Let be an abelian category.
A short exact sequence in is called split if either of the following equivalent conditions hold
It is clear that the third condition implies the first two: take the section/retract to be given by the canonical injection/projection maps that come with a direct sum.
Conversely, suppose we have a retract of . Write for the corresponding idempotent.
Then every element can be decomposed as hence with and . Moreover this decomposition is unique since if while at the same time then . This shows that is a direct sum and that is the canonical inclusion of . By exactness it then follows that and hence that with the canonical inclusion and projection.
The implication that the second condition also implies the third is formally dual to this argument.
There is a nonabelian analog of split exact sequences in semiabelian categories. See there.
A long exact sequence is split exact precisely if the weak homotopy equivalence from the 0-chain complex, namely the quasi-isomorphism is actually a chain homotopy equivalence, in that the identity on has a null homotopy.
Every exact sequence of finitely generated free abelian groups is split.
Every exact sequence of free modules which is bounded below is split.
Every short exact sequence of vector spaces is split.
Consider the first case. The other is formally dual.
By the properties of a short exact sequence the morphism here is a monomorphism. By definition of injective object, if is injective then it has the right lifting property against monomorphisms and so there is a morphism that makes the following diagram commute:
For instance section 1.4 of