Normal morphism

Normal morphism

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.

Definition

A monomorphism is normal if it is the kernel 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.[citation needed]

Examples

In the category of groups, 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.[citation needed]

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.

References

  • Section I.14 Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, 17, Academic Press, ISBN 978-0-124-99250-4, MR0202787 

Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • 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. In set theory, morphisms are functions; in linear… …   Wikipedia

  • Normal (mathematics) — In mathematics, normal can have several meanings:* Surface normal, a vector (or line) that is perpendicular to a surface. * Normal component, the component of a vector that is perpendicular to a surface. ** Normal curvature, of a curve on a… …   Wikipedia

  • List of category theory topics — This is a list of category theory topics, by Wikipedia page. Specific categories *Category of sets **Concrete category *Category of vector spaces **Category of graded vector spaces *Category of finite dimensional Hilbert spaces *Category of sets… …   Wikipedia

  • Outline of category theory — The following outline is provided as an overview of and guide to category theory: Category theory – area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as… …   Wikipedia

  • List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia

  • Conoyau — En mathématiques, le conoyau d un morphisme f : X → Y (par exemple un homomorphisme entre groupes ou bien un opérateur borné entre espaces de Hilbert) est la donnée d un objet Q et d un morphisme q : Y → Q tel que le morphisme composé… …   Wikipédia en Français

  • Frenet–Serret formulas — Binormal redirects here. For the category theoretic meaning of this word, see Normal morphism. In vector calculus, the Frenet–Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in… …   Wikipedia

  • Kernel (category theory) — In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel… …   Wikipedia

  • Cokernel — Coker (mathematics) redirects here. For other uses, see Coker (disambiguation). In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im(f) of the codomain of f by the image of f. Cokernels are… …   Wikipedia

  • Blowing up — This article is about the mathematical concept of blowing up. For information about the physical/chemical process, see Explosion. For other uses of Blow up , see Blow up (disambiguation). Blowup of the affine plane. In mathematics, blowing up or… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”