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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 of the
morphism ''f'' : ''X'' → ''Y'' is the "most general" morphism ''k'' : ''K'' → ''X'' which, when composed with ''f'', yields zero.
Note that
kernel pairs and
difference kernels (aka binary
equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article.
== Definition ==
Let
C be a
category.
In order to define a kernel in the general category-theoretical sense,
C needs to have
zero morphisms.
In that case, if
f :
X →
Y is an arbitrary
morphism in
C, then a kernel of
f is an
equaliser of
f and the zero morphism from
X to
Y.
In symbols:
:ker(
f) = eq(
f, 0
''XY'')
To be more explicit, the following
universal property can be used. A kernel of
f is any morphism
k :
K →
X such that:
*
f o k is the zero morphism from
K to
Y;
*
Given any morphism
k′ :
K′ →
X such that
f o k′ is the zero morphism, there is a
unique morphism
u :
K′ →
K such that
k o u =
k'.
Note that in many
concrete contexts, one would refer to the object
K as the "kernel", rather than the morphism
k.
In those situations,
K would be a
subset of
X, and that would be sufficient to reconstruct
k as an
inclusion map; in the nonconcrete case, in contrast, we need the morphism
k to describe ''how''
K is to be interpreted as a
subobject of
X. In any case, one can show that ''k'' is always a
monomorphism (in the categorical sense of the word). One may prefer to think of the kernel as the pair (
K,
k) rather than as simply
K or
k alone.
Not every morphism needs to have a kernel, but if it does, then all its kernels are isomorphic in a strong sense: if ''k'' : ''K'' → ''X'' and ''l'' : ''L'' → ''X'' are kernels of ''f'' : ''X'' → ''Y'', then there exists a unique
isomorphism φ : ''K'' → ''L'' such that ''l'' o φ = ''k''.
== Examples ==
Kernels are familiar in many categories from
abstract algebra, such as the category of
groups or the category of (left)
modules over a fixed
ring (including
vector spaces over a fixed
field).
To be explicit, if
f :
X →
Y is a
homomorphism in one of these categories, and
K is its
kernel in the usual algebraic sense, then
K is a
subalgebra of
X and the inclusion homomorphism from
K to
X is a kernel in the categorical sense.
Note that in the category of
monoids, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes.
Therefore, the notion of kernel studied in monoid theory is slightly different.
Conversely, in the category of
rings, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms.
Nevertheless, there is still a notion of kernel studied in ring theory.
See '''Relationship to algebraic kernels''' below for the resolution of this paradox.
''We have plenty of algebraic examples; now we should give examples of kernels in categories from
topology and
functional analysis.''
== Relation to other categorical concepts ==
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.
As mentioned above, a kernel is a type of binary equaliser, or
difference kernel.
Conversely, in a
preadditive category, every binary equaliser can be constructed as a kernel.
To be specific, the equaliser of the morphisms
f and
g is the kernel of the
difference g −
f.
In symbols:
:eq (
f,
g) = ker (
g −
f).
It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphsims cannot be subtracted.
Every kernel, like any other equaliser, is a
monomorphism.
Conversely, a monomorphism is called ''
normal'' if it is the kernel of some morphism.
A category is called ''normal'' if every monomorphism is normal.
Abelian categories, in particular, are always normal.
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:
:im
f = ker coker
f (in an abelian category)
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.
In symbols:
:
m = ker (coker
m) (for monomorphisms in an abelian category)
== Relationship to algebraic kernels ==
Universal algebra defines a
notion of kernel for homomorphisms between two
algebraic structures of the same kind.
This concept of kernel measures how far the given homomorphism is from being
injective.
There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above.
In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of
kernel pair.
In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic terms.
