Kernel

Summary

In a category with zero morphisms, the kernel of a morphism $f$ is the universal morphism $k$ with $f \circ k = 0$. It is the equalizer of $f$ and the zero morphism. In categories of groups, modules or vector spaces it is the inclusion of the usual kernel, the set of elements sent to zero. Kernels and their dual notion, cokernels, are the building blocks of abelian categories.