Context question: “Differential on a object of an abelian category” Let $C$ be an abelian category and $x$ an object of $C$. Consider a morphism $f: x\to x$ such that $f\circ f=0.$ I can see two ways to define a morphism $\text{im}\, f\to \ker f:$ Consider the diagram where the morphisms $i$ and $j$ are […]

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## Induced morphism between image and ker (category theory)

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- Post date April 1, 2020
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- Tags $$ thereforeby universal property of $\ker f$, consider the sequence $$\text{im}\, Context question: "Differential on a object of an abelian category" Let $C$ be an abelian category and $x$ an object of $C$. Consider a morp, exists a morphism between $\text{im}\, f.$$ Since $f\circ f$, f\stackrel{i}{\to} x\stackrel{j}{\to} \text{coker}\, f\stackrel{k\circ i}{\to} \ker f.$ \ \ The text says: Since $f\circ f\equiv 0$, f\to \ker f:$ Consider the diagram where the morphisms $i$ and $j$ are obtained by definition and $t$ is obtained by the universal property, f\to \ker f$. Is any of these induced morphism a monomorphism?, f$ and $\ker f$. Again, f$. Note that $$f\circ i=t\circ f\circ i=t\circ 0=0, then by universal property of $\ker f$ exists a morphism $x\stackrel{k}{\to} \ker f$. Then $\text{im}\, then exists a monomorphism $\text{im}\