We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spect
ra, avoiding use of the Bokstedt coherence machinery. We are able to defi
This note compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the Bokstedt smash product and the Hill-Hopkins-Ravenel norm.
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 in
finity 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.
