Structural Properties of Equivariant Spectra with Incomplete Transfers

Blumberg and Hill defined categories of equivariant spectra interpolating between the equivariant stable categories indexed by universes. Indexed by an N∞ operad O, these O-spectra are characterized by what transfers they admit between their fixed points. We study structural properties of the O-incomplete equvariant stable categories. We first show an analog of a theorem of Guillou and May, giving an equivalence between O-spectra and spectrally enriched presheaves on a spectral enhancement of the incomplete Burnside category. Using this, we define the smash product and geometric fixed points of O-spectra in terms of Kan extensions and show that O-spectra can be recovered from gluing diagrams between their geometric fixed points. Finally, we apply this to give Mayer-Vietoris sequences describing the Picard group of O-spectra. In the presence of an appropriate Segal conjecture, we compare this to the Picard group of invertible abelian Mackey functors, giving a partial generalization of a theorem by Fausk, Lewis, and May.