Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits

Question

I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here has been futile. Especially since I don't have the contents of the book, I am wondering if there is another way, or even another explanation, to see that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits, perhaps without going through the explicit construction of coproduct in $\textbf{Set}$ by taking disjoint unions and modding out by equivalence class. If such explicit construction of the coproduct seems unavoidable, can you provide more thorough explanation of the construction, since I am finding it a little abstract? I'd also be grateful for an online reference where I can find the book.

Answer 1

Recall that filtered colimits are exactly those colimits that commute with finite limits in the category of sets, and that the category of Abelian groups can be described as a finite limit theory over the category of sets. These facts allow us to put an Abelian group structure on the set colimit of the filtered system of Abelian groups. For example, we can define $+ : (\varinjlim\limits_{i \in I} U G_i)^2 \to \varinjlim\limits_{i \in I} UG_i$ by using the isomorphism $(\varinjlim\limits_{i \in I} U G_i)^2 \cong \varinjlim\limits_{i \in I} (U G_i)^2$ and the functorial properties of colimits. The other two operations (identity and inverse) can be defined similarly, and it's straightforward to verify all the equations involved. The same strategy works for any finite limit theory (which includes all purely equational theories like modules, rings, etc.)