  In principle, Floer theory can be extended to define homotopy invariants of
families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms,
3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The
invariant of a family consists of a filtered chain homotopy type, which gives
rise to a spectral sequence whose E^2 term is the homology of B with twisted
coefficients in the Floer homology of the fibers. The filtered chain homotopy
type also gives rise to a "family Floer homology" to which the spectral
sequence converges. For any particular version of Floer theory, some analysis
needs to be carried out in order to turn this principle into a theorem. This
paper constructs the invariant in detail for the model case of finite
dimensional Morse homology, and shows that it recovers the Leray-Serre spectral
sequence of a smooth fiber bundle. We also generalize from Morse homology to
Novikov homology, which involves some additional subtleties.
