# Equivalence of Definitions of Abelian Category

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## Theorem

The following definitions of the concept of **Abelian Category** are equivalent:

### 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.

## Proof

### $(1)$ implies $(2)$

Let $C$ be an abelian category by definition $1$.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Thus $C$ is an abelian category by definition $2$.

$\Box$

### $(2)$ implies $(1)$

Let $C$ be an abelian category by definition $2$.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Thus $C$ is an abelian category by definition $1$.

$\Box$

This needs considerable tedious hard slog to complete it.In particular: definition 3 etc.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |