Editing Enriched category (section)

{{Short description|Category whose hom sets have algebraic structure}}
{{More citations needed|date=August 2019}}In [[category theory]], a branch of [[mathematics]], an '''enriched category''' generalizes the idea of a [[category (mathematics)|category]] by replacing [[hom-set]]s with objects from a general [[monoidal category]]. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a [[vector space]] of [[morphism]]s, or a [[topological space]] of morphisms.  In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an [[object (category theory)|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, there must be a binary operation on objects giving us at least the structure of a [[monoidal category]], though in some contexts the operation may also need to be commutative and perhaps also to have a [[right adjoint]] (i.e., making the category [[symmetric monoidal category|symmetric monoidal]] or even [[closed monoidal category|symmetric closed monoidal]], respectively).{{citation needed|date=December 2016}}

Enriched category theory thus encompasses within the same framework a wide variety of structures including
* ordinary categories where 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., [[preorder]]s where the composition rule ensures transitivity, or [[pseudoquasimetric space|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'''. Due to Mac&nbsp;Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as '''V-categories'''.