# Questions & Answers

## Category Theory For Beginners - 4 [entire topic moved to new forum]

tsakanikasnickos on Friday, November 06 2015, 08:28 PM
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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.
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