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 often has additional structure that should be respected, e.g., that of being a
vector space of
morphisms, 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 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, 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 or even
symmetric closed monoidal, respectively).