Definition:Abelian Category

Definition

Definition 1

An abelian category is a pre-abelian category in which:

every monomorphism is a kernel

every epimorphism is a cokernel

Definition 2

An abelian category is a pre-abelian category in which:

every monomorphism is the kernel of its cokernel

every epimorphism is the cokernel of its kernel

Definition 3

An abelian category is a pre-abelian category in which

for every morphism $f$, the canonical morphism from its coimage to its image $\map {\operatorname {coim} } f \to \Img f$ is an isomorphism.

Also see

• Equivalence of Definitions of Abelian Category
• Category of Modules is Abelian

• Results about abelian categories can be found here.