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 $𝖬$ be a monoidal category. A category $𝖢$ enriched over $𝖬$ , also called an $𝖬$ -category consists of cat

a Collection of objects $O b (𝖢$ ;
for every ordered pair of objects $𝑎, 𝑏 \in 𝖢$ , a hom-object $𝖢 (𝑎, 𝑏) \in 𝖬$ ;
a morphism $i d 𝑎 : 𝟙 \to 𝖢 (𝑎, 𝑎)$ in $𝖬$ designating the identity; and
fir evert ordered triple of object $𝑎, 𝑏, 𝑐 \in 𝖢$ , a morphism $(\circ) 𝑎, 𝑏, 𝑐 : 𝖢 (𝑏, 𝑐) \otimes 𝖢 (𝑎, 𝑏) \to 𝖢 (𝑎, 𝑐)$ in $𝖬$ designating composition;

such that we have associativity

and unitality

Note it does not necessarily follow from this definition that $𝖢$ is a category. Nevertheless, usually $𝖬$ is a concrete category and we have some compatible “ordinary category” structure for $𝖢$ 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 \Set-category.
A closed category is naturally enriched over itself.
An Additive category is enriched over $𝖠 𝖻$ , as are the stronger notions of Preäbelian category and Abelian category.

develop | en | SemBr

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Enriched category

Further terminology

Examples

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