# Kernel of a morphism in a category

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A concept generalizing that of the kernel of a linear transformation of vector spaces, the kernel of a homomorphism of groups, rings, etc. Let be a category with null (or zero) morphisms. A morphism is called a kernel of a morphism if and if every morphism for which can be uniquely represented as . A kernel of a morphism is denoted by .

If and are both kernels of , then for a unique isomorphism . Conversely, if and if is an isomorphism, then is a kernel of . Thus, the kernels of a morphism form a subobject of , denoted by .

If , then is a monomorphism. In general, the converse is not true; a monomorphism which occurs as a kernel is called a normal monomorphism. The kernel of the null morphism is the identity morphism . The kernel of exists if and only if contains a null object (cf. Null object of a category).

Kernels do not always exist in a category with null morphisms. On the other hand, in a category with a null object a morphism has a kernel if and only if a pullback of and exists in .

The concept of the "kernel of a morphism" is the dual to that of the "cokernel of a morphism" .