Categorical proof that subgroups of free groups are free?

Question

Is there a categorical proof that the subgroups of free groups are free? Also that abelian subgroups of free abelian groups are free.

Answer 1

It depends on what you mean by a categorical proof. In any case, there won't be a proof which is so abstract that it works in any algebraic category, because (as was already mentioned in the comments) the claim fails in many algebraic categories, for example modules, (commutative) rings, Lie algebras.

Among beginners of category there might be a conception that, as soon as we have described some known objects and constructions via category theory, we can use this big machinery of category theory to solve all problems. Well, category theory helps to solve all trivial problems at once (or reveal that problems are trivial) and helps to organize the mathematics which is needed to solve problems, but this does not mean that category theory alone will do our job. In the end, there will still be some concrete work to do.

My favorite proof of the Nielsen-Schreier Theorem is the characterization of free groups as those groups which act freely on a tree. In categorical terms, we have found a class of groups $\mathcal{S}$ such that a group $G$ is free if and only if $G$ is a subgroup of a group in $\mathcal{S}$. From this it follows immediately that subgroups of free groups are free. But the existence of $\mathcal{S}$ is no abstract nonsense.