Enriched category
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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.
Definition
We will define what it means for C to be an enriched category over a 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 idA 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 idA 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) 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 idB 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 idA 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.
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 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. Every monoidal category M has a monoidal functor M(I, –) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties.