Riehl’s Category Theory in Context – Exercise 1.3.ix.

category-theory

I'm working through the following exercise from Emily Riehl's Category Theory in Context,

Exercise 1.3.ix. For any group $G$, we may define other groups:

  • the center $Z(G) = \{h \in G | hg = gh \forall g ∈ G\}$, a subgroup of $G$,
  • the commutator subgroup $C(G)$, the subgroup of $G$ generated by elements $ghg^{-1}h^{-1}$ for any $g, h \in G$, and
  • the automorphism group $\operatorname{Aut}(G)$, the group of isomorphisms $\phi: G \rightarrow G$ in $\operatorname{Group}$.

Trivially, all three constructions define a functor from the discrete category of groups (with
only identity morphisms) to $\operatorname{Group}$. Are these constructions functorial in

  • the isomorphisms of groups? That is, do they extend to functors $\operatorname{Group}_{iso} \rightarrow \operatorname{Group}$?

  • the epimorphisms of groups? That is, do they extend to functors $\operatorname{Group}_{epi} \rightarrow \operatorname{Group}$?

  • all homomorphisms of groups? That is, do they extend to functors $\operatorname{Group} \rightarrow \operatorname{Group}$?

I've concluded that for any group morphism $f : G \rightarrow H$ in $\operatorname{Group}$, $f(Z(G)) \subseteq Z(H)$ if $f$ is epi and $f(C(G))\subseteq
C(H)$ for all $f$, and thus we have functors

$$
F : \operatorname{Group}_{epi} \rightarrow \operatorname{Group} \\
G \mapsto Z(G) \\
f \mapsto f\restriction_{Z(G)}^{Z(H)}
\\
$$

$$
F' : \operatorname{Group} \rightarrow \operatorname{Group} \\
G \mapsto C(G) \\
f \mapsto f\restriction_{C(G)}^{C(H)}
$$

Now for the automorphisms, the trivial functor from the discrete category of groups to $\operatorname{Group}$ can be extended to $\operatorname{Group}_{iso}$ by taking the following functor:

$$
H : \operatorname{Group}_{iso} \rightarrow \operatorname{Group} \\
G \mapsto \operatorname{Aut}(G) \\
\phi \mapsto (f \mapsto \phi f \phi^{-1})
$$

so my question is:

Can we extend this construction any further?

It is not even clear to me if having a morphism $f : G \rightarrow H$ guarantees the existence of a (non-trivial) morphism $\operatorname{Aut}(G) \rightarrow \operatorname{Aut}(H)$ when $f$ is not an iso, and I'm thinking this is probably not the case. As pointed out in the comments, split epis seem to be a problem for an extension to $\operatorname{Group}$, although I haven't been able to find a counterexample yet and I am not sure that this will behave badly in $\operatorname{Group}_{epi}$, since split epis will be isomorphisms.

Best Answer

In a pointed category $\mathcal{C}$ such that every morphism which is a mono and an epi is an iso, every functor $F:\mathcal{C}_{\text{iso}} \to \mathcal{C}$ extends to a functor $F:\mathcal{C}_{\text{epi}} \to \mathcal{C}$. (Just note that in such a category the composite of two epis is an isomorphism if and only if both are and hence we can define $F$ on a non-isomorphism as a zero morphism and as before on isomorphisms). Therefore, since the category of groups is such a category - epis are surjective homomorphisms (see Maclane's Categories for a working mathematician, Exercise 5 of Section 5 of Chapter I for a hint how to prove that) - only the last possibility remains. However that is solved here: Taking the automorphism group of a group is not functorial.

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