Endomorphisms of forgetful functor $\mathbf{Grp}\to \mathbf{Set}$

Question

It is well known endomorphisms of faithful functor form a monoid. I was trying to determine monoid of endomorphisms of forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, and found it to be multiplicative monoid of $\mathbb{Z}$. Proof goes like this:

Let $U$ be the forgetful functor, and $\theta\colon U \to U$ a natural transformation. Consider infinite cyclic group $A=\left<a\right>$. We have $\theta_A(a) = a^k$ for some integer $k$. Now take arbitrary group $G$ and $g\in G$, and consider homomorphism $f\colon A\to G$ determined by $f(a) = g$. Clearly, $$ \theta_G\left(g\right) = \theta_G\left(f\left(a\right)\right) = \left(\theta_G \circ f\right)\left(a\right) = \left(f\circ \theta_A\right)\left(a\right) = f\left(\theta_A(a)\right) = f(a^k)=\left(f(a)\right)^k = g^k $$ the only nontrivial equality true by naturality of $\theta$, and so endomorphisms of $U$ correspond to integers, composition given by multiplication.

Now, for the questions...

• Is the above correct? It surely seems to be, but I find it kind of unsettling, especially since...

• ... the same reasoning seems to apply to $\mathbf{Ab}\to \mathbf{Set}$. Is the same indeed true for $\mathbf{Ab}$? I'd intuitively expect richer structure in the nonabelian case.

• The role played by the cyclic group ($\mathbb{Z}$) seems worth further consideration. I know $\mathbb{Z}$ is a generator (separator) in $\mathbf{Grp}$, but it doesn't seem to immediately lead to any useful generalizations. Is there some connection between endomorphisms of forgetful functor and separators? Or some other related notion?

• What about some "more abstract" ways to answer such questions? That is, instead of element chasing perhaps some variant of Yoneda lemma for suitably enriched categories?

Answer 1

Your proof is correct. Here is a shorter proof using the Yoneda Lemma (actually you have proven the Yoneda Lemma in a special case, we do this all the time without knowing it):

The functor $U$ is representable by $\mathbb{Z}$ (since an element of a group is the same as a homomorphism from $\mathbb{Z}$). Hence, the Yoneda Lemma tells us $$\mathrm{Hom}(U,U) \cong \mathrm{Hom}(\mathrm{Hom}(\mathbb{Z},-),U) \cong U(\mathbb{Z}),$$ the set of integers.

More generally, if $C$ is any variety of algebraic structures with forgetful functor $U : C \to \mathsf{Set}$, then $\mathrm{Hom}(U^n,U)$ corresponds to the underlying set of the free $C$-algebra on $n$ generators. You may imagine these as "universal $n$-ary operations". For example, for $C=\mathsf{Ring}$ and $n=2$ such an operation is $x^2 - xy + y^2$.

More interesting things happen for non-algebraic categories. For example, consider the forgetful functor $U : \mathsf{FinGrp} \to \mathsf{Set}$ from the category of finite groups. Here, $U$ is not representable, but it is ind-representable: We have a canonical isomorphism $$U \cong \varinjlim_n \,\mathrm{Hom}(\mathbb{Z}/n,-)$$ Therefore: $$\mathrm{Hom}(U,U) \cong \varprojlim_n \,\mathrm{Hom}(\mathrm{Hom}(\mathbb{Z}/n,-),U) \cong \varprojlim_n U(\mathbb{Z}/n),$$ which is the underlying set of the pro-finite completion $\widehat{\mathbb{Z}}$. If $(\overline{z_n})_n$ is an element in $\widehat{\mathbb{Z}}$, the corresponding operation on finite groups maps an $n$-torsion element $g$ to $g^{z_n}$. This is well-defined precisely because we have $z_n \equiv z_m \bmod n$ for $n | m$. We get the same results when we work with the larger category of torsion groups.

In general, if you want to determine $\mathrm{Hom}(U,U)$ for any functor, try to write $U$ as a colimit of representable functors $\mathrm{Hom}(X_i,-)$ (this is always possible, though in general somewhat tautological), then $\mathrm{Hom}(U,U)$ is the limit of the $U(X_i)$.