Normal morphism
Encyclopedia
In category theory
and its applications to mathematics
, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism
.
A normal category is a category in which every monomorphism
is normal. A conormal category is one in which every epimorphism
is conormal.
of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism.
A category 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.
, a monomorphism f from H to G is normal if and only if
its image is a normal subgroup
of G. In particular, if H is a subgroup
of G, then the inclusion map
i from H to G is a monomorphism, and 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 group
s is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
and its applications to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
.
A normal category is a category in which every monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
is normal. A conormal category is one in which every epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
is conormal.
Definition
A monomorphism is normal if it is the kernelKernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...
of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism.
A category 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
In the category of groupsCategory of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
, a monomorphism f from H to G is normal if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its image is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of G. In particular, if H is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of G, then the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
i from H to G is a monomorphism, and 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
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
, 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 group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.