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.

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.

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.

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.

More generally, the cokernel of a morphism f: X → Y in some category ( e. g. a homomorphism between groups or a bounded linear operator between Hilbert spaces ) is an object Q and a morphism q: Y → Q such that the composition q f is the zero morphism of the category, and furthermore q is universal with respect to this property.

A Fredholm operator is a bounded linear operator between two Banach spaces whose kernel and cokernel are finite-dimensional and whose range is closed.

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

The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.

Here, an operator is Fredholm if its range is closed and its kernel and cokernel are finite-dimensional.

Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain ( it maps to the domain ), while the cokernel is a quotient object of the codomain ( it maps from the codomain ).

As with all universal constructions the cokernel, if it exists, is unique up to a unique isomorphism, or more precisely: if q: Y → Q and q ‘: Y → Q ‘ are two cokernels of f: X → Y, then there exists a unique isomorphism u: Q → Q ‘ with q ‘ = u q.

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.

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

The order of PSL ( n, q ) is the above, divided by, the number of scalar matrices with determinant 1 – or equivalently dividing by, the number of classes of element that have no nth root, or equivalently, dividing by the number of nth roots of unity in .< ref group =" note "> These are equal because they are the kernel and cokernel of the endomorphism formally, More abstractly, the first realizes PSL as SL / SZ, while the second realizes PSL as the kernel of

Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as and: the kernel and image of are the cokernel and coimage of.

Every subgroup of a free abelian group is itself free abelian, which is important for the description of a general abelian group as a cokernel of a homomorphism between free abelian groups.

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

Embedding A into some injective object I < sup > 0 </ sup >, the cokernel of this map into some injective I < sup > 1 </ sup > etc., one constructs an injective resolution of A, i. e. an exact ( in general infinite ) complex

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.

( This is the cokernel of the morphism f – g ; see the next section ).

Consider the tensor product of modules H < sub > i </ sub >( X, Z ) ⊗ A. The theorem states that there is an injective group homomorphism ι from this group to H < sub > i </ sub >( X, A ), which has cokernel Tor ( H < sub > i-1 </ sub >( X, Z ), A ).

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

The cokernel can be thought of as the space of constraints that an equation must satisfy, as the space of obstructions, just as the kernel is the space of solutions.

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

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.