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

Understand

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.