# The Unapologetic Mathematician

## Representable Functors

Now that we have a handle on hom functors, we can use them to define other functors.

Since $\hom_\mathcal{C}$ is a functor from the product category $\mathcal{C}^{\rm op}\times\mathcal{C}$ to sets, it is functorial in each variable separately. This means that we can take any object $A$ of $\mathcal{C}$ and build the covariant functor $B\mapsto\hom_\mathcal{C}(A,B)$ and the contravariant functor $B\mapsto\hom_\mathcal{C}(B,A)$. We call $A$ the representing object of these functors.

Now not every functor $\mathcal{C}\rightarrow\mathbf{Set}$ is exactly of this form, but a good many of them are naturally isomorphic to such a functor. We call such a functor “representable”. Representable functors turn out to be extremely useful, as we’ll see.

As an example, consider the “underlying set” functor $U:\mathbf{Grp}\rightarrow\mathbf{Set}$. This takes a group to its underlying set, and a homomorphism of groups to its underlying function. You should check that it is, indeed, a covariant functor. I say that it is represented by the group $\mathbb{Z}$. That is, $U(\underline{\hphantom{X}})\cong\hom_\mathbf{Grp}(\mathbb{Z},\underline{\hphantom{X}})$.

Given an element $g$ of a group $G$, there is a unique homomorphism $\phi_g:\mathbb{Z}\rightarrow G$ so that $1\mapsto g$. Conversely, every homomorphism sends $1$ to some element of $G$, so this sets up a bijection between the set of elements of $G$ and the set of homomorphisms from $\mathbb{Z}$ to $G$. These bijections $\eta_G:U(G)\rightarrow\hom_\mathbf{Grp}(\mathbb{Z},G)$ are the components of the natural isomorphism we’re looking for. We need to check that for any group homomorphism $f:G\rightarrow H$ we have naturality: $\eta_H\circ U(f)=\hom(\mathbb{Z},f)\circ\eta_G$.

Since $U(f)$ is just the function (forgetting that it preserves a group structure), the left side of the naturality equation sends an element $g\in G$ to the unique homomorphism $\phi_{f(g)}:\mathbb{Z}\rightarrow H$ with $\phi_{f(g)}(1)=f(g)$. On the right, we first send $g$ to the homomorphism $\phi_g:\mathbb{Z}\rightarrow G$, and then compose with $f$. This also satisfies $f(\phi_g(1))=f(g)$, so $f\circ\phi_g=\phi_{f(g)}$, and the naturality equation is satisfied.

Now, we have a bunch of other “forgetful” functors sitting around. All of the algebraic structures we’ve considered — semigroups, monoids, abelian groups, rings, $R$-modules (left and right), and even quandles — they all have an “underlying set”. First you should be able to verify that each assignment of an underlying set is actually a covariant functor. Then, try to find a representing object for each functor.

This particular use of representable functors to describe forgetful functors to the category of sets is the first solid step towards moving away from viewing algebraic concepts as “sets with extra structure”. For instance, we now see that we don’t really need to describe a group as a set with a multiplication function, inverse function, and identity. A group is just an object in the category of groups, and within the category it has all sorts of interrelationships with all the other groups. We can reconstruct the older picture of a group from these relationships, and here we’ve seen how to recover the set of elements of $G$ as $\hom_\mathbf{Grp}(\mathbb{Z},G)$.