Abelian category

An abelian category is a category $\mathcal A$ in which one can do “linear algebra + exact sequences” abstractly.

Definition

A category $\mathcal A$ is abelian if:

1. $\mathcal A$ is an additive category (in particular, hom-sets are abelian groups and finite biproducts exist);
2. every morphism has a kernel and a cokernel ;
3. every monomorphism is a kernel of its cokernel, and every epimorphism is a cokernel of its kernel.

Equivalently, $\mathcal A$ satisfies the standard abelian category axioms .

Consequences (often used as “working facts”)

In an abelian category:

• one can define exact sequences and do homological algebra;
• every morphism $f$ admits an “image–coimage” comparison, and the canonical map $\mathrm{coim}(f)\to \mathrm{im}(f)$ is an isomorphism;
• kernels are monomorphisms and cokernels are epimorphisms, mirroring the familiar situation in $\mathbf{Ab}$ and $R$-modules.

Examples

1. $\mathbf{Ab}$. The category of abelian groups is abelian.

2. $R\text{-}\mathbf{Mod}$. The category of left modules over a ring $R$ is abelian.

3. $\mathrm{Ch}(R)$. The category of chain complexes of $R$-modules is abelian (kernels and cokernels are computed degreewise).

Non-examples (useful to remember)

• $\mathbf{Set}$ is not abelian (not additive).
• $\mathbf{Grp}$ is not abelian (kernels exist, but cokernels and exactness do not satisfy the abelian axioms).
• $\mathbf{Top}$ is not abelian (again, not additive).