Kernel (category theory)

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In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f : XY is the "most general" morphism k : KX that yields zero when composed with (followed by) f.

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

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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 : XY 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 any morphism k : KX such that:

  • f o k is the zero morphism from K to Y;
  • Given any morphism k′ : K′ → X such that f o k′ is the zero morphism, there is a unique morphism u : K′ → K such that k o 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 : KX and l : LX are kernels of f : XY, then there exists a unique isomorphism φ : KL such that l o φ = k.

Examples

Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field). To be explicit, if f : XY is a homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then K is a subalgebra of X and the inclusion homomorphism from K to X is a kernel in the categorical sense.

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