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

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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 퐾-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 퐾-theory of such objects as spectral Mackey functors. Finally, we employ this technology to introduce fully functorial versions of 퐴-theory, upside-down 퐴-theory, and the algebraic 퐾-theory of derived stacks. We use this to give
what we think is the first general construction of 휋ét1 -equivariant algebraic 퐾-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.

Original languageEnglish
Pages (from-to)646-727
Number of pages60
JournalAdvances in Mathematics
Volume304
Early online date15 Sep 2016
DOIs
Publication statusPublished - 2 Feb 2017

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