Orbital presheaves in equivariant infinite loop space theory
arXiv:2607.12141
The paper develops an operadic infinite loop space machine that builds genuine G‑spectra from orbital presheaves, providing a new operadic view of Mackey functors and defining associated topological Mackey, Picard, Azumaya, and Brauer G‑spectra.
Abstract
Let be a finite group. Using a new kind of operad, we axiomatize and explore an infinite loop space machine that constructs (genuine) -spectra from suitably structured functors on orbital presheaves, which are just contravariant functors from the orbit category of to based spaces. The theory leads unexpectedly to a new operadic description of Mackey functors and hence to a definition of ``topological Mackey functors" and a construction of their associated -spectra. It also leads to Picard -spectra, Azumaya ring -spectra, and Brauer -spectra. These constructions raise many unanswered questions.
50 pages