Direct limit commutes with direct sums in the category of abelian groups

Question

I want to show that direct limit commutes with direct sums in the category of abelian groups. (I need this when studying algebraic topology) To do this, I first need a concise setting to start a proof, but I got stuck with this.

Let $\{G_\alpha , f_\alpha^\beta \}$ and $\{H_i,g_i^j\}$ be directed systems of abelian groups, with directed index sets $A$ and $I$, respectively. Then we have to show that the direct sum of the direct limit groups of these two directed systems are isomorphic to the direct limit group of the system $\{G_\alpha \oplus H_i, f_\alpha^\beta \oplus g_i^j \}$.

My question is

1. Actually I'm not sure that my assertion is true. Is it true?

2. Is my setting valid?

3. If 1 is true, then it seems to be this isomorphism is natural. Is it right?

Thanks in advance

Answer 1

Denote the direct limit of $\left\{G_\alpha , f_\alpha^\beta \right\}$ as $G$, and the direct limit of $\left\{H_i,g_i^j\right\}$ as $H$. Then we claim that $G\oplus H$ is the direct limit of $\left\{G_\alpha \oplus H_i, f_\alpha^\beta \oplus g_i^j \right\}$. And your questions are answered by this claim.

The morphisms $G_\alpha\oplus H_i\rightarrow G\oplus H$ are induced by the morphisms $G_\alpha\rightarrow G$ and $H_i\rightarrow H$.

Suppose there is a compatible system of morphisms $G_\alpha\oplus H_i\rightarrow M$. Since $\operatorname{Hom}(G_\alpha\oplus H_i,M)\cong\operatorname{Hom}(G_\alpha, M)\oplus\operatorname{Hom}(H_i, M)$, this induces compatible systems of morphisms $G_\alpha\rightarrow M$ and $H_i\rightarrow M$. Then by the universal property of the direct limit, these morphisms factor through $G_\alpha\rightarrow G$ and $H_i\rightarrow H$. This immediately implies that the morphisms $G_\alpha\oplus H_i\rightarrow M$ factor through $G_\alpha\oplus H_i\rightarrow G\oplus H$. Therefore $G\oplus H$ is the direct limit of $\left\{G_\alpha \oplus H_i, f_\alpha^\beta \oplus g_i^j \right\}$ as claimed.

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Hope this helps.