nLab normal subobject

The notion of a normal subobject is the proper generalization of a normal subgroup to other algebraic categories. The notion was found relatively late. In the category of groups, there are two equivalent descriptions of a normal subgroup: as the kernel of a homomorphism of groups and as the equivalence class of the unit of some (necessarily unique) congruence.

Given a category $C$ admitting finite limits, one says that a morphism $f:X\to Y$ in $C$ is normal to the internal equivalence relation $r: R\hookrightarrow Y\times Y$ if $f \times f$ factors through the monomorphism $r$ (i.e. $\exists \tilde{f}:X \times X \to R$ such that $r\circ\tilde{f}=f \times f$) and such that

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is a pullback diagram.

A normal subobject is a monomorphism which is normal to some equivalence relation.

A protomodular category is defined in such a way that it possesses an intrinsic notion of normal subobject.

See also normal monomorphism