David Roberts local epimorphism

A local epimorphism $p:A \to B$ for a given pretopology $J$ is a map such that there is a covering family $\{j_i:U_i \to B\}$ and maps $s_i:U_i \to A$ such that $p\circ s_i = j_i$.

When $J$ is the open cover pretopology $\mathcal{O}$ on $Top$, then a local epimorphism is a map admitting local sections in the usual sense.