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 : X → Y 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.
Note that in the category of monoids, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes. Therefore, the notion of kernel studied in monoid theory is slightly different. Conversely, in the category of rings, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms. Nevertheless, there is still a notion of kernel studied in ring theory. See Relationship to algebraic kernels below for the resolution of this paradox.
In the category of pointed topological spaces, if f :X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.
Read more about this topic: Kernel (category Theory)
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