Classifying topos of integral domains
= Classifying topos of integral domains
On the opposite of finitely presented commutative rings, cover the zero ring by the empty family and cover $A$ by $A/(a),A/(b)$ whenever $ab=0$. The generated topology classifies nontrivial integral domains, with generic model the sheafified tautological ring. It is not subcanonical: at $\mathbb Z/4$, the covering quotient to $\mathbb Z/2$ identifies the distinct sections $0$ and $2$ of the polynomial-ring representable.