Abelian category

Unpacking "additive", $\mathcal A$ is abelian exactly when it satisfies the following.

1. Preadditivity. For all objects $A,B$, the hom-set $\operatorname{Hom}_{\mathcal A}(A,B)$ is an abelian group, and composition is bilinear (see preadditive category).
2. Zero object. $\mathcal A$ has a zero object $0$, both initial and terminal, so between any two objects there is a zero morphism.
3. Finite biproducts. Finite products and coproducts exist and coincide, so $A\oplus B$ is both.
4. Kernels and cokernels. Every morphism $f\colon A\to B$ has a kernel $\ker f\to A$ and a cokernel $B\to\mathrm{coker}\, f$.
5. Normal monomorphisms. Every monomorphism $m$ is a kernel; equivalently, $m\cong\ker(\mathrm{coker}\, m)$.
6. Normal epimorphisms. Every epimorphism $e$ is a cokernel; equivalently, $e\cong\mathrm{coker}(\ker e)$.

In (5) and (6), $\cong$ is the canonical isomorphism over the common target for kernels and under the common source for cokernels.

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.

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

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

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

• $\mathbf{Set}$ is not abelian (not additive).
• $\mathbf{Grp}$ is not abelian. It is not even additive, since finite products (direct products) and coproducts (free products) differ, and the inclusion of a non-normal subgroup is a monomorphism that is not a kernel.
• $\mathbf{Top}$ is not abelian (again, not additive).