# Definition:Abelian Category

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## 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.