# The Unapologetic Mathematician

## Enriched Categories

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 $\mathbb{Z}$. More generally, the set of homomorphisms between two $R$-modules naturally has the structure of a $\mathbb{Z}$-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 $\mathcal{V}$. When this happens we say that our category is “enriched” over $\mathcal{V}$. So to rephrase what I said above, the category of $R$-modules is enriched over $\mathbf{Ab}$. Similarly, locally small categories are enriched over $\mathbf{Set}$.

When we talk about categories — which usually for us means locally small categories — we are implicitly using a number of properties of $\mathbf{Set}$. In particular, to set up compositions we need to be able to take pairs of morphisms, which the cartesian product handles for us nicely: $\hom_\mathcal{C}(B,C)\times\hom_\mathcal{C}(A,B)$. 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 $\hom_\mathcal{C}(C,C)$ 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:
$(\hom_\mathcal{C}(C,D)\times\hom_\mathcal{C}(B,C))\times\hom_\mathcal{C}(A,B)\cong$
$\hom_\mathcal{C}(C,D)\times(\hom_\mathcal{C}(B,C)\times\hom_\mathcal{C}(A,B))$
We also need to pair a morphism with a (unique) identity morphism:
$\hom_\mathcal{C}(A,B)\cong\hom_\mathcal{C}(A,B)\times\{*\}\rightarrow\hom_\mathcal{C}(A,B)\times\hom_\mathcal{C}(A,A)$

What are the important properties of the category of sets that make it useful for these purposes? It’s just the fact that $\mathbf{Set}$ equipped with finite products (including a singleton set as terminal object) is a monoidal category! So let’s take a monoidal category $\mathcal{V}$ — a useful example to have always at hand is $\mathbf{Ab}$ — and try to use it for enrichment. As we proceed, we’ll write $\mathcal{V}_0$ for the underlying regular category (that is, forget that $\mathcal{V}$ is monoidal).

So, given such a monoidal category $\mathcal{V}$ we’ll define a $\mathcal{V}$-category $\mathcal{C}$ to consist of a class of objects $\mathrm{Ob}(\mathcal{C})$, and for each pair $(A,B)$ of objects a “hom-object” $\hom_\mathcal{C}(A,B)\in\mathcal{V}_0$. For each triple of objects $(A,B,C)$ there is a composition $\circ:\hom_\mathcal{C}(B,C)\otimes\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(A,C)$. For each object $A$ there is an “identity”, described by an arrow $i:\mathbf{1}\rightarrow\hom_\mathcal{C}(A,A)$.

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 $\mathbf{Ab}$-category, and see what the definition says such a thing should look like.

August 13, 2007 Posted by | Category theory | 11 Comments