Enriched category

An enriched category is a certain generalization of an ordinary category, for which the hom-sets may be given additional structure, namely the structure of objects of another category.

Let be a monoidal category. A category enriched over , also called an -category consists of #m/def/cat

a Collection of objects ;
for every ordered pair of objects , a hom-object ;
a morphism in designating the identity; and
fir evert ordered triple of object , a morphism in designating composition;

such that we have associativity

and unitality

Note it does not necessarily follow from this definition that is a category. Nevertheless, usually is a concrete category and we have some compatible “ordinary category” structure for arising from the underlying sets of hom-objects.

Further terminology

The appropriate notion of functor is an Enriched functor.

Examples

An ordinary locally small category is the same as a -category.
A closed category is naturally enriched over itself.
An Additive category is enriched over , as are the stronger notions of Preäbelian category and Abelian category.

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