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Let $X$,$Y$,$Z$ be objects of R-Mod. ($\oplus$ denotes the direct sum or coproduct of modules).

The question is, are the following statements equivalent?

  • $X$ is a retraction of Y.
  • $\exists Z$ : $X\oplus Z \cong Y$

If yes, why? What is the most general category that satisfies such a relation?

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Do you know (pre-) Abelian categories? – Berci Jan 30 '13 at 23:21
It's true in the category of vector spaces over a field, but I suspect not in general. – Zhen Lin Jan 30 '13 at 23:23
@Zhen: Why not modules over an arbitrary ring? – Martin Brandenburg Jan 31 '13 at 22:54
Oops, I think I must have misread the question... as stated this is true in any abelian category, as t.b. explained here. – Zhen Lin Jan 31 '13 at 23:03
up vote 4 down vote accepted

Given a linear category with kernels, let $i : X \to Y$ be a split monomorphism. Choose $p : Y \to X$ with $pi=id_X$. Let $K \to Y$ be the kernel of $p$. Then $K \hookrightarrow Y \hookleftarrow X$ is a coproduct diagram. In fact, it extends to a biproduct diagram using $K \xleftarrow{q} Y \xrightarrow{p} X$, where $q$ is defined by $\mathrm{id}_Y-ip : Y \to Y$, which factors through $K$ since $p(\mathrm{id}_Y-ip)=p-pip=p-p=0$. It is easy to check the defining 5 equations.

So the statement holds in every linear category with kernels. But it is also true in the category of sets. I think the most general case is a a category in which every idempotent splits and has a complement.

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