Normal monomorphism

A normal monomorphism is a monomorphism that is the kernel in a category of some morphism. In a pointed category with kernels and cokernels, a monomorphism is normal exactly when it is the kernel of its own cokernel.

In an additive category with finite categorical limits and colimits in which every epimorphism is a conormal epimorphism, closure of normal monomorphisms under composition forces every monomorphism to be normal. To prove this, write a monomorphism $m$ as $kg$, where $k=\ker \mathrm{coker}m$, and put $l=\ker \mathrm{coker}g$. If $kl=\ker t$ were normal, $tm=0$ would force $tk=0$, hence $l$ would be an isomorphism. This makes $\mathrm{coker}g=0$, so $g$ is epic as well as monic. An epic monomorphism which is a cokernel is invertible, forcing $m$ to be normal. Contrapositively, any failure of normality provides two normal monomorphisms whose composite is not normal.