Enriched category

In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner.

Table of contents

1 Definition
2 Examples
3 A property

Definition

We will define what it means for C to be an enriched category over the monoidal category M.

We require the following structures:

Let Ob(C) be a set (or proper class, if you prefer). Then an element of Ob(C) is an object of C.
For each pair (A,B) of objects of C, let Hom(A,B) be an object of M. Then Hom(A,B) is the hom-object of A and B.
For each object A of C, let id_A be a morphism in M from I to Hom(A,A), where I is a fixed identity object of the monoidal operation of M. Then id_A is the identity morphism of A.
For each triple (A,B,C) of objects of C, let _° be a morphism in M from Hom(B,C) ⊗ Hom(A,B) to Hom(A,C), where ⊗ is the monoidal operation in M. Then _° is the composition morphism of A, B, and C.

We require the following axioms:

Associativity: Given objects A, B, C, and D of C, we can go from Hom(C,D) ⊗ Hom(B,C) ⊗ Hom(A,B) 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_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.
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_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.

We should include some commutative diagrams illustrating these axioms.

Then C (consisting of all the structures listed above) is a category enriched over M.

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 hom-sets 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 Cartesian 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).

If M is a category of abelian groups with tensor product as the monoidal operation, then C is a preadditive category.

A property

If there is a monoidal functor from a monoidal category M to a monoidal category N, then any category enriched over M can be reinterpreted as a category enriched over N. In each of the examples above, there is such a functor from M to the category of sets, so each kind of enriched category in the examples can also be described as an ordinary category with certain additional structure or properties. Some more abstract situations will not have this feature, however.