In category theory, a branch of mathematics, an enriched category generalizes the idea of a locally small 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 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).

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

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 Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as V-categories.

Let (M, ⊗, I, α, λ, ρ) be a monoidal category. Then an enriched category C (alternatively, in situations where the choice of monoidal category needs to be explicit, a category enriched over M, or M-category), consists of

The first diagram expresses the associativity of composition:

That is, the associativity requirement is now taken over by the associator of the monoidal category M.

For the case that M is the category of sets and (⊗, I, α, λ, ρ) is the monoidal structure (×, {•}, ...) given by the cartesian product, the terminal single-point set, and the canonical isomorphisms they induce, then each C(a, b) is a set whose elements may be thought of as "individual morphisms" of C, while °, now a function, defines how consecutive morphisms compose. In this case, each path leading to C(a, d) in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms a → b → c → d, i.e. elements from C(a, b), C(b, c) and C(c, d). Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.