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There are abelian groups $A$ such that $A\cong A\oplus A \oplus A$ but $A\not\cong A\oplus A$. Let $E=\operatorname{End}(A)$. The functor $F=\operatorname{Hom}(A,-)$ is an equivalence from the catego …

Here's a simpler, but less consequential, example. Take the category of "eventually constant" functors from ordinals (considered as a category with a single morphism $\alpha\to\beta$ when $\alpha\leq …

$\mathbb{Z}$ is cancellable for abelian groups. This was proved in the 1950s by Walker and Cohn (independently) and is often called "Walker's cancellation theorem". The proof is only a few lines. So …

This is probably an absurdly over-complicated answer, but ... Let $$J(X)=\left\{\sum_{x\in X}a_xx\in GX: \sum_{x\in X}a_x=0\right\}.$$ I claim that $J$ is not of the form $H\circ G\circ F$. Suppos …

The category of countable abelian groups is an essentially small abelian category, and has enough projectives and injectives (the countable free abelian groups and the countable divisible groups respe …

that $A\cong B\oplus B$ and $B\cong B\oplus B\oplus B$, so $B$ is an example of an abelian group $B$ with $B\cong B\oplus B\oplus B\not\cong B\oplus B$, I think rather simpler to describe than other examples … with $A\oplus A\cong B\oplus B$, which is one of Kaplansky's "test problems" for abelian groups in his famous 1954 book on Infinite Abelian Groups, which also seems to be simpler to describe than other examples …