Abstract Algebra – Understanding Coproducts in Abelian Groups

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I am currently trying to understand why finite products and coproducts in the category $\text{Ab}$ coincide. In fact, I'm not even sure I can show it. My question is the following:

Is there an intuitive way to understand why finite products and coproducts in $\text{Ab}$ coincide, while the same is not true in $\text{Grp}?$

As for the formal direction, I'm not sure if I have shown that the two coincide. The idea is to show that $G\times H$ satisfies the universal property for coproducts for $G,H \in \text{Obj}(\text{Ab})$.

Attempt

I defined the inclusion functions $\iota_G:G\longrightarrow G\times H$ and $\iota_H:H\longrightarrow G\times H$ by $\iota_G(g)=(g,1_G)$ and $\iota_H(h)=(1_H,h)$. Suppose $\varphi:G\longrightarrow K$ and $\psi:H\longrightarrow K$ are homomorphisms. To show $G\times H$ is a coproduct in $\text{Ab}$, I then need to construct a unique function $\tau:G\times H\longrightarrow K$ such that $\varphi=\tau\circ\iota_G$ and $\psi=\tau\circ\iota_H$. As such, I defined by $\tau$ by $\tau(g,1_H)=\varphi(g)$ and $\tau(1_G,h)=\psi(h)$. I think this should work since for all $(g,h) \in G\times H$ we have $(g,h)=(g,1_H)*(h,1_H)$, so by defining $\tau$ so that $\tau(g,h)=\tau(g,1_H)*\tau(h,1_H)$ everything should work.

Best Answer

The category $\mathsf{Ab}$ of abelian groups and their homomorphisms is an example of a preadditive category. In a preadditive category, the notions of (finitary) product and coproduct coincide, i.e. $$A \times B \cong A + B \cong A \oplus B$$ where $\oplus$ denotes the biproduct of $A$ and $B$.

In an arbitrary category, the biproduct is defined as follows: $X$ is a biproduct of $A$ and $B$ if there exist maps $$A \overset{\pi_A}{\underset{\iota_A}{\leftrightarrows}} X \overset{\pi_B}{\underset{\iota_B}{\rightleftarrows}} B$$ making $(A \overset{\pi_A}{\leftarrow} X \overset{\pi_B}{\rightarrow} B)$ into a product and $(A \overset{\iota_A}{\rightarrow} X \overset{\iota_B}{\leftarrow} B)$ into a coproduct.

Now, preadditive categories have an additive structure on their homsets. For example, in $\mathsf{Ab}$, if $f, g : A \rightrightarrows B$ are homomorphisms between abelian groups $(A,+)$ and $(B,+)$ then we can obtain another homomorphism $f+g : A \to B$ defined by $$(f+g)(a)=f(a)+g(a)$$ Then the binary operation $+$ gives $\text{Hom}_{\mathsf{Ab}}(A,B)$ an abelian group structure. The identity element is the zero map $0 : A \to B$, which is a zero morphism in the category theoretic sense.

There is a result which states that $(A \overset{\pi_A}{\underset{\iota_A}{\leftrightarrows}} X \overset{\pi_B}{\underset{\iota_B}{\rightleftarrows}} B)$ is a biproduct if and only if all of the following are satisfied:

  • $\pi_A \circ \iota_A = \text{id}_A$ and $\pi_B \circ \iota_B = \text{id}_B$
  • $\pi_B \circ \iota_A = 0 : A \to B$ and $\pi_A \circ \iota_B = 0 : B \to A$
  • $\iota_A \circ \pi_A + \iota_B \circ \pi_B = \text{id}_X$

This hints at why it works in $\mathsf{Ab}$ but not in $\mathsf{Grp}$: in $\mathsf{Grp}$ we don't have additive structure on homsets like $\mathsf{Ab}$ does. [And, indeed, the coproduct of two groups in $\mathsf{Grp}$ is the free product; in $\mathsf{Ab}$ it is the direct sum $\equiv$ Cartesian product.]

It's worth trying to prove the above result and work through the proof with a (more) concrete example.

P.S. Thanks Alexander Thumm for reminding me about preadditive categories.

Edit: See comments.

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