Spectral Mackey functors and equivariant algebraic K-theory (I)

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable infinity-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantly, we introduce the unfurling of certain families of Waldhausen infinity-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors. Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups and to study fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks.