# Category Theory/Abelian categories

Proposition (object in abelian category is decomposed into sum by subobject):

Let ${\displaystyle {\mathcal {C}}}$ be an abelian category, and let ${\displaystyle X\in {\mathcal {C}}}$ be an object. Let ${\displaystyle f:Y\to X}$ be a subobject, and let ${\displaystyle Z=\operatorname {Coker} f}$ be the corresponding quotient object. Moreover, denote ${\displaystyle g=\operatorname {coker} f}$. Then there exists a unique isomorphism ${\displaystyle \theta :X\to Y\oplus Z}$ such that

${\displaystyle \pi _{Z}\circ \theta =g}$ and ${\displaystyle \theta ^{-1}\circ \iota _{Y}=f}$.

Proof: ${\displaystyle Y\oplus Z}$ is a biproduct. First we apply the universal property of a product in order to obtain a morphism

${\displaystyle \phi :X\to Y\oplus Z}$ such that ${\displaystyle \pi _{Z}\circ \phi =g}$ and ${\displaystyle \pi _{Y}\circ \phi =0}$.

Then we apply the universal property of a coproduct in order to obtain a morphism

${\displaystyle \chi :Y\oplus Z\to X}$ such that ${\displaystyle \chi \circ \iota _{Y}=f}$ and ${\displaystyle \chi \circ \iota _{Y}=0}$.

Moreover, we get a morphism ${\displaystyle Y\otimes X\to Y\otimes 0}$ from the projection to ${\displaystyle Y}$, and a morphism ${\displaystyle 0\otimes Z\to Y\oplus Z}$ from the inclusion of ${\displaystyle Z}$. The latter morphism is the kernel of ${\displaystyle {\tilde {f}}:Y\oplus Z\to X}$, and the cokernel of that kernel is ${\displaystyle \Box }$