Kernel (category theory)

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.

Note that kernel pairs and difference kernels (a.k.a. binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article.

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:

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

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.

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