nLab
kernel

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Category theory

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homological algebra

and

nonabelian homological algebra

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Contents

Idea

The kernel of a morphism is that part of its domain which is sent to 0.

Definition

There are various definitions of the notion of kernel, depending on the properties and structures available in the ambient category. We list a few definitions and discuss (in parts) when they are equivalent.

As a pullback

Definition

In a category with an initial object 00 and pullbacks, the kernel ker(f)ker(f) of a morphism f:ABf: A \to B is the pullback ker(f)Aker(f) \to A along ff of the unique morphism 0B0 \to B

ker(f) 0 p A f B. \array{ ker(f) &\to& 0 \\ {}^{\mathllap{p}}\downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \,.
Remark

More explicitly, this characterizes the object ker(f)ker(f) as the object (unique up to unique isomorphism) that satisfies the following universal property:

for every object CC and every morphism h:CAh : C \to A such that fh=0f\circ h = 0 is the zero morphism, there is a unique morphism ϕ:Cker(f)\phi : C \to ker(f) such that h=pϕh = p\circ \phi.

As an equalizer

Definition

In a category with zero morphisms (meaning: enriched over the category of pointed sets), the kernel ker(f)ker(f) of a morphism f:cdf : c \to d is, if it exists, the equalizer of ff and the zero morphism 0 c,d0_{c,d}.

As a weighted limit

In any category enriched over pointed sets, the kernel of a morphism f:cdf:c\to d is the universal morphism k:ack:a\to c such that fkf \circ k is the basepoint. It is a weighted limit in the sense of enriched category theory. This applies in particular in any (pre)-additive category.

This is a special case of the construction of generalized kernels in enriched categories.

As a representing object

Let AbAb be the category of abelian groups. It is a category with kernels. In every AbAb-enriched category AA, for every morphism f:XYf: X\to Y in AA there is a subfunctor

kerf:A opAbker f : A^{op}\to Ab

of the representable functor hom(,X)hom(-,X), defined on objects by

(kerf)(Z)=ker(hom(Z,X)hom(Z,Y)), (ker f)(Z) = ker(hom(Z,X)\to hom(Z,Y)),

where kerker on the right-hand side is the kernel n the category of abelian groups.

If the category is in fact preabelian, kerfker f is also representable with representing object KerfKer f. One has to be careful with CokerfCoker f which does not represent the functor naive cokerfcoker f defined as (cokerf)(Z)=coker(hom(Z,X)hom(Z,Y))(coker f)(Z) = coker(hom(Z,X)\to hom(Z,Y)) in AbAb, which is often not representable at all, even in the simple example of the category of abelian groups. Instead, as a colimit construction, one should corepresent another functor, namely, the covariant functor Zker(hom(Y,Z)hom(X,Z))Z\mapsto ker(hom(Y,Z) \to hom(X,Z)) (which is a quotient of the corepresentable functor hom(X,)hom(X,-)). In short, CokerfCoker f is defined by the double dualization using the kernel in AbAb: Cokerf=(Kerf op) opCoker f = (Ker f^{op})^{op}. This is a particular case of the dualization involved in defining any colimit from its corresponding limit.

In an (,1)(\infty,1)-category

The kernel of a morphism in an (∞,1)-category with \infty-categorical zero object is the homotopy pullback as in the pullback definition above: the homotopy fiber.

See also stable (∞,1)-category.

Other meanings

In some fields, the term ‘kernel’ refers to an equivalence relation that category theorists would see as a kernel pair. This is especially important in fields such as monoid theory where both notions exist but are not equivalent (while in group theory they are equivalent).

In ring theory, even when one assumes that rings have units preserved by ring homomorphisms, the traditional notion of kernel (an ideal) exists in the category of non-unital rings (and is not itself a unital ring in general). A purely category-theoretic theory of unital rings can be recovered either by using the kernel pair instead or (to fit better the usual language) moving to a category of modules.

In universal algebra, this may be handled in the framework of Malʹcev varieties.

Properties

Property

Let CC be a category with pullbacks and zero object.

In CC, the kernel of a kernel is 0.

Proof

By the pasting law for pullbacks we have that the total square

kerkerf kerf 0 0 c f d \array{ ker ker f &\to& ker f &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& c &\stackrel{f}{\to}& d }

is a pullback. Since 0c0 \to c is a monomorphism and the pullback of a monomorphism along itself is the domain of the monomorphis, we have kerkerf0ker ker f \simeq 0.

Remark

This statement crucially fails to be true in higher category theory. There, the kernel of a kernel is the based loop space object of dd. For this reason where one has short exact sequences in 1-category theory, there are instead long fiber sequences in higher category theory.

Proposition

In a category CC with pullbacks and pushouts and zero object, kernel and cokernel form a pair of adjoint functors on the arrow categories

(cokerker):Arr(C)kercokerArr(C). (coker \dashv ker) : Arr(C) \stackrel{\overset{coker}{\leftarrow}}{\underset{ker}{\to}} Arr(C) \,.
Proof

We check the hom-isomorphism of a pair of adjoint functors. An element in the hom-set Arr C(g,kerf)Arr_C(g,ker f) is a diagram

c ker(f) 0 g d a f b. \array{ c &\to& ker(f) &\to& 0 \\ {}^{\mathllap{g}}\downarrow && \downarrow && \downarrow \\ d &\to& a &\stackrel{f}{\to}& b } \,.

By the universal property of the pullback, this is the same as a diagram

c 0 g d a f b. \array{ c &\to& &\to& 0 \\ {}^{\mathllap{g}}\downarrow && && \downarrow \\ d &\to& a &\stackrel{f}{\to}& b } \,.

By the dual reasoning, an element in Arr C(cokerg,f)Arr_C(coker g, f) is a diagram

c g d a f 0 cokerg b. \array{ c &\stackrel{g}{\to}& d &\to& a \\ \downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ 0 &\to& coker g &\to& b } \,.

By the universal property of the pushout this is equivalently a diagram

c g d a f 0 b. \array{ c &\stackrel{g}{\to}& d &\to& a \\ \downarrow && && \downarrow^{\mathrlap{f}} \\ 0 &\to& &\to& b } \,.

(This also follows from the general theory of generalized kernels.)

Examples

Example

In the category Ab of abelian groups, the kernel of a group homomorphism f:ABf : A \to B is the subgroup of AA on the set f 1(0)f^{-1}(0) of elements of AA that are sent to the zero-element of BB.

Example

More generally, for RR any ring, this is true in RRMod: the kernel of a morphism of modules is the preimage of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.

Revised on June 1, 2013 00:42:03 by Zoran Škoda (94.250.138.220)