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

Abstract

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 ∞-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 ∞-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 introduce fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks. We use this to give what we think is the first general construction of π1ét-equivariant algebraic K-theory for profinite étale fundamental groups. This is key to our approach to the “Mackey functor case” of a sequence of conjectures of Gunnar Carlsson.