Extremal, but not regular monomorphism

Question

Is there an example of a category, and a monomorphism $m:X\to Y$ between two objects such that $m$ is extremal, but not regular? (A monomorphism $m:X\to Y$ is said to be extremal if whenever $m=g\circ e$ with $e$ an epimorphism, then $e$ is an isomorphism.)

Answer 1

The Joy of Cats by Adámek, Herrlich, and Strecker should be your go-to book for this type of question. They note in Proposition 7.62 that if $f: X \to Y$ is extremal and $g: Y \to Z$ is regular, then $g \circ f$ is extremal. Since regular monomorphisms are extremal, it will suffice to find composable regular monos $f, g$ whose composite is not regular; that composite is extremal. They give an example of this in 7J(a).

A simpler example (found in Paul Taylor's Practical Foundations of Mathematics, p. 289) is in $Cat^{op}$. In $Cat$, let $\mathbb{2}$ be the arrow category $\bullet \to \bullet$ and let $\mathbb{N}$ be the monoid (one-object category) of natural numbers. Then the coequalizer of the two object inclusions $1 \rightrightarrows \mathbb{2}$ is the evident (regular) epi $\mathbb{2} \to \mathbb{N}$, and there is a regular epi $\mathbb{N} \to \mathbb{Z}/(3)$, but their composite is not regular in $Cat$.