I'm reading in Borceux' book Basic Category Theory about pullbacks. It turns out that the pullback of an epimorphism is not necessarily an epimorphism. On the linked page, Borceux gives a counterexample in the category Haus.
I was wondering whether there also exist easy counterexamples in Ring, the category of commutative rings with a 1 element (or in Grp, Rng, etc.). I have been playing around with the embedding of $\mathbb{Z}$ in $\mathbb{Q}$, which is an epimorphism, but I couldn't find anything.
Best Answer
This is supposed to be an answer to a tiny fraction of your question: In $\mathsf{Grp}$ epis are stable under pullbacks (therefore you can stop searching for a counterexample there).
This is because in $\mathsf{Grp}$ every epi is regular and regular epis are stable under taking pullbacks, because $\mathsf{Grp}$ (and more generally every category of universal algebras) is a regular category.
The same applies for example to $\mathsf{Set}$ and to every abelian category, in particular $R\mathsf{Mod}$ for every ring $R$.
Actually, in a (concrete) category of universal algebras $(\mathcal{A}, U : \mathcal{A} \to \mathsf{Set})$ a morphism is a regular epi if and only, if it is surjective (its underlying function is surjective). We can conclude, that the following are equivalent for $\mathcal{A}$:
Hence, every concrete "algebraic" category satisfying one of the equivalent conditions above is off the table for counterexamples (again: regular epis are stable under pullbacks in a regular category)