A monic epi that is not an iso

A monic epi that is not an iso

Consider the ring map  i : \mathbb{Z} \to \mathbb{Q}. It is an epimorphism because  \mathbb{Z} is initial in Ring. It is a monomorphism because maps into  \mathbb{Z} that agree after composition with  i agree everywhere.

Another example would be the inclusion  \mathbb{Q} \to \mathbb{R} in the category Haus. This follows from the density of  \mathbb{Q} in  \mathbb{R} .

Jargon. A category where all monic epis are isos is called balanced. A famous (and pleasant!) property of toposes is that all toposes are balanced.

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