Kernel (category theory)

In 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 (also known as 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 . 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, 0XY) To be more explicit, the following universal property can be used. A kernel of f is an K together with a morphism k : K → X such that: f ∘k is the zero morphism from K to Y; Given any morphism : → X such that f ∘ is the zero morphism, there is a unique morphism u : → K such that k∘u = . As for every universal property, there is a unique isomorphism between two kernels of the same morphism, and the morphism k is always a monomorphism (in the categorical sense). So, it is common to talk of the kernel of a morphism. In concrete categories, one can thus take a subset of for K, in which case, the morphism k is the inclusion map. This allows one to talk of K as the kernel, since f is implicitly defined by K. There are non-concrete categories, where one can similary define a "natural" kernel, such that K defines k implicitly. 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 → X are kernels of f : X → Y, then there exists a unique isomorphism φ : K → L such that ∘φ = k. 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).

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