# 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.