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Suppose $C$ is an additive category and $X_i,i\in I$ are objects of $C$.

Question

If $X_i$ has direct sum $X$ in $C$, then $\forall i \in I$, does there exist an object $Y_i\in \newcommand{\Obj}{\operatorname{Obj}}\Obj C$ such that $X_i\coprod Y_i\cong X$?

My thoughts

If $A,B\in \Obj C$, $\alpha :A\rightarrow B$ and $\beta: B\rightarrow A$ are two morphisms such that $\alpha\beta=\mathrm{Id}_A$, then is $A$ a direct summand of $B$?

If the question in my thoughts is true, then my original question is true. But it seems that it is not right.

Of course, if the category is an Abelian category, then the question is obviously true.

Any help will be appreciated.

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  • $\begingroup$ @QiaochuYuan as my thoughts have said.$\beta\alpha=Id$,if $C$ is Abelian,then $\alpha$ is split monomorphism. $\endgroup$ – Sky Jan 6 '18 at 1:29
  • $\begingroup$ @QiaochuYuan Since $X $ is direct sum of $X_i$,there are morphisms $e_j:X_j\rightarrow X$(about direct sum).Difine $\delta_{ij}:X_j\rightarrow X_i$.by definition of direct sum, there is a unique morphism $\beta:X\rightarrow X_i$ such that $\beta e_j=\delta_{ij}$. $\endgroup$ – Sky Jan 6 '18 at 4:21
  • $\begingroup$ @QiaochuYuan is this not the definition of infinite direct sum(coproduct)? $\endgroup$ – Sky Jan 6 '18 at 6:35
  • $\begingroup$ @QiaochuYuan Do you mean the construction in my second reply has problem? $\endgroup$ – Sky Jan 6 '18 at 10:20
  • $\begingroup$ Okay, never mind, I understand your construction now. Sorry for the confusion. $\endgroup$ – Qiaochu Yuan Jan 6 '18 at 19:57
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No, not necessarily. For instance, let $C$ be the full subcategory of $Ab$ consisting of all groups that are either finite or contain an element of order $p$ for all prime numbers $p$. It is easy to see that $C$ is closed under finite direct sums, so it is additive. In $C$, the direct sum $X=\bigoplus \mathbb{Z}/p\mathbb{Z}$ exists where $p$ ranges over all primes. However, if $p$ is any prime, there does not exist any object $Y$ of $C$ such that $Y\oplus \mathbb{Z}/p\mathbb{Z}\cong X$ (such a $Y$ would have to be infinite but could not have any elements of order $p$). In fact, with a bit more work you can show that $X$ cannot be written as a binary direct sum in $C$ in any nontrivial way at all.

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  • $\begingroup$ Thank you!I like this example. $\endgroup$ – Sky Jan 6 '18 at 0:52

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