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

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