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In
category theory and its applications to
mathematics, a '''normal monomorphism''' or '''normal epimorphism''' is a particularly well-behaved type of
morphism.
A '''normal category''' is a category in which morphisms are normal, whenever reasonable.
==Definition==
A category
C must have
zero morphisms for the concept of normality to make complete sense.
In that case, we say that a
monomorphism is ''normal'' if it is the
kernel of some morphism, and an
epimorphism is ''normal'' (or ''conormal'') if it is the
cokernel of some morphism.
C itself is ''normal'' if every monomorphism is normal.
C is ''conormal'' if every epimorphism is normal.
Finally,
C is ''binormal'' if it's both normal and conormal.
But note that some authors will use only the word "normal" to indicate that
C is actually binormal.
==Examples==
Suppose that
G is a
group and
H is a
subgroup of
G.
Then the
inclusion map i from
H to
G is a monomorphism.
i will be normal
if and only if H is a
normal subgroup of
G.
(In fact, this is the origin of the term "normal" for monomorphisms.)
On the other hand, every epimorphism in the category of groups is normal (since it is the cokernel of its own kernel), so this category is conormal.
In an
abelian category, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel.
Thus, abelian categories are always binormal.
The category of
abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.
----
'''Binormal''' has another meaning in the
Frenet-Serret formulas for
curves in three dimensions.
