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Just a quick question. Suppose a monad on $\mathbf{Set}$ (in particular monad's endofunctor preserves epimorphisms), are epimorphisms in the category of algebras also surjective?

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up vote 5 down vote accepted

No. A well-known example is the category of rings, in which epimorphisms may fail to be surjective (e.g. $\mathbb{Z} \to \mathbb{Q}$).

What is true is that, for any monad on $\mathbf{Set}$, the regular epimorphisms in the category of algebras are precisely the surjective homomorphisms.

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