Coimage
Encyclopedia
In algebra
, the coimage of a homomorphism
is the quotient
of domain
and kernel
.
The coimage is canonically isomorphic to the image
by the first isomorphism theorem
, when that theorem applies.
More generally, in category theory
, the coimage of a morphism
is the dual notion of the image of a morphism
. If f : X → Y, then a coimage of f (if it exists) is an epimorphism
c : X → C such that
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the coimage of a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
- f: A → B
is the quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
- coim f = A/ker f
of domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
and kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
.
The coimage is canonically isomorphic to the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
by the first isomorphism theorem
Isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
, when that theorem applies.
More generally, in category theory
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...
, the coimage of a 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...
is the dual notion of the image of a morphism
Image (category theory)
Given a category C and a morphismf\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:#There exists a morphism g\colon X\to I such that f = hg....
. If f : X → Y, then a coimage of f (if it exists) is an epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
c : X → C such that
- there is a map fc : C → Y with f = fcc,
- for any epimorphism z : X → Z for which there is a map fz : Z → Y with f = fzz, there is a unique map π : Z → C such that both c = πz and fz = fcπ.