# 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

• Results about abelian categories can be found here.