In category theory and its applications to mathematics, an enriched category is a category whose homsets are replaced by objects from some other category, in a wellbehaved manner.
Contents
Definition
We define here what it means for C to be an enriched category over a monoidal category .
The following structures are required:
 Let Ob(C) be a set (or proper class). An element of Ob(C) is called an object of C.
 For each pair (A,B) of objects of C, let Hom(A,B) be an object of M, called the homobject of A and B.
 For each object A of C, let id_{A} be a morphism in M from I to Hom(A,A), called the identity morphism of A.
 For each triple (A,B,C) of objects of C, let
be a morphism in M called the composition morphism of A, B, and C.
The following axioms are required:
 Associativity: Given objects A, B, C, and D of C, we can go from Hom(C,D) ⊗ Hom(B,C) ⊗ Hom(A,B) to Hom(A,D) in two ways, depending on which composition we do first. These must give the same result.
 Left identity: Given objects A and B of C, we can go from I ⊗ Hom(A,B) to just Hom(A,B) in two ways, either by using id_{A} on I and then using composition, or by simply using the fact that I is an identity for ⊗ in M. These must give the same result.
 Right identity: Given objects A and B of C, we can go from Hom(A,B) ⊗ I to just Hom(A,B) in two ways, either by using id_{B} on I and then using composition, or by simply using the fact that I is an identity for ⊗ in M. These must give the same result.
Given the above, C (consisting of all the structures listed above) is a category enriched over M.
Examples
The most straightforward example is to take M to be a category of sets, with the Cartesian product for the monoidal operation. Then C is nothing but an ordinary category. If M is the category of small sets, then C is a locally small category, because the homsets will all be small. Similarly, if M is the category of finite sets, then C is a locally finite category.
If M is the category 2 with Ob(2) = {0,1}, a single nonidentity morphism (from 0 to 1), and ordinary multiplication of numbers as the monoidal operation, then C can be interpreted as a preordered set. Specifically, A ≤ B iff Hom(A,B) = 1.
If M is a category of pointed sets with smash product for the monoidal operation, then C is a category with zero morphisms. Specifically, the zero morphism from A to B is the special point in the pointed set Hom(A,B).
Full article ▸
