Equivalence of Definitions of Abelian Category

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 $1$.

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

$\Box$

$(2)$ implies $(1)$

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

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

$\Box$

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