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Concept# Kernel (category theory)

Summary

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.
Definition
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 

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