Empty-cover criterion for the domain-classifying site
= Empty-cover criterion for the domain-classifying site
For a finitely presented ring $A$, its sheafified representable is initial exactly when $A$ is the zero ring. The zero ring has a generating empty cover. Every nonzero ring maps to a field by quotienting by a maximal ideal, and that set-based domain gives a point at which the representable has a section, precluding initiality.