Regular epimorphism

A regular epimorphism is a morphism out of some object $𝑋$ which occurs as the coëqualizer of some parallel pair of morphisms into $𝑋$ . cat In particular by the universal property of the coëqualizer it is an epimorphism.

Proof

Let $𝑓, 𝑔 : 𝑌 \to 𝑋$ and $𝑞 : 𝑄 \to 𝑋$ be their equalizer. Let $𝑎, 𝑏 : 𝑋 \to 𝑍$ so that $𝑎 𝑡 = 𝑏 𝑡 : = ℎ$ . Since the universal property demands the factorization of $ℎ$ via $𝑞$ be unique, it follows that $𝑎 = 𝑏$ .

See Regular monomorphism for the dual notion.

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Regular epimorphism

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