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.
A normal subobject is a monomorphism which is normal to some equivalence relation.
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A protomodular category is defined in such a way that it possesses an intrinsic notion of normal subobject.
See also normal monomorphism