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Let $A$ be an additive category, with objects $X$ and $Y$. If there exists direct sum decompositions $X = Y\oplus Y'$ and $Y = X \oplus X'$ in $A$, must $X$ and $Y$ be isomorphic?

If one only assumes that there exist monomorphisms $X\to Y$ and $Y\to X$, the answers is No in the category of abelian groups, as explained in answers linked to in the comments. However, the question about direct summands is not resolved there.

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    This fails in $\mathsf{Ab}$ (have you thought about this "generic" example before asking?) This has also been asked a couple of times on math.SE. – Martin Brandenburg Jan 06 '14 at 01:53
  • I agree that the question about monomorphisms is answered in the question linked to above, but the example there uses the inclusion of $\mathbb{Z}/2\mathbb{Z}$ into $\mathbb{Z}/4\mathbb{Z}$, which is not a direct summand. So I'm still not sure about the question regarding direct sums. –  Jan 06 '14 at 02:40
  • Yeah the first version wasn't about direct sums .... maybe you can ask a new question, but I also vote for reopen then. – Martin Brandenburg Jan 06 '14 at 12:13
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    There are (http://mathoverflow.net/questions/10128) abelian groups $A$ such that $A \cong A^3$, but $A \not\cong A^2$. Then $A$ is a direct summand von $A^2$, and $A^2$ is a direct summand von $A^3 \cong A$. – Martin Brandenburg Jan 06 '14 at 13:07
  • @MartinBrandenburg Thanks for the example! If the question reopens, I'll be happy to accept it as an answer. –  Jan 06 '14 at 14:32

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