The splitting lemma in abelian categories is a basic tool for decomposing objects into biproducts. It is at the heart of some powerful structure theorems, such as the ones for finitely generated abelian groups and more generally finitely generated modules over a PID.

In this short post we state the basic decomposition result that any morphism in an abelian category decomposes canonically into an epimorphism followed by a monomorphism, and derive some very useful consequences.

In this short post we define abelian categories (which takes some work), which were introduced as abstractions of some core properties of categories like abelian groups and modules over a ring; so for example most of homological algebra can be carried over any abelian category, which is neat.