Axioms for an abelian category

An abelian category is most often defined as an additive category in which kernels/cokernels exist and monomorphisms/epimorphisms are “normal.” One standard axiom list is:

Axiom list

Let $\mathcal A$ be a category .

1. Preadditivity. For all objects $A,B$, the hom-set $\mathrm{Hom}_{\mathcal A}(A,B)$ is an abelian group, and composition is bilinear in each variable.

2. Zero object. $\mathcal A$ has a zero object $0$ (both initial and terminal). Equivalently, for all $A,B$ there are distinguished zero morphisms $0_{A,B}:A\to B$.

3. Finite biproducts. $\mathcal A$ has finite products and finite coproducts , and for each finite family they coincide (so one can write $A\oplus B$ as both product and coproduct).

4. Kernels and cokernels. Every morphism $f:A\to B$ has a kernel $\ker(f)\to A$ and a cokernel $B\to \mathrm{coker}(f)$.

5. Normal monos. Every monomorphism $m:A\to B$ is a kernel of some morphism; equivalently,

   $$m \cong \ker(\mathrm{coker}(m)).$$

6. Normal epis. Every epimorphism $e:A\to B$ is a cokernel of some morphism; equivalently,

   $$e \cong \mathrm{coker}(\ker(e)).$$

A category satisfying (1)–(6) is abelian. Conversely, any abelian category satisfies these properties.

Why these axioms matter

They guarantee that exactness behaves “like modules,” so one can define and use exact sequences , images/coimages, and homological algebra in $\mathcal A$.

Examples and non-examples

• Example: $\mathbf{Ab}$. The category of abelian groups is abelian: kernels/cokernels are the usual subgroup kernel and quotient cokernel.

• Example: $R\text{-}\mathbf{Mod}$. The category of left $R$-modules is abelian for any ring $R$.

• Example: $\mathrm{Ch}(R)$. The category of chain complexes of $R$-modules is abelian (kernels/cokernels are taken degreewise).

• Non-example: $\mathbf{Set}$. Not additive: $\mathrm{Hom}(A,B)$ has no natural abelian group structure in general.

• Non-example: $\mathbf{Grp}$. Not abelian: cokernels exist, but kernels/cokernels do not interact in the way required by (5)–(6).