Abelian Category

An abelian category is the natural setting for homological algebra. It axiomatizes the properties of module categories that make diagram chasing, exact sequences, and derived functors possible. The Freyd-Mitchell embedding theorem guarantees that every small abelian category embeds into a module category, so element-wise arguments remain valid.

Definition

Definition3.4Abelian Category

An additive category $\mathcal{A}$ is abelian if:

Every morphism has a kernel and a cokernel.
Every monomorphism is the kernel of some morphism (equivalently, of its cokernel).
Every epimorphism is the cokernel of some morphism (equivalently, of its kernel).

RemarkEquivalent formulations

Condition (2) can be rephrased: the canonical morphism $\mathrm{coim}\, f \to \mathrm{im}\, f$ is an isomorphism for every morphism $f$ . Here $\mathrm{coim}\, f = \mathrm{coker}(\ker f)$ and $\mathrm{im}\, f = \ker(\mathrm{coker}\, f)$ .

Definition3.5Grothendieck Category

An abelian category $\mathcal{A}$ is a Grothendieck category if:

$\mathcal{A}$ is cocomplete (has all small colimits).
Filtered colimits are exact (the AB5 condition).
$\mathcal{A}$ has a generator.

Grothendieck categories have enough injectives, making them ideal for derived functor theory.

Examples

ExampleR-Mod is abelian

For any ring $R$ , the category $R\text{-}\mathbf{Mod}$ of left $R$ -modules is abelian. Kernels and cokernels are the usual module-theoretic ones, and monomorphisms/epimorphisms correspond to injective/surjective homomorphisms. It is also a Grothendieck category with generator $R$ .

ExampleAb is abelian

$\mathbf{Ab} = \mathbb{Z}\text{-}\mathbf{Mod}$ is abelian. It is the prototypical abelian category and a Grothendieck category with generator $\mathbb{Z}$ .

ExampleVect_k is abelian

$\mathbf{Vect}_k$ is abelian: every linear map has a kernel (null space) and cokernel (quotient by the image). Every injective map is a kernel, and every surjective map is a cokernel.

ExampleSheaves of abelian groups

For a topological space $X$ , the category $\mathbf{Sh}(X)$ of sheaves of abelian groups on $X$ is abelian. Kernels are computed as presheaf kernels; cokernels require sheafification. $\mathbf{Sh}(X)$ is a Grothendieck category.

ExampleCoherent sheaves on a scheme

For a Noetherian scheme $X$ , $\mathbf{Coh}(X)$ (coherent sheaves) is abelian. The category $\mathbf{QCoh}(X)$ of quasi-coherent sheaves is a Grothendieck abelian category.

ExampleFunctor category into an abelian category

If $\mathcal{A}$ is abelian and $\mathcal{C}$ is small, then the functor category $[\mathcal{C}, \mathcal{A}]$ is abelian with pointwise kernels, cokernels, and exact sequences.

ExampleCh(A) is abelian

If $\mathcal{A}$ is abelian, then the category of chain complexes $\mathbf{Ch}(\mathcal{A})$ is abelian. Kernels and cokernels are computed degreewise: $(\ker f)^n = \ker(f^n)$ and $(\mathrm{coker}\, f)^n = \mathrm{coker}(f^n)$ .

ExampleCategory of finite abelian groups

$\mathbf{FinAb}$ is abelian. However, it is NOT a Grothendieck category (it does not have all small coproducts: the direct sum of infinitely many copies of $\mathbb{Z}/2\mathbb{Z}$ is not a finite group).

ExampleGrp is NOT abelian

$\mathbf{Grp}$ is not abelian. The cokernel of $f : H \to G$ is $G/N(f(H))$ (quotient by the normal closure), but not every monomorphism is a kernel: the inclusion $A_3 \hookrightarrow S_3$ is monic but is not the kernel of any homomorphism out of $S_3$ (since $A_3$ is normal, it is, but this fails for non-normal subgroups like $\{e, (12)\} \hookrightarrow S_3$ ).

ExampleTop is NOT abelian

$\mathbf{Top}$ is not even additive (no abelian group structure on hom-sets), hence far from abelian.

ExampleOpposite of an abelian category is abelian

If $\mathcal{A}$ is abelian, then $\mathcal{A}^{\mathrm{op}}$ is abelian. Kernels in $\mathcal{A}^{\mathrm{op}}$ are cokernels in $\mathcal{A}$ , and vice versa. This gives the principle of abelian duality: every theorem in an abelian category has a dual.

ExampleRepresentations of a quiver

For a quiver $Q$ and a field $k$ , the category $\mathrm{Rep}_k(Q)$ of finite-dimensional representations is abelian. It is equivalent to $\mathrm{mod}\text{-}kQ$ (finitely generated modules over the path algebra), hence abelian by the general theory.

Key Properties

Theorem3.2Properties of abelian categories

In an abelian category $\mathcal{A}$ :

A morphism is an isomorphism if and only if it is both monic and epic.
Every morphism $f : A \to B$ factors as $A \twoheadrightarrow \mathrm{im}\, f \rightarrowtail B$ .
The Snake Lemma and Five Lemma hold.
Finite limits and colimits exist and can be expressed in terms of kernels, cokernels, and biproducts.

RemarkThe Freyd-Mitchell embedding

The Freyd-Mitchell Embedding Theorem states that every small abelian category admits an exact, fully faithful embedding into $R\text{-}\mathbf{Mod}$ for some ring $R$ . This means that diagram chasing with elements is always valid in small abelian categories — a crucial tool for homological algebra.

RemarkLooking ahead

Abelian categories are the starting point for:

Exact sequences and homological algebra
Chain complexes and cohomology
Derived categories and derived functors