  We introduce a general theory of parametrized objects in the setting of
infinity categories. Although spaces and spectra parametrized over spaces are
the most familiar examples, we establish our theory in the generality of
objects of a presentable infinity category parametrized over objects of an
infinity topos. We obtain a coherent functor formalism describing the
relationship of the various adjoint functors associated to base-change and
symmetric monoidal structures.
  Our main applications are to the study of generalized Thom spectra. We obtain
fiberwise constructions of twisted Umkehr maps for twisted generalized
cohomology theories using a geometric fiberwise construction of Atiyah duality.
In order to characterize the algebraic structures on generalized Thom spectra
and twisted (co)homology, we characterize the generalized Thom spectrum as a
categorification of the well-known adjunction between units and group rings.
