We will define what it means for C to be an enriched category over the monoidal category[?] M.
We require the following structures:
We require the following axioms:
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 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 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.
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.
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