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 CC admitting finite limits, one says that a morphism f:XYf:X\to Y in CC is normal to the internal equivalence relation r:RY×Yr: R\hookrightarrow Y\times Y if it factors through the monomorphism rr (i.e. f¯\exists \bar{f} such that rf¯=fr\circ\bar{f}=f and …)

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

Revised on August 16, 2016 10:48:53 by David Corfield (