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 $𝒪$ on $\mathrm{Top}$, then a local epimorphism is a map admitting local sections in the usual sense.