composition rule provides the triangle inequality).

In the case where the hom-object category happens to be the category of sets with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory.

An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in M, or simply an M-category, with the last terminology being used in some influential texts, such as MacLane's.