Axioms for an abelian category

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.

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

1. 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).

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

1. Normal monos. Every monomorphism $m:A\to B$ is a kernel of some morphism; equivalently,
   $$m \cong \ker(\mathrm{coker}(m)).$$

1. 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.

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

• 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).