Table of contents |

2 Examples 3 A property |

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*.

- 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.

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.