Definition 12.3.9. Let $\mathcal{A}$ be a preadditive category. Let $f : x \to y$ be a morphism.

A

*kernel*of $f$ is a morphism $i : z \to x$ such that (a) $f \circ i = 0$ and (b) for any $i' : z' \to x$ such that $f \circ i' = 0$ there exists a unique morphism $g : z' \to z$ such that $i' = i \circ g$.If the kernel of $f$ exists, then we denote this $\mathop{\mathrm{Ker}}(f) \to x$.

A

*cokernel*of $f$ is a morphism $p : y \to z$ such that (a) $p \circ f = 0$ and (b) for any $p' : y \to z'$ such that $p' \circ f = 0$ there exists a unique morphism $g : z \to z'$ such that $p' = g \circ p$.If a cokernel of $f$ exists we denote this $y \to \mathop{\mathrm{Coker}}(f)$.

If a kernel of $f$ exists, then a

*coimage of $f$*is a cokernel for the morphism $\mathop{\mathrm{Ker}}(f) \to x$.If a kernel and coimage exist then we denote this $x \to \mathop{\mathrm{Coim}}(f)$.

If a cokernel of $f$ exists, then the

*image of $f$*is a kernel of the morphism $y \to \mathop{\mathrm{Coker}}(f)$.If a cokernel and image of $f$ exist then we denote this $\mathop{\mathrm{Im}}(f) \to y$.

## Comments (2)

Comment #4525 by Aniruddh Agarwal on

Comment #4528 by Johan on

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