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

Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map