Editing Kernel (category theory) (section)

==Examples==
Kernels are familiar in many categories from [[abstract algebra]], such as the category of [[group (algebra)|group]]s or the category of (left) [[module (mathematics)|modules]] over a fixed [[ring (mathematics)|ring]] (including [[vector space]]s over a fixed [[field (mathematics)|field]]). To be explicit, if ''f'' : ''X'' → ''Y'' is a [[homomorphism]] in one of these categories, and ''K'' is its [[kernel (algebra)|kernel in the usual algebraic sense]], then ''K'' is a [[subobject]] of ''X'' and the inclusion homomorphism from ''K'' to ''X'' is a kernel in the categorical sense.

Note that in the category of [[monoid]]s, 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 (see [[#Relationship to algebraic kernels]] below).

In the [[Category of rings|category of unital 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 that corresponds to kernels in the [[Category_of_rings#Rings_without_identity|category of non-unital rings]].

In the category of [[pointed space|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''.
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