Kernel (category Theory) - Definition

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. In symbols:

ker(f) = eq(f, 0_XY)

To be more explicit, the following universal property can be used. A kernel of f is any morphism k : K → X such that:

• f k is the zero morphism from K to Y;

• Given any morphism k′ : K′ → X such that f k′ is the zero morphism, there is a unique morphism u : K′ → K such that k u = k'.

Note that in many concrete contexts, one would refer to the object K as the "kernel", rather than the morphism k. In those situations, K would be a subset of X, and that would be sufficient to reconstruct k as an inclusion map; in the nonconcrete case, in contrast, we need the morphism k to describe how K is to be interpreted as a subobject of X. In any case, one can show that k is always a monomorphism (in the categorical sense of the word). One may prefer to think of the kernel as the pair (K,k) rather than as simply K or k alone.

Not every morphism needs to have a kernel, but if it does, then all its kernels are isomorphic in a strong sense: if k : K → X and l : L → X are kernels of f : X → Y, then there exists a unique isomorphism φ : K → L such that l o φ = k.