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## Category Theory for beginners (4)

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Tsakanikas Nickos
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### Category Theory for beginners (4)

Note: The terms "monic,epic,section,retraction" used in the post "Category Theory For Beginners - 2" and the terms "monomorphism,epimorphism,split monomorphism,split epimorphism" used in this one are respectively equivalent.

Let $\mathcal{C}$ be a category. Show that

1. An isomorphism is a monomorphism and an epimorphism. Show that the inverse is not true in arbitrary categories. However, show that a morphism is an isomorphism if and only if it is a split monomorphism and a split epimorphism.
2. Let $\displaystyle f : X \longrightarrow Y$ be a morphism in $\mathcal{C}$. If $\displaystyle \left( K, \phi \right)$ is the kernel of $\displaystyle f$ and if $\displaystyle \left( C, \psi \right)$ is the cokernel of $\displaystyle f$ (assuming that both exist in $\mathcal{C}$, show that $\displaystyle \phi$ is a monomorphism and that $\displaystyle \psi$ is an epimorphism.