Now that we’ve done a whole lot about enriched categories in the abstract, let’s look at the very useful special case of categories enriched over — the category of abelian groups.
We know that is a monoidal category, with the tensor product of abelian groups as its monoidal structure and the free abelian group as the monoidal identity. Even better, it’s symmetric, and even closed. That is, for any two abelian groups and we have an isomorphism , and there is a natural abelian group structure on the set of homomorphisms satisfying the adjunction .
So let’s read the definitions. An -category has a collection of objects, and between objects and there is an abelian hom-group .
For each object we have a homomorphism of abelian groups which picks out the “identity morphism” from to itself at the level of the underlying sets. Remember that we’re no longer thinking of an abelian group as having elements — only its underlying set has elements anymore, and the underlying set of an abelian group is the set of abelian group homomorphisms .
Given three objects we have a “composition” arrow in : . This is associative and the identity morphism acts as an identity in the sense that the appropriate diagrams commute. Of course, since the composition arrows are morphisms in they are linear functions in each input.
An -functor between -categories and is defined by a function from the objects of to the objects of , and for each pair of objects a homomorphism of abelian groups . Two diagrams are required to commute, saying that these linear functions preserve the composition and identity functions.
An -natural transformation is one of two forms. In one we’re given two -functors and . Then a natural is a collection of linear functions making one diagram commute. In the other we’re given an object and a bifunctor . Then is a collection of linear functions making another diagram commute.
Together, -categories, -functors between them, and -natural transformations (of the first kind) form a 2-category. We can pair off -categories and to get the product category (in fact we already did once above) and we can take the opposite category . Thus -categories form a symmetric monoidal 2-category with a duality involution.
There’s a whole lot of structure here, but ultimately it boils down to “the hom-sets all have the structure of abelian groups, and everything in sight is -linear”. And that’s the usual definition given, that I decided to forgo back when I started in on enriched categories.