I’d like to move on now to another way of blending various structures. We’ve seen that in certain situations the set of morphisms between two objects in a category naturally has deeper structure itself. For example, the set of homomorphisms between two abelian groups is itself an abelian group, because abelian groups are modules over the commutative ring . More generally, the set of homomorphisms between two -modules naturally has the structure of a -module, and sometimes more.
We need a good way of talking about this sort of thing, where we replace hom sets by “hom objects” in some other category . When this happens we say that our category is “enriched” over . So to rephrase what I said above, the category of -modules is enriched over . Similarly, locally small categories are enriched over .
When we talk about categories — which usually for us means locally small categories — we are implicitly using a number of properties of . In particular, to set up compositions we need to be able to take pairs of morphisms, which the cartesian product handles for us nicely: . We also need to be able to pick out a special morphism in each set of endomorphisms to be the identity, which we can take to be the image of a function from a one-point set to the set of endomorphisms sort of like we did for monoid objects.
For setting up the relations a category must satisfy we need to be able to build triples from pairs in two ways:
We also need to pair a morphism with a (unique) identity morphism:
What are the important properties of the category of sets that make it useful for these purposes? It’s just the fact that equipped with finite products (including a singleton set as terminal object) is a monoidal category! So let’s take a monoidal category — a useful example to have always at hand is — and try to use it for enrichment. As we proceed, we’ll write for the underlying regular category (that is, forget that is monoidal).
So, given such a monoidal category we’ll define a -category to consist of a class of objects , and for each pair of objects a “hom-object” . For each triple of objects there is a composition . For each object there is an “identity”, described by an arrow .
I’ll be spending some time on this, so let’s leave it at the definition for now. Go through and unpack it for the case of an -category, and see what the definition says such a thing should look like.