Editing Kernel (category theory)

{{short description|Generalization of the kernel of a homomorphism}}
{{other uses|Kernel (disambiguation)}}

{{More citations needed|date=December 2009}}
In [[category theory]] and its applications to other branches of [[mathematics]], '''kernels''' are a generalization of the kernels of [[group homomorphism]]s, the kernels of [[module homomorphism]]s and certain other [[kernel (algebra)|kernels from algebra]]. Intuitively, the kernel of the [[morphism]] ''f'' : ''X'' → ''Y'' is the "most general" morphism ''k'' : ''K'' → ''X'' that yields zero when composed with (followed by) ''f''.

Note that [[kernel pair]]s and [[difference kernel]]s (also known as binary [[Equaliser (mathematics)|equaliser]]s) 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 theory|category]].
In order to define a kernel in the general category-theoretical sense, '''C''' needs to have [[zero morphism]]s.
In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary [[morphism]] in '''C''', then a kernel of ''f'' is an [[Equaliser (mathematics)|equaliser]] of ''f'' and the zero morphism from ''X'' to ''Y''.
In symbols:
:ker(''f'') = eq(''f'', 0<sub>''XY''</sub>)

To be more explicit, the following [[universal property]] can be used. A kernel of ''f'' is an [[Object (category theory)|object]] ''K'' together with a morphism ''k'' : ''K'' → ''X'' such that:
* ''f''{{Hair space}}∘''k'' is the zero morphism from ''K'' to ''Y'';

<div style="text-align: center;">[[File:First_property_of_the_kernel.svg|100px|class=skin-invert]]</div>

* Given any morphism ''{{prime|k}}'' : ''{{prime|K}}'' → ''X'' such that ''f''{{Hair space}}∘''{{prime|k}}'' is the zero morphism, there is a unique morphism ''u'' : ''{{prime|K}}'' → ''K'' such that ''k''∘''u'' = ''{{prime|k}}''.

<div style="text-align: center;">[[File:Properties_of_a_kernel.svg|200px|class=skin-invert]]</div>

As for every universal property, there is a unique isomorphism between two kernels of the same morphism, and the morphism ''k'' is always a [[monomorphism]] (in the categorical sense). So, it is common to talk of ''the'' kernel of a morphism. In [[concrete categories]], one can thus take a [[subset]] of ''{{prime|K}}'' for ''K'', in which case, the morphism ''k'' is the [[inclusion map]]. This allows one to talk of ''K'' as the kernel, since ''k'' is implicitly defined by ''K''. There are non-concrete categories, where one can similarly define a "natural" kernel, such that ''K'' defines ''k'' implicitly.

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 {{math|''{{ell}}'' : ''L'' → ''X''}} are kernels of ''f'' : ''X'' → ''Y'', then there exists a unique [[isomorphism]] φ : ''K'' → ''L'' such that {{math|1=''{{ell}}''}}∘φ = ''k''.

==Examples==
Kernels are familiar in many categories from [[abstract algebra]], such as the category of [[group (algebra)|group]]s or the category of (left) [[module (mathematics)|modules]] over a fixed [[ring (mathematics)|ring]] (including [[vector space]]s over a fixed [[field (mathematics)|field]]). To be explicit, if ''f'' : ''X'' → ''Y'' is a [[homomorphism]] in one of these categories, and ''K'' is its [[kernel (algebra)|kernel in the usual algebraic sense]], then ''K'' is a [[subobject]] of ''X'' and the inclusion homomorphism from ''K'' to ''X'' is a kernel in the categorical sense.

Note that in the category of [[monoid]]s, 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 (see [[#Relationship to algebraic kernels]] below).

In the [[Category of rings|category of unital 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 that corresponds to kernels in the [[Category_of_rings#Rings_without_identity|category of non-unital rings]].

In the category of [[pointed space|pointed topological spaces]], if ''f'' : ''X'' → ''Y'' is a continuous pointed map, then the preimage of the distinguished point, ''K'', is a subspace of ''X''.  The inclusion map of ''K'' into ''X'' is the categorical kernel of ''f''.
<!-- ''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 [[subtraction|difference]] ''g'' &minus; ''f''.
In symbols:
:eq (''f'', ''g'') = ker (''g'' &minus; ''f'').
It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphisms cannot be subtracted.

Every kernel, like any other equaliser, is a [[monomorphism]].
Conversely, a monomorphism is called ''[[normal morphism|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 (category theory)|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 [[kernel (universal algebra)|notion of kernel]] for homomorphisms between two [[algebraic structure]]s 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.

== Sources ==
* {{cite book |first=Steve |last=Awodey |authorlink=Steve Awodey |title=Category Theory |edition=2nd |url=http://angg.twu.net/MINICATS/awodey__category_theory.pdf |orig-year=2006 |year=2010 |publisher=Oxford University Press |isbn=978-0-19-923718-0 |series=Oxford Logic Guides |volume=49 |access-date=2018-06-29 |archive-date=2018-05-21 |archive-url=https://web.archive.org/web/20180521155021/http://angg.twu.net/MINICATS/awodey__category_theory.pdf |url-status=dead }}
* {{nlab|id=kernel|title=Kernel}} 

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[[Category:Category theory]]