The persistence of a function f: X -> R is a collection of measurements, one for each open interval of the real line.  We call each measurement a persistent homology group.  If f is stratifiable, then its persistence can be visualized by something called the persistence diagram.  The persistent homology group is special because it is stable to perturbations of the function f.  Through the lens of intersection theory, the persistent homology group is simply the image of a cap product.  This interpretation leads to a definition of the persistent homology group for Whitney stratifiable maps g: X -> M to manifolds M.