 Short exact sequence in an abelian category

A short exact sequence in an abelian category has one nonzero morphism entering the middle object and one leaving it: $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$. Equivalently, $i$ is the kernel in a category of the epimorphism $p$. It splits if $p$ has a section; in that case the additive category splitting argument identifies $B$ with the biproduct $A\oplus C$. Categories of modules give familiar examples.