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Homenormal category

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# normal category

A monomorphism is a category is said to be *normal* if it is a kernel (of a morphism). A subobject of an object is *normal* if any (and hence all) of its representing monomorphisms is normal.

For example, in Grp, the category of groups, the inclusion of a subgroup $H\subseteq G$ into $G$ is normal iff $H$ is a normal subgroup of $G$.

A category is said to be *normal* if every monic is a kernel. Equivalently, a normal category is a category in which every subobject of every object is normal.

Dually, an epimorphism is *conormal* if it is a cokernel (of a morphism). A quotient object of an object is *conormal* if any (and hence all) of its representing epimorphisms is conormal. A category is said to be *conormal* if every epimorphism is conormal.

The category AbGrp of abelian groups, and more generally, any abelian category, is normal and conormal.

# References

- 1
C. Faith
*Algebra: Rings, Modules, and Categories I*, Springer-Verlag, New York (1973)

## Mathematics Subject Classification

18E10*no label found*

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