With an analogous construction, the cokernel of ƒ can be seen as an initial object of a suitable category.

One can define the cokernel in the general framework of category theory.

If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.

This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data.

However the jumps in the dimensions of the kernel and cokernel are the same, so the index, given by the difference of their dimensions, does vary continuously, and can be given in terms of topological data by the index theorem.

( The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually ; the index theorem shows that we can usually at least evaluate their difference.

More generally, one can consider the image, kernel, coimage, and cokernel, which are related by the fundamental theorem of linear algebra.

In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism.

In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference:

One can also define the degree to be order of the cokernel of the corresponding linear transformation on weight lattices.

In this situation, the kernel of the cokernel of any morphism ( which always exists in an abelian category ) turns out to be the image of that morphism ; in symbols:

When m is a monomorphism, it must be its own image ; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel.

This is a group homomorphism ; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

* the cokernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.

The kernel may be expressed as the subspace < math >( x, 0 ) < V :</ math > the value of x is the freedom in a solution – while the cokernel may be expressed via the map given a vector the value of a is the obstruction to there being a solution.

If G is a group, the Whitehead group Wh ( G ) is defined to be the cokernel of the map G ×

The left null space of A is the orthogonal complement to the column space of A, and is the cokernel of the associated linear transformation.

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. The dimension of the cokernel is called the corank of f.

The dimension of the cokernel plus the dimension of the image ( the rank ) add up to the dimension of the target space, as the dimension of the quotient space is simply the dimension of the space minus the dimension of the image.

More generally, if W is a linear subspace of a ( possibly infinite dimensional ) vector space V then the codimension of W in V is the dimension ( possibly infinite ) of the quotient space V / W, which is more abstractly known as the cokernel of the inclusion.

the cokernel measures the constraints that y must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the degrees of freedom in a solution, if one exists.

Any Fredholm operator has an index, defined as the difference between the ( finite ) dimension of the kernel of D ( solutions of Df = 0 ), and the ( finite ) dimension of the cokernel of D ( the constraints on the right-hand-side of an inhomogeneous equation like Df

In the category of groups, the cokernel of a group homomorphism f: G → H is the quotient of H by the normal closure of the image of f. In the case of abelian groups, since every subgroup is normal, the cokernel is just H modulo the image of f:

In such a category, the coequalizer of two morphisms f and g ( if it exists ) is just the cokernel of their difference:

To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone – Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

* An additive category is preabelian if every morphism has both a kernel and a cokernel.

This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.

the notion of kernel and cokernel

Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.

There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets.

* A pre-abelian category is an additive category in which every morphism has a kernel and a cokernel.

# given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists ( this is the kernel ), as does the coequaliser ( this is the cokernel ).

Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.

We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f ; similarly, their coequaliser is the cokernel of their difference.

That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel.

* Abelian category, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel

The dual concept to that of kernel is that of cokernel.

That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.

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.

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.

equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism.