  Numerical algebraic geometry has a close relationship to intersection theory
from algebraic geometry. We deepen this relationship, explaining how rational
or algebraic equivalence gives a homotopy. We present a general notion of
witness set for subvarieties of a smooth complete complex algebraic variety
using ideas from intersection theory. Under appropriate assumptions, general
witness sets enable numerical algorithms such as sampling and membership. These
assumptions hold for products of flag manifolds. We introduce Schubert witness
sets, which provide general witness sets for Grassmannians and flag manifolds.
