Category of abelian groups

Example

The category of abelian groups $\mathbf{Ab}$ has the abelian groups as objects and the group homomorphisms between them as morphisms, with composition of maps. It is an abelian category. Each set $\Hom(A, B)$ is an abelian group under pointwise addition $(f + g)(a) = f(a) + g(a)$, and composition is bilinear. The trivial group is a zero object, and $A \oplus B$ is both the product and the coproduct of $A$ and $B$. Every homomorphism $f \colon A \to B$ has a kernel $\set{a \in A : f(a) = 0}$ and a cokernel $B / f(A)$. An injective homomorphism $f$ is the kernel of the projection $B \to B/f(A)$. A surjective homomorphism is the cokernel of the inclusion of its kernel. So every monomorphism is a kernel and every epimorphism is a cokernel.