Short exact sequence in an abelian category ($0\to A\to B\to C\to 0$)

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.

Two projective presentations of the same object in an abelian category, with kernels $K,K^{\prime }$ and middle objects $P,P^{\prime }$, satisfy $K\oplus P^{\prime }\cong K^{\prime }\oplus P$. Form their pullback in a category. Its two induced short exact sequences in an abelian category split because each quotient is a projective object in a category. The two resulting biproduct descriptions of the pullback give the isomorphism.

Pulling back the final arrow of a short exact sequence in an abelian category preserves its kernel object and yields another short exact sequence in an abelian category. The new kernel inclusion is specified by its old kernel component and zero component into the new quotient. The pullback stability of epimorphisms in an abelian category ensures the new final arrow remains epic.