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

1. 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 $i' = i \circ g$.

2. If the kernel of $f$ exists, then we denote this $\mathop{\mathrm{Ker}}(f) \to x$.

3. 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 $p' = g \circ p$.

4. If a cokernel of $f$ exists we denote this $y \to \mathop{\mathrm{Coker}}(f)$.

5. If a kernel of $f$ exists, then a coimage of $f$ is a cokernel for the morphism $\mathop{\mathrm{Ker}}(f) \to x$.

6. If a kernel and coimage exist then we denote this $x \to \mathop{\mathrm{Coim}}(f)$.

7. If a cokernel of $f$ exists, then the image of $f$ is a kernel of the morphism $y \to \mathop{\mathrm{Coker}}(f)$.

8. If a cokernel and image of $f$ exist then we denote this $\mathop{\mathrm{Im}}(f) \to y$.

Comment #4525 by Aniruddh Agarwal on

This is a very minor point, but is there a reason for using the terminology "a cokernel", etc. instead of "the cokernel", etc. when these objects are unique upto unique iso?


Comment #4528 by on

Usually, in a situation like this when we are definining a blah which is unique up to isomorphism in the definiition we say "a blah" and then in a comment after the definition we say that because the thing is unique up to unique isomorphism we will from now on use the terminology "the blah". See the text following this definition in Section 12.3.

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