In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X that yields zero when composed with (followed by) f.
Kernel pairs and difference kernels (also known as binary equalisers) sometimes go by the name "kernel", and are not discussed in this article.
Definition
Let C be a category.
In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms.
In that case, if f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y.