  We introduce a new notion of persistence modules endowed with operators. It
encapsulates the additional structure on Floer-type persistence modules coming
from the intersection product with classes in the ambient (quantum) homology,
along with a few other geometric situations. We provide sample applications to
the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian
diffeomorphisms, that go beyond spectral invariants and traditional persistent
homology.
