Abelian categories

#category_theory/abelian_categories

Kernels and Cokernels

Definition of kernel

Let C be a category with a null object. A kernel of a morphism f:a→b is an equalizer of the morphisms f,0:a→b. In other words, it is a morphism k:s→a such that fk=0 and every h:c→a with fh=0 factors uniquely through k:

Since k:s→a is an equalizer, the morphism k is a monomorphism (i.e., left cancellable). It is thus common to think of the kernel k:s→a as a subobject of a, although technically subobjects of a are equivalence classes of monomorphisms.

Example in Grp

In Grp, the kernel of a group morphism ϕ:G→H is the inclusion k:ker⁡(ϕ)→G, where ker⁡(ϕ)={g∈G∣ϕ(g)=eH} is the usual kernel.

Example in Ab-categories

In any Ab-category A, all equalizers are kernels. This is because the hom-set HomA(b,c) is an abelian group, and so for arrows f,g:b→c and h:a→b we have fh=gh exactly when (f−g)h=0. The arrow h can therefore be described either as the equalizer of f and g, or as the kernel of f−g. This is why we usually deal with kernels (and not equalizers) in R-Mod, Ab, etc.

If we dualize the notion of kernel we obtain the notion of cokernel:

Definition of cokernel

Let C be a category with a null object. A cokernel of a morphism f:a→b is a coequalizer of the morphisms f,0:a→b. In other words, it is a morphism u:b→e such that uf=0 and every h:b→c with hf=0 factors uniquely through u:

Example in Ab

In Ab, the cokernel of f:A→B is the projection u:B→B/f(A).

Abelian categories

Definition of an abelian category

An abelian category is an Ab-category satisfying the following conditions:

  1. A has a null object.
  2. A has binary biproducts.
  3. Every morphism in A has a kernel and a cokernel.
  4. Every monomorphism is a kernel and every epimorphism is a cokernel.

The first two conditions ensure A is an additive category. The existence of kernels (and binary products) then implies A has all finite limits, while the existence of cokernels (and binary coproducts) implies the existence of all finite colimits.

The fourth condition is strong. It implies that any morphism f that is both monomorphism and epimorphism is an isomorphism.

Examples of abelian categories

The categories Ab, R-Mod, and many others are abelian, with the usual kernels and cokernels.

If A is an abelian category, then so is any functor category AJ. See page 199 in Mac Lane for the short proof.

Examples of additive categories that are not abelian

The category of even-dimensional vector spaces over a field F is additive but not abelian, since linear transformations with odd rank do not have kernels in that category.

The category of finitely generated modules over a non-Noetherian ring (such as Q[x1,x2,…]) is also additive but not abelian.